phy131h1s practicals manual - department of physicsjharlow/teaching/phy131s09/practicals... ·...

PHY131 Practicals Manual Introduction

PHY131H1S Practicals Manual

Department of

Physics

January to April, 2009 University of

Toronto Welcome to the Physics Practicals! We have devised a number of Activities and Projects which will help you to learn a lot of Physics. They will also help you to do well on the tests and exam of the course. We are very excited about this new way of helping you to learn Physics, and hope you find your time in the Practicals to be fun and productive. The course web-site has the most up-to-date contact information, handouts, schedules and information:

http://www.physics.utoronto.ca/~jharlow/teaching/phy131s09 The course coordinator and lecturer is

Jason Harlow, Office: MP129-A, Phone 416-946-4071. The materials in this book were mainly developed by

David Harrison, Office: MP121-B, Phone 416-978-2977 The co-coordinator of the course is

Pierre Savaria, Office: MP129-E, Phone 416-978-4135. The course administrator is April Seeley, Office: MP129-E, Phone 416-946-0531. Email addresses are listed on the course web-site and on the Physics Department directory at http://www.physics.utoronto.ca/people . Course staff will endeavour to respond to email inquiries from students within 2 days. If you do not receive a reply within this period, please resubmit your question(s) and/or phone (leave a message if necessary). Please note that some servers (such as hotmail) can be unreliable in both sending and receiving messages.

Table of Contents Section Page

Introduction 1 Mechanics Module 1 7

Scientific Method Module 15 Mechanics Module 2 17 Mechanics Module 3 28 Mechanics Module 4 37

Teamwork Module 42 Mechanics Module 5 50 Mechanics Module 6 59 Oscillations Module 69

Numerical Approximation Module 85 Fluids Module 94


2

Schedule (preliminary)

Students will meet once per week in MP126-P on Thursdays or Fridays for two hours.

Practical Session # Dates

Topics, Activities

Schedule is Subject to Change Jan 8, 9 NO PRACTICALS THIS WEEK

1 Jan 15, 16 Introduction Mechanics Module 1

2 Jan 22, 23

Error Analysis Assignment due Scientific Method Module Mechanics Module 1 continued Mechanics Module 2

3 Jan 29, 30 Written Homework #1 due Mechanics Modules 2 continued

4 Feb 5, 6 Mechanics Module 3

Feb 12, 13 NO PRACTICALS THIS WEEK – Extra office hours for test prep.

Feb 19, 20 Reading week. Then test on Feb.24 evening.

5 Feb 26, 27 Scrambling teams Teamwork module Numerical Approximation Module

6 Mar 5, 6 Mechanics Module 4

7 Mar 12, 13 Mechanics Module 5

8 Mar 19, 20 Written Homework #2 due Mechanics Module 6

9 Mar 26, 27 Oscillations Module

10 Apr 2, 3 Fluids Module The Error Analysis Assignment is a series of online tutorials which must be viewed with a computer. There is a link from the Practicals tab of the main course web page. PHY131 should please write the answers on a paper print-out of the 3-page “Answer Sheet”, which is linked as a PDF on the first page of the assignment. The answers, done by individual students, are due in the 2nd practicals session.


3

How the Practicals Work You will be meeting for 2 hours every week in room MP126P, which is in the back of MP126. Each Group will have a maximum of 36 students. You will be working in a Team with up to three of your classmates. There will be two Teaching Assistant Instructors present for each Practical. Your Team will keep a single lab book, which is to be a complete record of everything you did, what you and your teammates thought it meant, and what conclusions you have drawn from your work. Each Practical session will include time for student questions and discussion. However the “heart” of the Practicals will be a series of Activities. Every week you will be doing Activities based on the material currently being discussed in class. Often the Activities will be based on material that has already been discussed in class, but sometimes the Activities may be used to introduce material that has not yet been talking about in class. In addition, you will be doing three “value added” Modules, that we believe are important for your overall learning about science in general and Physics in particular. These are:

• A Module on the Scientific Method • A Module on effective Teamwork • A Module on Numerical Approximation

For each Practical session two members of each Team will serve the following roles:

Facilitator. This person, a different individual each week, is responsible for keeping the Team on track with the Activities. When the entire Practical group discusses some topic, the Facilitator will be the Team’s primary spokesperson.

Recorder. This person, also a different individual each week, takes primary responsibility for recording all work, speculations, conclusions etc. in the lab notebook.

Evaluation and Marks The Practicals will count for 20% of your mark in PHY131. All marks will be given on an integer scale from 0 to 4:

0. Missing work. 1. Seriously deficient. 2. Requires improvement. 3. The standard mark indicating good work 4. Exceptional. We will be very stingy in awarding marks of 4.

Each mark component has a weight, and the mark times the weight will be added to generate a Practical mark. The total number of weights of all components is 20. The one exception to this marking system is the Error Analysis Assignment. It is marked out of 100.


4

Attendance at the Practical is vital for your learning. We will deduct the cube of the number of un-excused absences from the final Practical mark. Here are the components and their weights:

1. Notebook Mark 1 (0 Weights). After the first Practical the lab books will be collected and marked. However, this mark will not count towards your Practical mark. Instead it is intended to make our standards and requirements clear to you.

2. Error Analysis Assignment (1 Weight). You will do this assignment individually. It is due at Practical Session #2.

3. Scientific Method Module (1 Weight). This Module will be done during the second practical, and will be marked.

4. Written Homework #1 (2.5 Weights). You will do this assignment outside of class in collaboration with the members of your first Practicals team. It is due at Practical Session #3.

5. Notebook Mark 2 (6 Weights). After the last Practical before Test, a selection of Activities from Practical sessions completed so far will be chosen to be marked. The decision of which Activities will be marked will be chosen more-or-less randomly after the books have been collected. All Teams will have the same Activities marked.

6. Numerical Approximation Module (1 Weight). 7. Written Homework #2 (2.5 Weights). You will do this assignment outside of

class in collaboration with the members of your second Practicals team. It is due at Practical Session #8.

8. Notebook Mark 3 (6 Weights). At the end of the term a selection of the Mechanics, Oscillations and Fluids Activities you have done since the Test will be chosen to be marked. The decision of which Activities will be marked will be chosen more or less randomly after the books have been collected. All Teams will have the same Activities marked.

Computers and Networks The Practical server is: feynman.physics.utoronto.ca. You will access the server using your UTORid and password. You will have access to three folders on this server:

Your home directory. You have read and write privileges for this directory. Your team directory. All members of your team have read and write privileges here. public. This is an area of the server containing documents, computer programs, etc.

Everyone has read privileges for this directory. Note: you should never save work on the local PC. These discs will be ruthlessly purged on a regular basis. Remote Access You may access the server at: https://feynman.physics.utoronto.ca. You may upload and download files from your computer to the server.


5

Printing There is a colour printer in the Practical Room. You may choose to print either in colour or black and white by choosing the appropriate printer in the print dialog. We charge for printing using your TCard. We charge:

10 cents per page for black and white printing. 15 cents per page of colour printing.

We do not (yet) have facilities in the building to add dollar values to your card. The locations of cash-to-card locations is at: http://content.library.utoronto.ca/finance-admin/photo/cash-to-card At present the nearest location is the Main Floor of the Earth Science building, just across Huron Street.

Datasets All datasets in the Practicals have a standard uniform format. This section describes that format. The dataset file is text.

1 The first line of the file is the title of the dataset. 2 The second line of the file names the variables of the data. The names are

separated by tabs. In the examples below we represent a tab with: <TAB> 3 The third and subsequent lines of the file contain the data. Each datapoint is on a

separate line and the values are separated by tabs. Thus, the dataset can be edited with a text editor or a spreadsheet program such as Excel. There are four cases for the number of variables in the dataset.

One Variable If only one value is given for each datapoint, it is the dependent (i.e. y) variable. In this case the values of the independent (x) variable are assumed to be 1, 2, 3, … in order. Here is an example of such a dataset: Balonium decay values Counts per second 50 32 27 15 11 8


6

Two Variables In this case the first column contains the values for the independent (x) variable and the second column the values for the dependent (y) variables. For example: Student collected data on pressure-temperature values Pressure (cm Hg)<TAB>Temperature (C) 65<TAB>-10 75<TAB>17 86<TAB>42 In the above <TAB> denotes the TAB character.

Three Variables If there are three variables, the third one is the error in the dependent (y) variable. Thermocouple calibration data Temp (C)<TAB>Voltage (Volts)<TAB>errV0<TAB>-0.89<TAB>0.05 5<TAB>-0.69<TAB>0.05 10<TAB>-0.53<TAB>0.05

Four Variables Now there are explicit errors in both coordinates of the data. The first column contains the name and values of the independent (x) variable, the second column contains the name and values of the error in the independent variable, the third column contains the name and values of the dependent (y) variable and the fourth column the name and values of the error in the dependent variable. Pearson’s Data with York’s WeightsX<TAB>errX<TAB>Y<TAB>errY 0<TAB>0.0316<TAB>5.9<TAB>1 0.9<TAB>0.0316<TAB>5.4<TAB>0.746 1.8<TAB>0.0447<TAB>4.4<TAB>0.5

7

PHY131 Practicals Manual Mechanics Module 1

Mechanics Module 1 Student Guide

Concepts of this Module • Scaling • Dimensions • Fermi Problems • Introduction to Experimental Uncertainties • Kinematics in One Dimension • Motion Diagrams • Setting up Newtonian Dynamics

The Activities

Activity 1

A sculptor is making a statue of a duck. She first creates a model. To make the model requires exactly 2 kg of bronze. The final statue will be 5 times the size of the model in all three dimensions. How much bronze, in kg, will she require to cast the final statue? You may find it helpful to think about the model being constructed of Lego blocks, with the final statue made of Lego blocks that are 5 times the size in each dimension as the ones used to make the model.

Activity 2

When the sculptor finished making her model of the duck statue, she gave it 2 coats of varnish. This took exactly one can of varnish. How many cans of varnish will she need to give the final statue 2 coats of varnish?

Activity 3

Surprisingly, the units of all physical quantities can be defined in terms of combinations of only four fundamental units: a unit for length, mass, time, and electric current. In the SI system the units are:


8

• Second, s: the time required for 9,192,631,770 oscillations of the radio wave absorbed by the cesium-133 atom.

• Meter, m: the distance traveled by light in a vacuum in 1/299,792,458 of a second. • Kilogram, kg: the mass of the international standard kilogram, a polished

platinum-iridium cylinder stored in Paris. • Ampere, A: the constant current which, if maintained in two straight parallel

conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per meter of length.

Mort the politician has a not-so bright idea that we could save money by simplifying the standards for units. Instead of having a unit of length be fundamental, the politician suggests having a unit of volume as fundamental. Of course this unit of volume would be called a mort. Then, instead of a difficult to measure and expensive separate standard for length we could define the volume of the standard kilogram to be exactly 1 mort.

In this system of units, what is the unit of density? What is the density of the standard kilogram in kg/mort? The density of the standard kilogram is about 21,500 kg/m3. The density of water is

1000 kg/m3. What is the density of water in kg/mort? In this system of units length is now a derived quantity. What is its relation to the

mort? You have a replica of the standard kilogram and an object of unknown material with a

similar volume. How might you actually measure the volume of this object to determine if its volume is greater than, less than, or equal to one mort?

Activity 4

The ancient Greeks built a temple to Apollo on the island of Delos. It was 11 m wide, 24 m long, and 10 m high. In 427 B.C. a plague ravaged Athens, and the Athenians consulted the oracle on Delos, who demanded that they double the size of the temple.

(a) What is the original volume of the temple? (b) The Athenians re-built the temple by doubling the size of each dimension of the

temple. What was the volume of the new temple? (c) The Athenians consulted to oracle again, who said “You have not doubled the size

of god’s temple, as he demanded of you.” What mistake did the Athenians make? (d) What would be the dimensions of the temple that the oracle wanted the Greeks to

build?


9

Activity 5

How many musical notes are played on an average radio station in a given year?

Activity 6

A useful visualization technique in studying motion is called a motion diagram. We will be using these diagrams frequently in this course. For example, consider an apple that is dropped from rest at some height above the ground. For many objects in translational motion we can ignore the details of the object itself and model the object as an ideal particle and draw it as a simple dot. We number each dot to show the order in which the apple was at the positions indicated. The same amount of time elapses between each dot and the next one. The figure to the right shows the motion diagram for the apple in free fall. Four motion diagrams are shown below. One is of a car moving to the right at constant speed, one is of a car moving to the left at constant speed, one is a car accelerating to the right away from a stop light that has just turned green, and one is a car moving to the right and slowing down as it approaches a stop sign. Which motion diagram corresponds to which case?


10

Activity 7

If a motion diagram represents the position of an object every second, then the distance between each dot and the next is numerically equal to the average speed of the object during that one second interval. For the motion diagrams of Activity 6, draw a line from each dot to the next representing the magnitudes of these speeds. Put an arrowhead on each line indicating the direction of the motion.

Imagine that two of the dots in the motion diagram are separated by 0.15m. If the second dot is the position of the object 1.0 second after the position of dot 1, what is the average speed of the object during this one second interval?

Imagine that the two dots of Part B, 0.15 m apart, represent the positions of the objects for a time interval of 0.50 seconds. Now what is the average speed of the object during the half-second interval?

In Part A you “connected to dots” of the motion diagram. If the motion diagram represents the position of the object every millisecond, what is the relationship between the length of the line from each dot to the next and the average speed of the object during that millisecond?

Activity 8

Here are some made up data for the x component of the position of an object at various times:

Time (s) Position (m) 0.00 0.002 0.10 0.111 0.20 0.385 0.30 0.892 0.40 1.613 0.50 2.501 0.60 3.612

Sketch a graph of position vs. time. . Make the horizontal axis the time and the

vertical axis the position. Is it reasonable to “connect the dots” with a smooth line in the graph you sketched? If

yes, what assumption is being made about the motion of the object? If no, why? Sketch a motion diagram of the motion of the object. Calculate the displacements of the object for each 0.1 s interval. How does the number of displacements you calculated compare to the number to the

number of data points in the position-time data?


11

From Part D, calculate the x components of the average velocities of the object for each 0.1 s interval.

Consider the first of the average velocity values from Part F. At what time does the object have this value of the average speed? Is the value of the time a single value or a range of values? Why?

Sketch a graph of the average velocity versus time. Make the time the horizontal axis. From your result from Part F calculate the x component of the average acceleration of

the object for each 0.1 s interval. How does the number of calculated values of the average acceleration compare t the number of data points in the position-time data?

Sketch a graph of the average acceleration versus time. Make the time the horizontal axis.

What does the data indicate about the acceleration of the object?

Activity 9

Imagine that the data from Activity 8 were taken with a computerized data acquisition system. The system has nearly perfect accuracy, but the precision of each distance measurement is ± 0.020 m. What are the corresponding uncertainties in the calculated values of the displacements, velocities, and accelerations/

Activity 10

An experiment to determine whether Energizer or Duracell batteries last longer could measure the number of hours two AA batteries from each brand will run a tape player. Here is some made up data: Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Average Duracell (hours)

11.4 12.2 7.8 5.3 10.3 9.4

Energizer (hours)

11.6 7.0 10.6 11.9 9.0 10.0

A. From the data, what brand of battery would you choose for your tape player? B. Why do you think there is such a large variation for the different trials of the same

brand of battery?


12

Preparing for Activities 11 - 13 The next few Activities will involve a Track and Collision Cart. The Track should be leveled, but you should check to make sure.

1. Push the Cart and let it run up and down the Track a few times to warm up the bearings in its wheels.

2. Place the Cart near one end of the Track and give it a very gentle push. It should drift a few centimeters and stop. Give the Cart a very gentle push in the opposite direction: it should drift a few centimeters and stop. If the Cart has a tendency to stop and reverse its direction then the Track needs leveling.

The feet under the Track are adjustable by loosening the lock nut and rotating the feet. Be sure to tighten the lock nut when you have the Track level. The Instructors have a level, which may help. The level will be required if you suspect that the Track is not level along the axis perpendicular to its length. Please do not adjust the positions where the feet are mounted on the Track. Note that although the Carts have low friction, the fact that they do slow down and stop means the friction is not zero. At this time, you will find it convenient to measure and record the distance between the feet. The mounts for the feet provide a convenient way to do this. Estimate the position of one of the mounts with the scale mounted on the Track and the corresponding position of the other mount. You will notice that there is a Cart Launcher mounted on one end of the Track. When the Launcher is used the Track tends to recoil. Thus the bracket for the feet closest to the Launcher is braced with double-rod assembly connected fixed to the tabletop with two table clamps. You are provided with a set of blocks which will be placed under the feet tilt the Track. There are blocks that are 1.000 cm, 0.500 cm, and 0.100 cm thick. In addition, for one of the Activities you will need finer adjustments than these blocks provide. It turns out that good quality playing cards are carefully controlled in all their dimensions, and are typically 0.029 cm thick. You are provided a deck of playing cards with the card thickness written on the box.


13

Activity 11

A Cart Launcher is mounted on one end of the Track. Raise the other end of the Track by raising the feet 3.000 cm. The Launcher may be cocked by pulling back the horizontal rod until the disc mounted on it latches to the “finger” on the base. Cock the Launcher and place the Cart against it. Fire the launcher. You want the Cart to travel almost but not quite all the way up the Track. You want the highest position to be at least a few cm away from the magnetic bumper mounted on the far end of the Track, so the cart does not interact with the bumper. You may need to adjust the Launcher to achieve this. There is a disc mounted on the rod that pushes the Cart whose position can be adjusted to get the desired force.

Sketch a motion diagram of the movement of the Cart up the Track from a moment after it leaves the launcher until it comes to rest. It should have some resemblance to one of the motion diagrams of Activity 6.

Roughly, what is the time between each successive dot of Part A? Remember that best laboratory practice is to record everything. The Launcher includes a scale that reads how far the spring has been compressed when it is cocked. You should record this value.

Activity 12

This Activity uses the same setup as Activity 11. Note and record the position of the Cart as measured by the scale on the Track when it is resting against the Launcher when it is not cocked.

A. Launch the Cart and note the position on the scale of the Track where the Cart is at its maximum distance. Repeat a few times, recording each position. Are the values exactly the same for each launch?

B. What are all the reasons you can think of to explain why the positions are not exactly repeatable? The manufacturer of the Launcher says it will launch the cart “with the same force each time.” Is this statement correct?

C. How can you quantitatively characterize the spread in values of the positions that you measured?


14

D. Is it possible to have an apparatus similar to this one for which the positions would be exactly the same each time?

E. What is the mean, i.e. average, value of the positions you measured? What is the mean value of the total distance the Cart travels up the Track between launch and momentarily coming to rest at the top of the Track?

Activity 13

Now raise the feet on the end of the Track opposite the Launcher by 3.500 cm. Measure the distance the Cart travels up the Track. Although you may do a careful measurement like you did in Activity 12, just estimating the position of the Cart at its greatest distance to the nearest centimeter will be sufficient. Remember to keep the end of the Track with the Launcher against the U-shaped rod to minimize rebound.

A. The total distance the Cart travels is less than in Activity 11. So the angle of the Track and the distance the Cart travels are both different. Is anything the same? If so, what?

B. It is unlikely that your answer to Part A came out numerically perfect. What are all the reasons you can think of to account for the small variation from perfection?

This Guide was written in May, 2007 by David M. Harrison, Dept. of Physics, Univ. of Toronto. Activity 10 is from R.L. Kung, Am. J. Phys. 73 (8), 771 – 777 (2005). Last revision: September 25, 2008.

15

PHY131 Practicals Manual Scientific Method Module

Scientific Method Module Student Guide.

“God is subtle but he is not malicious.”

-- Einstein The Scientific Method is a set of techniques and assumptions used to try to discover the organizing principles of the physical universe. In this Module, we will explore the method using a “universe” of a set of playing cards. Here your Instructor will present cards from a deck to you; your task will be to figure out what ‘law’, if any, controls the pattern by testing hypotheses (i.e. through conjecture and refutation). The basis of the Scientific Method is that one must be prepared to “dare to be wrong.” If we are not prepared to be wrong, then we are not able to increase our understanding.

A. List five different patterns that might be true for the cards in the deck. These should be possible general patterns, not predictions of what the next card might be. For each pattern, list what cards would support the hypothesis and what cards would falsify the hypothesis.

B. It is likely that in Part A you made one or more assumptions about that nature of the “universe” of the deck of cards. These could be:

• The deck contains four suits: clubs, diamonds, hearts and spades. • The deck contains aces, cards numbered between 2 and 10, plus jacks,

queens and kings. • There are 52 different cards in the deck. • Etc.

Identify as many of those assumptions that you have made as possible. C. Your Instructor will show you the first three cards of the deck. For the patterns of

Part A: • Which have been proven to be correct? • Which have been proven to be incorrect? • Which have not been proven to be either correct or incorrect?

Have any of your patterns been proven to be correct? What would be necessary for a pattern to be proven to be correct? If all of your patterns have been proven to be incorrect, try to choose two or more patterns that might be true for the cards of the deck based on the three cards that you can see.

D. Your Instructor will show you the next three cards of the deck. For the patterns not proven to be incorrect in Part C:

• Which have now been proven to be correct? • Which have now been proven to be incorrect? • Which have not been proven to be either correct or incorrect?

PHY131 Practicals Manual Scientific Method Module

16

E. Can you now say what the pattern of the cards in the deck is? What would be necessary for you to be 100% sure that you know what the pattern is?

This Student Guide was written by David M. Harrison, Dept, of Physics, Univ. of Toronto, in August 2008. It is based on materials developed by Allen Journet, Dept. of Biology, Southeast Missouri State University, http://cstl-csm.semo.edu/journet/BS107/LabManual/BS107-LAB3%20F2008.pdf. Last revision: September 25, 2008.

17



Concepts of this Module

• Vectors • Relative Speeds • Newton’s First and Second Laws

The Activities

Activity 1

If we subtract a vector Ar

from itself, there are at least two ways to write the result:

1. 0rrr

=− AA 2. 0=− AA

rr

The right hand side of the first form is a vector, while the right hand side of the second form is not. Which form is correct, 1 or 2? Why?

Activity 2

Here are two position vectors, A

r and B

r.

Vector Ar

has a magnitude of 5.0 cm and makes an angle of 30 degrees with the vertical as shown. Vector B

rhas a magnitude of 7.5 cm and makes an angle of 45 degrees with the

vertical as shown.


18

Sketch the two vectors as shown onto the graph paper in your Lab Notebook. Note that sketch means a rough version: draftsman-like precision is not required. Leave at least 5 cm space around the sketch in all directions.

A. In the Notebook sketch and label the sum BArr

+ B. On the same sketch draw and label the sum AB

rr+ . Compare your result to Part A.

C. Sketch the two vectors into your Lab Notebook again. Leave at least 5 cm of space around the sketch in all directions. Draw and label the difference BA

rr− .

D. On the same sketch as Part C, draw and label the difference ABrr

− . Compare your results to Part C.

A simple little Flash animation illustrating addition of two vectors is available at: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Vectors/Add2Vectors.html There is also an animation of subtracting two vectors at: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Vectors/Subtract2Vectors.html

Activity 3

Here are three position vectors Ar

, Br

and Cr

. Ar

and Br

are the same vectors as in Activity 2. Vector C

r has a magnitude of 7.0 cm at makes an angle of 30 degrees with the

horizontal as shown.

Sketch the three vectors as shown into your Lab Notebook. Leave at least 5 cm space around the sketch in all directions.


19

A. In the Notebook sketch and label the sum CBArrr

++ )( . B. On the same sketch draw and label the sum )( CBA

rrr++ . Compare your result to

Part A A Flash animation illustrating the addition of three vectors is at: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/Vectors/Add3Vectors.html

Activity 4

A. Two vectors have different magnitudes. Can their sum be zero? Explain. B. If one component of a vector is nonzero, can the vector have zero magnitude?

Explain.

Activity 5 Assume that the speed of sound is exactly 344 m/s relative to the air. Assume that the speed of light is exactly 3 x 108 m/s relative to the observer.

A. If you are pursuing a sound wave at a speed of 99% of the speed of sound, what is the speed of the sound wave relative to you?

B. If you are moving through the air at 99% of the speed of sound in the opposite direction to the velocity of a sound wave, what is the speed of the sound wave relative to you?

C. If you are pursuing a light wave at 99% of the speed of sound, what is the speed of the light wave relative to you?

D. If you are pursuing a light wave at 99% of the speed of light, what is the speed of the light wave relative to you?


20

Activity 6

We model Sponge Bob Square Pants as a simple sponge of width w, height h, and depth d. He is 6 cm wide, 12 cm high, and 4 cm deep. It is raining. The raindrops are falling straight down at a constant speed of 9 m/s. Each raindrop has a diameter of 5 mm, and we can treat them as perfect spheres. There are 8000 raindrops per cubic meter.

A. Bob is stationary in the rainstorm. How many raindrops per second fall on the top

of his head, i.e. the upper horizontal surface of the sponge? Do any raindrops strike his vertical surfaces?

B. Bob is now walking forward at 1.3 m/s. What is the velocity of the raindrops relative to Bob?

C. Now how many raindrops per second fall on the top of his head? D. Bob is initially 50 m from a shelter. How many raindrops fall on the top of his

head until he reaches the shelter? E. How many raindrops per second strike his “face” i.e. the vertical surface of width

w and height h? F. How many raindrops strike his face before he reaches the shelter? G. Instead of walking, Bob runs for the shelter at 2.5 m/s. What is the velocity of the

raindrops relative to Bob? H. Now how many raindrops per second strike his “face” i.e. the vertical surface of

width w and height h? I. Now how many raindrops strike his face before he reaches the shelter? J. If it is raining, is it worth running for shelter instead of walking?

Activity 7

You can swim at a speed v relative to the water. You are swimming across a river which flows at a speed V relative to the shore. The river is straight and has a constant width.

A. If you wish to swim directly across the river, in what direction should you swim relative to the water in the river?

B. If you wish to get across the river as quickly as possible and don’t care where you land on the opposite bank, in what direction should you swim relative to the water?


21

Activity 8

Joe is stationary on the ground, and sees an airplane moving to the right with a speed of 200 m/s and accelerating at 5 m/s2. Suzy is driving to the left at a constant 40 m/s and Latoya is driving to the right at a constant 40 m/s.

A. Rank in order, from the largest to the smallest, the airplane’s speed according to Suzy, Joe, and Latoya at the moment shown in the figure. Explain.

B. Rank in order, from the largest to the smallest, the magnitude of the airplane’s acceleration according to Suzy, Joe, and Latoya. Explain.

Activity 9

This Activity uses the Cart and Track that were introduced in Module 1. Remember to:

• Push the Cart and let it run up and down the Track a few times to warm up the bearings in its wheels.

• Check that the Track is level. Now use the thin blocks to raise the side of the Track closest to the wall a few millimeters.

A. Place the Cart on the Track near the end closest to the wall, place the supplied wooden block on the Cart, and give the Cart a very gentle push. Does it move at a constant speed down the track? If it is slowing down, raise the height a bit more. If it is speeding up, reduce the height. At what height does the Cart move at approximately constant speed? The playing cards are a good way to make small changes in the height.

B. When the Cart is moving at constant speed down the Track, sketch a Motion Diagram of its motion.

C. Treat the Cart plus the block on top of it as a single system. When the Cart is moving at constant speed down the Track, sketch a Free Body Diagram of all the forces acting on the system when it is about half-way down the Track.

D. How much can you vary the height of the track and not see any difference in the motion of the Cart? The playing cards are a good way to introduce small changes in the height.

E. Express your result from Part A as a single value. Include your result from Part D by adding a ± error term to the value.


22

F. Place the wooden block in front of the Cart so the cart will push it down the Track. Now there will be more friction. Now what height must you raise the Track to have the Cart moving at approximately constant speed?

G. Again treat the Cart plus the wooden block as the system. Sketch a Free Body Diagram of all the forces acting on the system for Part F.

H. If you could completely eliminate the friction of the Cart and Track, what height would the end of the Track be raised for the Cart to move at constant speed?

I. Is it ever possible to completely eliminate friction? J. Remove the wooden block but keep the Track at the same angle as Part F. Give

the Cart a gentle push. Draw a Motion Diagram of its motion down the Track. K. Draw a Free Body Diagram of all the forces acting on the Cart in Part J when it is

about half-way down the Track. Compare to your Free Body Diagram of Part C.

Activity 10

For this Activity you will be using a computer-based laboratory system with an ultrasonic motion sensor and motion software. The motion sensor acts like a stupid bat when hooked up with a computer-based laboratory system. It sends out a series of sound pulses that are too high frequency to hear. These pulses reflect from objects in the vicinity of the motion sensor and some of the sound energy returns to the sensor. The computer is able to record the time it takes for the reflected sound waves to return to the sensor and then, by knowing the speed of sound in air, figure out how far away the reflecting object is. There are a few points to be aware of when using the sensor:

1. The sensor cannot detect distances less than about 0.15 meters because it cannot record reflected pulses than come back too soon after they are sent.

2. The ultrasonic waves spread out in a cone of about 15° as they travel. They will “see” the closest object. Be sure there is a clear path between the object you are tracking and the motion sensor.

For further details, see the manual on the detector and software at: http://faraday.physics.utoronto.ca/Practicals/Equipment/MotionSensor.pdf Set the detector to collect about 40 samples per second. Set the switch on top of the sensor to the wide beam, which on some sensors is indicated by an icon of a person. Use the system to take position-time data of one of your Team as he/she walks towards and away from the sensor. Try to glide as smoothly as possible at constant speed.

Loose clothing like bulky sweaters are good sound absorbers and may not be “seen” very well by the motion sensor.


23

The software will compute the average velocity and acceleration, just as you did by hand in Activity 8 of Module 1. Use the software to do those computations. Does the plot of average velocity look smooth? If not, why? What about the plot of average acceleration?

Activity 11

Mount the motion sensor on the end of the Track closest to the wall, and use the hardware and software to repeat Part D of Activity 9. Use the switch on top of the sensor to select the narrow beam, which on some sensors is indicated by an icon of a cart. Set the angle of the sensor to 0 so it is “looking” straight down the track. Setup the sensor so it takes 5 samples per second. Do not try to measure distances less than about 0.15 m.

A. Does this do a better job than the estimates by eye that you did in Activity 10? Explain. In particular, is the ± error term using this method smaller than the result for Part E of Activity 10?

B. Save your distance-time data for one of your trials to the server by using the File tab of the Motion Sensor vi. Use a descriptive name for the file. Since this is Module 2, Activity 11, if you have raised the track by 3 mm the file name could be: M2A11_3mm. Write down the name of the file and the path in the lab book.

C. The datafile is a tab separated text file. Look at the file using either Excel or a text editor, but do not change the contents. For constant acceleration a, the distance d depends on the initial distance d0, initial speed v0 according to:

200 2

1 attvdd ++=

Use the PolynomialFit program, which is available on your computer’s desktop, to fit the dataset to a second order polynomial (Powers 0 1 2). Is the acceleration of the Cart zero within errors? How does using the Motion Sensor compare to you doing it by eye as in Activity 9?

Activity 12

In this Activity you will use a Force Sensor. When connected to appropriate software, this device measures forces exerted on it. The device uses a piezoelectric material, which generates a voltage proportional to the force exerted on it. Other uses of piezoelectrics


24

include contact microphones, the motion sensor capabilities of the Sony Playstation 3 and Nintendo Wii controllers. Pick out one of the Number 24 rubber bands as your standard rubber band. You may want to identify it by marking it with a pen or pencil. Loop the rubber band loose around your fingers as shown. Slowly separate your hands until the rubber band is not slack. Now separate your hands by some further predetermined “standard” length that you choose. You can feel that the rubber band is exerting forces on both of your fingers. How do the magnitudes of these two forces compare? Each member of your Team should do this simple little experiment.

A. When stretched by the standard length the rubber band is exerting a standard force on your fingers. Decide what name you wish to give to this standard force.

B. Now loop the rubber band around the hook on the Force Sensor that is mounted on the vertical rod and start the Force Sensor program on the computer. Push the Tare button on the Force Sensor to zero its reading. Stretch the rubber band by the standard length and determine the force in newtons corresponding to your standard force.

C. If you were to attach the rubber band to a Cart on the Track and kept the rubber band stretched by your predetermined length, the Cart would accelerate. (This would take some physical dexterity to achieve.) What other ways can you think of to apply an equivalent standard force to the Cart?

D. How would you test to determine that these forces really are equivalent to your standard force?

E. Which of the methods you thought of in Part D do you think is the best one?

Activity 13

A. Loop the standard rubber band around your fingers and stretch it by your standard length to refresh your


25

memory about what the standard force feels like. Now loop two rubber bands around your fingers and stretch them by your standard length. How does the force exerted on your fingers with two rubber bands compare to just one?

B. Repeat with three rubber bands. C. Use the Force Sensor to check your feelings about the magnitudes of the forces. D. Is there any difference between the forces exerted on the Force Sensor by a rubber

band and an equal force exerted on it when you just hold the hook and pull? Explain.

Activity 14

In this Activity you will use a Fan Cart. This has a two-speed motor which causes the fan to rotate.

A. Keep your fingers away from the moving fan blade! B. Avoid a runaway Cart falling off the Track.

Level the Track and leave the Motion Sensor mounted on one end. Warm up the bearings of the wheels of the Fan Cart by rolling it up and down the Track a few times. Set the fan angle at zero degrees.

A. Place the Fan Cart on the Track close to the Motion Sensor but at least 0.15 m away from it. You will want the direction of the air from the fan to blow towards the Motion Sensor. Turn the fan on low and use the Motion Sensor to measure the acceleration of the Cart.

B. Sketch a motion diagram of the Cart. C. Consider the Cart, motor, fan and the housing for the fan as the system under

consideration. Sketch a Free Body Diagram of all the force acting on the system when the Cart was accelerating in Part A.

D. Use the Force Sensor to measure the net horizontal force acting on the system when it is not moving. Is this the force acting on the system when it is moving?

E. Repeat Parts A – D with the fan on high. F. Sketch a graph of acceleration versus force, with the force on the horizontal axis.

Be sure to include the origin on the graph. Although you only have two data points, what do you think the shape of the graph is for an arbitrary number of data points?

G. Is there a “free” third data point that you can include in your graph? Hint: what is the acceleration of the Fan Cart when the fan is off?

H. Sketch a straight line that “fits” the two data points. Should the line go through the origin?

I. How much can you vary the slope of the line and still more-or-less “fit” the data? Graphical estimation of slopes and their errors was discussed in Section 14 – Graphical Analysis of the document on error analysis that was also the assignment due at the first Practical.


26

J. Write down the relation between the force F and acceleration a as an equation including any necessary constants. Include as estimate of the error in those constants.

K. Place the sail on the magnetic pad of the Cart with the plane of the sail parallel to the plane of the fan. Predict the motion of the Cart when the fan is blowing.

L. Check your prediction. Explain the result.

Activity 15

A key aspect of the scientific method is that often when a physical system has many variables we can keep all but two of the variables constant, and can investigate how those two variables relate to each other. In Activity 14 you varied the force applied to the Cart and saw how different forces cause different accelerations of the Cart. In this Activity you will apply the same constant force to the Cart but will vary its mass.

A. Measure the mass of the Fan Cart, the non-fan Cart, and the two black rectangular masses.

B. The Fan Cart stacks on top of the non-fan one. In addition, the black rectangular masses can be placed on the magnetic pad of the Fan Cart. How many possible values of the total mass are possible using the Fan Cart with and without the rectangular masses both by itself and when stacked on top of the non-fan Cart?

C. Measure the acceleration three or four of the values of the mass you determined in Part D. Which masses did you choose to measure and why?

D. Sketch a graph of acceleration versus total mass, with the mass on the horizontal axis. What is the shape of the graph?

E. Sketch a graph of acceleration versus one over the mass, with one over the mass on the horizontal axis. Include the origin in the graph. Is this graph simpler than the one in Part F?

F. For the graph of Part E, draw a straight line that “fits” the data. Should the line go through the origin? Why?

G. Write down the relation between mass m and acceleration a including any necessary constants and their errors.

H. Combine your result for Part G and Activity 14 Part H into a single equation involving F in newtons, m in kg, and a in m/s2 and any necessary constants. You may find the following useful: • What is the value of constant you found in Part J of Activity 15 in terms of

any physical parameters of the system? • What is the value of the constant you found in Part G of this Activity in terms

of any physical parameters of the system? I. Repeat Part H when the force is expressed in the unit you chose for the standard

force in Activity 12 Part A. Are there any constants required now? Explain.


27

Activity 16

For one-dimensional motion in the x direction, here are three ways to write Newton’s 2nd Law:

1. xx maF =

2. xx Fm

a 1=

3. x

x

aF

m =

Although these three forms are mathematically equivalent, in terms of using mathematics as a language to describe the relation between forces, masses and acceleration they are not. Which form best describes the central idea of Newton’s 2nd Law? Explain. Hint: when you write that some variable y is a function of another variable x, such as:

y = f(x)

one variable is called the independent variable and another is called the dependent variable. Which is which, and why is this terminology used? This Guide was written in July 2007 by David M. Harrison, Dept. of Physics, Univ. of Toronto. Some parts are based on Priscilla W. Laws et al, Workshop Physics Activity Guide (John Wiley, 2004) Unit 5. The figures in Activities 12 and 13 are modified from Randall D. Knight, Physics for Scientists and Engineers (Pearson Addison-Wesley, 2004), Figure 4.16. Last revision: October 3, 2008

28



Concepts of this Module • Equilibrium • Mass and weight • Two dimensional motion • Projectile motion • Circular motion • Tensions and Ropes

Preparation for this Module In each of the four elevators in the tower of the Physics building is mounted a spring scale with a mass hanging from it. Before this Practical take a ride in one of the elevators and note what happens to the reading of the scale for the six cases listed in Activity 2 below.

The Activities

Activity 1

A round table is supported by three legs. If you are going to push down on the top of the table to make it unstable, where is the best place to push? Explain.

Activity 2

As preparation for this Module you took a ride on one of the elevators in the tower, paying attention to the reading of the spring scale for six different cases:

a) Starts from rest and starts moving to a higher floor. b) Is moving uniformly up. c) Approaches the higher floor and starts slowing down. d) Starts from rest and starts moving to a lower floor. e) Is moving uniformly down. f) Approaches the lower floor and starts slowing down.

For each of the six cases:


29

A. Describe the reading of the scale. B. Sketch a Free Body Diagram of all the forces acting on the mass during the

motion being investigated. Use the diagram to explain the reading of the scale. C. Suppose that instead of a single mass suspended

from a spring scale, the apparatus consisted of a pan balance with two masses with equal values on the pans. What would be the motion of this balance for each of the six cases you investigated? Explain.

Activity 3

In Jules Verne’s From the Earth to the Moon (1865) a huge cannon fires a projectile at the moon. Inside the projectile was furniture, three people and two dogs. The figure is from the original edition. Verne reasoned that at least until the projectile got close to the Moon it would be in the Earth’s gravitational field during its journey. Thus the people and dogs would experience normal gravity, and be able to, for example, sit on the chairs just as if the projectile were sitting on the Earth’s surface. One of the dogs died during the trip. They put the dog’s body out the hatch and into space. The next day the people looked out the porthole and saw that the dog’s body was still floating just beside the projectile.

A. Is there a contradiction between the inhabitants inside the projectile experiencing normal gravity and the dog’s body outside the projectile not falling back to the Earth?

B. If your answer to Part A is yes, where did Verne make his mistake? If your answer is no, explain.

Activity 4

A bucket of water has a one end of a spring soldered to the bottom, as shown. A cork is attached to the other end of the spring and is suspended motionless under the surface of the water. You are holding the bucket so that it is stationary

A. Draw a Free Body diagram of all the forces acting on the cork.


30

B. As Archimedes realized a long time ago, the upward “buoyant” force on the cork is equal to the weight of the water that the cork has displaced. Imagine an identical bucket-spring-cork system is stationary on the surface of the Jupiter where the acceleration due to gravity is 2.65 times greater than on Earth. Compared to the bucket-spring-cork on Earth, is the cork closer to the surface of the water, closer to the bottom of the bucket, or in the same relative position?

C. Imagine that you take the Earth bucket-spring-cork onto an elevator. The elevator starts accelerating upwards. While it is accelerating does the cork move closer to the surface of the water, closer to the bottom of the bucket, or stay in the same relative position?

D. Imagine that you take the Earth bucket-spring-cork up on the roof of a tall building. Still holding the bucket you step off. While you are in free fall towards the ground, does the cork move closer to the surface of the water, closer to the bottom of the bucket, or stay in the same relative position?

Activity 5

Wilma, queen of the drag strip, is about to race her Corvette Z06. She is stationary on the track, waiting for the lights to go green so she can accelerate down the strip. For luck, she always has a pair of fuzzy dice of mass m hanging from the rear view mirror. We will model the dice hanging from the rear view mirror with the supplied ball and string. One of your Team should hold the string with the ball hanging down. This person then begins walking forward at a fairly high speed.

A. Before the person started walking sketch a Free Body Diagram of all the forces acting on the ball.

B. Initially the ball was at rest for all of you. Newton’s First Law says that bodies at rest remain at rest until a force causes their state of motion to change. When the person holding the ball begins walking what does he/she see the ball do? Is this what Wilma would see the fuzzy dice do? Are these consistent with Newton’s First Law? Explain.

C. For those of you who were not holding the ball and string, what did you see the ball do when the person holding the string began walking? Is this consistent with Newton’s First Law? Explain.

D. Assume Wilma is accelerating at a constant rate a. For you, standing beside the track, the dice reach a steady state where they are not hanging straight down, but make an angle θ with the vertical as shown. Draw the Free Body diagram of all external forces acting on the dice.

E. What is the angle θ?


31

Activity 6

Wilma, queen of the drag strip, has taken the kids to the zoo in her SUV. They are going home, and the kids are sitting in the back seat while the SUV is stopped at a stop light. Wilma bought them a Helium-filled balloon, which they are holding by the string so it is not touching the roof of the SUV. The balloon “floats” in the air because of a buoyant force on it, which Archimedes realized long ago is equal to the weight of the displaced air. The windows of the car are all rolled up. The light turns green and Wilma accelerates the SUV, but certainly at a lower rate than when she races her ‘vette at the drag strip. Describe the motion of the balloon as seen by the kids after the light turns green.

Activity 7

A “funnel cart” has a ball on top of a funnel. Inside the funnel is an apparatus that fires the ball straight up at a pre-determined time. If the cart is stationary, when the ball is fired it goes straight up and then lands back in the funnel.

A. The cart is moving to the right at constant speed. When the ball is fired, does it land in the funnel? If not where does it land? Why?

B. Now the cart is being pulled to the right and is accelerating. When the ball is fired, does it land in the funnel? If not where does it land? Why?

C. Now the cart is rolling down a frictionless inclined track. Assume that the track is longer than is shown in the figure. When the ball is fired, does it land in the funnel? If not where does it land? Why?


32

Activity 8

In Module 2 Activity 14 you used a Fan Cart with a mass sitting on the magnetic pad, as shown. Assume, as you did in that Activity, that the friction of the wheels is negligible. The mass on the pad has a value m, and the mass of the cart, fan, motor etc. is M. A total force F

r is

exerted on the system. As you showed, the acceleration a of the system is:

mMFa+

= .

A. If the pad on the Cart was not magnetic and was also super slippery, when you released the Cart what would have been the motion of the mass m?

B. For this case what would have been the acceleration of the Cart? C. In the actual case, the mass m moves along with the Cart with the same

acceleration. Sketch a Free Body Diagram of all the forces acting on the mass m for this case.

D. What is the magnitude and direction of the horizontal force exerted on mass m? What is the cause of this force?

E. Sketch a Free Body Diagram of all the forces acting on the mass M. F. From Part E calculate the acceleration of mass M. Is your value reasonable?

Activity 9

Whirl the supplied ball on a string in a horizontal circle, being careful not to hit anybody or thing with it. Try to maintain the ball at constant speed.

A. What is the net vertical force on the ball? B. Sketch a Free Body Diagram of the forces acting on the

ball for some point in its circular orbit. There is a common convention for indicating vectors that are going out of or into the page, illustrated to the right. It is like an arrow: when it is moving towards us we see the tip, but when it is moving away from us we see the feathers at the other end of the arrow.


33

C. What must be the direction of the ball’s acceleration to keep it moving in a horizontal circle?

D. From you Free Body Diagram determine the net force acting on the ball. Does this agree with Part C?

E. To maintain the ball at constant speed you need to move your hand that is holding the string. Explain why this is so. What would be the necessary condition to maintain the ball in uniform circular motion without needing to move your hand?

F. If you suddenly let go of the string, what will be the motion of the ball? If you actually do this, be sure that you know in what direction the ball will go so that you don’t hit anybody or thing.

Activity 10

Tarzan is swinging back and forth on a vine. We will model his motion with the supplied ball and string, and will assume that air resistance is negligible. Fix the upper length of the string to a fixed point. Hold the ball so the string makes an angle of about 45° = π/4 radians and release it from rest so that it swings back and forth. A. Using the supplied graph paper, draw a Motion Diagram

for when the ball is released until it reaches its maximum swing on the other side. Use a total of 11 dots, with the 1st dot for the moment he steps off the branch, the 6th dot for when the vine is vertical, and 11th dot to the next position where the instantaneous speed is zero.

B. Imagine that the dots in the diagram of Part A were for Tarzan’s motion every second. Now draw an expanded scale Motion Diagram on another sheet of graph paper for the first second after he steps off the branch. Use 11 dots, each representing his position every 0.1 seconds. Connect the dots with vectors which are proportional the average velocity vectors.

C. Re-draw the velocity vectors from Part B from a common origin. What is the direction of Tarzan’s acceleration when he just steps off the branch?

D. Sketch a Free Body Diagram of all the forces acting on Tarzan when he just stepped off the branch. What is the direction of the total force acting on him?

E. Draw an expanded scale Motion Diagram for Tarzan’s motion from 0.5 seconds before he reaches the bottom of his swing to 0.5 seconds after, again using a total of 11 dots. Connect the dots with vectors pointing from one position to the next.

F. Re-draw the velocity vectors from Part E from the same origin. What is the direction of Tarzan’s acceleration at the moment that the vine is vertical?

G. Sketch a Free Body Diagram of all the forces acting on Tarzan when he is at the bottom of his swing? What is the direction of the total force acting on him?


34

Activity 11

Whirl the supplied ball on a string in a vertical circle. Have the ball moving fast enough that the string remains taut at all times.

A. Qualitatively how does the speed of the ball at the top of the circle compare to its speed at the bottom of the circle?

B. Sketch a Motion Diagram of the motion of the ball. C. Your hand can feel the tension the string is exerting on it. How does this tension

related to the force being exerted on the ball? Qualitatively how does the force exerted on the ball at the top of the circle compare to the force exerted on it at the bottom of the circle?

D. Allow the speed of the ball the decrease until the string is no longer taut at some point near the top of the circle. Sketch a Motion Diagram of the motion of the ball after this point in its motion.

Activity 12

Suppose you were to hang masses of m = 0.5 kg from the Force Sensors with light strings in the configurations shown below.

Predict the readings of the Force Sensors for each of A – G. Check your prediction by doing the measurements. The sensor tends to “drift” in time. Therefore, before each measurement you should:


35

1. Have zero force being exerted on the sensor. 2. Press the Tare button on the sensor.

Activity 13

In the figure professional wrestler Randy “Macho Man” Savage is suspending a 10 kg mass with a rope between his two hands. Is the strongest member of your team, or even the Macho Man, strong enough to keep a heavy mass stationary and the rope perfectly horizontal? Explain

Activity 14

A wooden rod is suspended by a string tied to one end; the other end of the string is tied to a fixed support. The other end of the rod is resting on a piece of Styrofoam that is floating on water. Which figure is closest to the equilibrium position of the system?

Explain your answer. Your Instructors will demonstrate this system. Was your prediction correct?


36

Activity 15 In the figure the Track is at an angle θ with the horizontal. The Cart has a mass M approximately equal to 0.5 kg. It is connected to a hanging mass m = 0.0500 ± 0.0001 kg by a massless string over a massless pulley.

A. Use the balance to measure the mass M of the Cart. B. For some angle θ the masses are in equilibrium, i.e. if they are at rest they remain

at rest and if they are moving at some speed the continue moving at that speed. Calculate the value of the angle θ. Express your result in radians.1

C. The end of the Track that has the pulley mounted on it can be moved up and down using the attached clamp and the vertical rod mounted to the table. You may find that to make changes in the angle of the Track it is easiest to adjust the position of the vertical rod. Verify your prediction of Part B. The digital level is a good way to measure the angle of the Track.

Activity 16

A. How much can you change the angle θ of the Track and not see any visible deviation from equilibrium. Express your result from Part C of Activity 15 and from Part A of this Activity by expressing the angle for equilibrium as θ ± Δθ, with both values in radians.

B. Imagine you are going to use this apparatus as a silly way of measuring the mass M of the Cart. From Part B what is the value and error of M determined this way? What is the dominant error in your measurements that has the greatest effect on your value of ΔM?

C. The string is not really massless. Can you think of an experimental procedure for which the mass of the string does not matter?

This Guide was written in July 2007 by David M. Harrison, Dept. of Physics, Univ. of Toronto. Some parts are based on Priscilla W. Laws et al., Workshop Physics Activity Guide (John Wiley, 2004), Unit 7. Christos Josephides and Andrew Zasowski have participated in development of the Mechanics Modules 1 – 4, and wrote much of Activity 9 of this Module. Last revision: October 16, 2008.

1 1 radian = 57.2958°, or 2π radians = 360°, or π radians = 180°.

37



Concepts of this Module • Impulse-momentum Theorem • Conservation of Momentum • Introduction to Conservation of Energy • Angular Momentum

The Activities

Activity 1

Setup:

• Make sure the Track is level. • The Motion Sensor should be mounted on the end of the Track closest to the wall

and connected to the DAQ board. • Clamp a vertical rod to the Table Clamp on the end of the Pod opposite the wall,

and place the Track so the end without the Motion Sensor is almost touching the rod.

• The Force Sensor with the spring bumper attached should be mounted on the Cart. Measure the total mass of the Cart and Force Sensor.

• Place the Cart on the Track with the bumper facing the vertical rod. Connect the Force Sensor to the DAQ board. Press the Tare button on the Force Sensor to zero its readings.

• Run the Cart up and down the Track a few times to warm up the bearings in the wheels.

• Set the sample rate for the Motion Sensor to 50 samples per second. Set the sample rate for the Force Sensor to 100 samples per second.

Now the Activities:

A. You want the spring bumper on the Force Sensor to collide with the vertical rod mounted on the Table Clamp. The Cart will drag the wire connecting the Force Sensor to the DAQ board; make sure it is as free to move as possible. You may need to adjust the position of the Track and/or vertical rod to get a nice “clean” collision.

B. Take data on the speed versus time using the Motion Sensor and force versus time using the Force Sensor from a second or so before the collision to a second or so after the collision.

C. From the speed versus time data, what is the momentum of the Cart plus Force Sensor just before the collision? What is the momentum just after the collision?


38

Use the convention that the speed of the Cart before the collision is a positive number, and the speed after the collision is a negative number.

D. What is the change in the momentum from just before the collision to just after? E. From the force-time data, visually estimate the total impulse exerted on the Cart

during the collision: this is the area under the force-time graph. Compare your result to Part D.

F. Use the software to calculate the total impulse exerted on the Cart. Which result do you think is better, the software’s calculation or your estimate in Part E? Explain.

G. Export your speed-time data into a file on the server. Since this is Module 4, Activity 1 Part G, the name of the file could be something like: M4A1PG_vt.

Activity 2

The Physics presented in textbooks almost always deals with ideal cases. For example, in that context we often say “ignore air resistance” or “assume that friction is negligible.” In the real world, often these idealizations are not correct. For example, we know that there is always friction between the Cart and the Track. In Activity 4, there was additional “friction” because the Cart was dragging the wire of the Force Sensor, and the effect of these two forces is easily seen in the data. . These extra forces are in the opposite direction to the motion of the Cart, so always slow down the magnitude of the velocity, regardless of whether the velocity is a positive or negative number. Here we will investigate some ways of dealing with these terms

A. Copy the file you saved in Activity 1 to some name like M4A5_vt_Edited. B. Use a text editor or spreadsheet program to:

• Remove all the speed-time data for when the collision is occurring. • Change the values of the speeds after the collision to positive numbers. If you

are using Excel but are not very familiar with this program, Appendix A discusses some ways to do this.

C. Fit the edited data to a straight line using the PolynomialFit program. What is the meaning the slope of the line? If the slope is found to be zero within errors what does this say about the total friction acting on the system?

D. What is the change in the momentum of the Cart plus Force Sensor during the collision due to these extra forces? Can you express your answer including an error term?

E. Considering your data and your experience with these Carts, which force do you think is largest, friction or dragging the wire?


39

F. What is the change in momentum of the Cart plus Force Sensor during the collision due to the collision itself? Is this a better value than you got in Part D of Activity 1?

G. We can eliminate the term due to dragging the wire by mounting the Force Sensor on the vertical rod mounted on the Table Clamp. Now the Force Sensor will measure the force exerted on it by the Cart. How is this related to the force exerted by the Force Sensor on the Cart?

H. Is this a better procedure than the one you used in Activity 4? Explain.

Activity 3

You are asleep in your room, but a fire has broken out in the hall and smoke is pouring in through the partially open door. You need to close the door as soon as possible. The room is so messy you cannot get to the door. You have a ball of clay and a super ball, each with the same mass. If you throw the clay at the door it will stick to it; if you throw the super ball at the door it will bounce off. You only have time to throw one thing at the door.

A. Which should your throw at the door, the clay or the super ball? Explain. B. Which ball will experience the largest impulse during the collision? C. From Newton’s 3rd Law the impulse that the door exerts on the ball during the

collision is equal in magnitude although opposite in direction to the impulse the ball exerts on the door. Which ball exerts the largest impulse on the door?

Activity 4

Three identical balls slide on a table and hit a block that is fixed to the table. In the figures we are looking down from above. In each case the ball is going at the same speed before it hits the block.

Rank in order from the largest to the smallest the magnitude of the force exerted on the block by the ball.


40

Activity 5

An air track is similar to the Track you have been using in the Practicals. The track has small holes drilled in it, and air blows out of the holes. Thus the carts for the air track float on the air and there is extremely low friction between the cart and the track. An animation of an air track is at: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/AirTrack/AirTrack.html There are six possible collisions that may be simulated: three different values for the mass of one of the carts, with elastic “bouncy” and inelastic “sticky” collisions for each value of the mass.

A. The momentum vmp rr= is also called the quantity of motion and sometimes just

the inertia. For Newton momentum was central to his thinking about dynamics. Use the animation to determine if the total momentum of the carts is conserved before and after each of the six possible collisions.

B. For Leibniz, a contemporary and rival of Newton, momentum was not central to his thinking. Instead he concentrated on the quantity mv2, which he called the vis viva (literally “living force”). Use the animation to determine if the total vis viva of the carts is conserved before and after each of the six possible collisions.

C. Which concept, momentum or vis viva, appears to be the most fundamental?

Appendix A – Using Excel Here is what the raw data file might look like:


41

We will delete rows 7 through 11. Click on the 7 row label. Hold down the Shift key and click on the 11 row label. This will select the rows. Right click on the highlighted area and choose Delete. Now change all the negative values for the speeds after the collision to positive values. The “brute force” way to do this is to double click on each cell with a negative velocity and manually remove the minus sign, although Excel experts will know of a more elegant way to do this. The result is shown to the right. Save the file. You may ignore Excel’s mumbling about losing information. This Guide was written in July 2007 by David M. Harrison, Dept. of Physics, Univ. of Toronto. Some parts are based on Priscilla W. Laws et al., Workshop Physics Activity Guide (John Wiley, 2004), Unit 8. Last revision: August 8, 2008.

42

PHY131 Practicals Manual Teamwork Module

Teamwork Module Student Guide

In the real world as well as in many aspects of your University studies you work in teams. This Module will help you to learn how to make your own teamwork more effective and pleasant. The Module has the following sections:

• Preparation for this Module. Things you should read and do before coming to the Practical.

• Teamwork Basics. This is a discussion of some of the issues that determine whether a team functions effectively, and ways to deal with any problems that arise. (Pages 1 – 5)

• Teamwork Activity. A set of characteristics of effective and ineffective teams. You both individually and as a team rank these characteristics. (Page 6)

• Teamwork Homework. After this Module you may be assigned to write a short paper about teamwork. The paper is described in this section. It will be due one week after the Practical that uses this Module. (Pages 7 – 8)

Preparation for this Module

1. Read the next section of this Module, Teamwork Basics. 2. Do the ranking exercise of the following section, Teamwork Activity.

Teamwork Basics Two things get accomplished in good teams: the task gets accomplished and the satisfaction of team members is high. In order to achieve both of these ends:

• Get to know other members of your group and their strengths • Set ground rules • Use a facilitator • Keep lines of communication open • Know how to avoid (or solve) common problems

Ground Rules Setting some basic ground rules helps to insure that everyone is in agreement about how the team will operate. You will want to establish norms about how work will be done, the role and responsibilities of a facilitator, how you will communicate with one another, and how your meetings will be run. Some of the ground rules can be decided on now; others will develop as the semester progresses.


43

1. Work Norms: How will work be distributed? Who will set deadlines? What happens if someone doesn't follow through on his/her commitment (for example, misses a deadline)? How will the work be reviewed? What happens if people have different opinions about the quality of the work? What happens if people have different work habits (e.g., some people like to get assignments done right away; others work better with the pressure of a deadline). 2. Facilitator Norms: Will you use a facilitator? How will the facilitator be chosen? Will you rotate the position? What are the responsibilities of the facilitator? (see below) 3. Communication Norms: When should communication takes place and through what medium (e.g., do some people prefer to communicate through e-mail while others would rather talk on the phone)? 4. Meeting Norms: What is everyone's schedule? Should one person be responsible for coordinating meetings? Do people have a preference for when meetings are held? Where is a good place to hold meetings? What happens if people are late to a meeting? What happens if a group member misses a meeting? What if he/ she misses several meetings? 5. Consideration Norms: Can people eat at meetings? smoke? What happens if someone is dominating the discussion? How can norms be changed if someone is not comfortable with what is going on in the team? About Goals: Often there is the unstated assumption in student teams that everyone wants to get an "A" in the course, and that should be the team's primary goal. But sometimes, as the semester progresses and everyone gets pressed for time, people have to make decisions about which courses take priority. If this course is a higher priority for some team members than for others, that can create dissension in the group. Talking about this will help to lessen that tension and help you find solutions to the problem. Keep communicating with one another! Also, there may be other goals you want to consider as you work together during the semester. These include: having a high level of camaraderie in the team, learning about how to work together on a team-based project, or learning how to interact with others as a member of a team.

The Responsibilities of the Facilitator The facilitator is not necessarily the group’s leader although he/she can be. It is better to think of the facilitator as the person who keeps the group progressing in the right direction (i.e., toward productivity). Therefore, the facilitator should:


44

• Focus the team on the task (both short term and long term) • Get participation from all team members • Keep the team to its agreed-upon time frame (both short term and long term) • Suggest alternative procedures when the team is stalled • Help team members confront problems • Summarize and clarify the team’s decisions

Hints for Handling Difficult Behavior Just one difficult personality in a group can make the group unproductive and the teamwork experience unpleasant. Here are some suggestions for resolving problems:

How the Person Description What to Do Acts Overly Talkative This person is usually one of Sometimes humor can be used to dis- four types: (a) an "eager beaver"; courage people from dominating the (b) a show-off; (c) very well- discussion; be sure when the person informed and anxious to show stops talking to direct the conversation it; (d) unable to read the responses to another person. of others and use the feedback to monitor his/her own behavior. If the person's behavior can't be changed subtly, one member of the group should speak to the person privately and explain that while his/her enthusiasm is appreciated, it's only fair to the whole group that every person gets an equal amount of air time. Too quiet The quiet person may be: shy, Make a special effort to draw this person bored, tired, unsure of himself/ out: ask for his/her opinion on something; herself, uninvolved in the group. ask him/her something about himself/herself; tell the person you appreciate his/her participation. Argues Is the person critical of ideas, If the person is critical of ideas, use that the group process, or other response to test the work the group is group members? doing--the person may be providing good feedback. If he/she is critical of others, tell him/her how the effect that is having on both the team or individual team members. Be explicit about the fact that his/her behavior is detrimental to the goals of the team.


45

Complains The person may have a pet peeve, Listen to the person's complaint; if it is or may complain for the sake of legitimate, set aside group time to solve complaining. the problem. Point out that part of your work this semester is to learn how to solve problems. Ask the person to join with you to improve whatever is disturb- ing him/her.

Hints for Handling Group Problems Besides problems with individual team members, the team as a whole may run into some difficulties. Here are some suggestions for dealing with teams that aren't functioning properly: Floundering Groups are often not as productive as they could be especially when people are just getting to know one another and how each person works. Drawing up a list of tasks to be accomplished can help. So can saying something like: "What do we need in order to move forward?" or "Let's see if we can all come to an agreement about what we're trying to accomplish." Going Off on Digressions and Tangents Group members may get caught up in chatting about things not central to the work at hand. A little of this can be O.K. because it helps to put people in contact with one another. But if that kind of conversation continues to dominate the group, it can be detrimental to progress. Things to say include: "Can we go back to where we were a few minutes ago and see what we were trying to do?" Making a Decision Too Quickly Sometimes there is one person in the group who is less patient and more action-oriented than other group members. This person may reach a decision more quickly than others and pressure people to move on before it is a good idea to do so. Someone could say: "Are we all ready to make a decision on this?" "What needs to be done on this before we can move ahead?" "Let's check and see where everyone stands on this."


46

Not Making a Decision The best way to make a decision is by consensus with all team members agreeing on the decision together. As you are discussing various ideas, try to be open to what each person is saying. Remember you are trying to come to the best decision for the group as a whole, not for any one person. If the team is having trouble reaching consensus, here are some tools to use: Multivoting--List all the ideas the group has generated. Have each person vote on his/her top four choices. Choose the three or four ideas that have gotten the most votes. Identify similarities and differences among the ideas, then the positive and negative aspects of each. Have each person vote again, this time for his/her top two choices. Tally the votes to see which idea has the most support. Plan A--List all the ideas the group has generated. Each person is given 100 points to allocate among the choices in any way he/she wants to. The alternate that receives the highest number is the team's choice. (NOTE: Use Plan A to reach a quick solution when the decision is not very important. Use Multivoting for more important decisions.) Feuding Between Group Members A conflict--either related to a work project or to something outside of the group--can erupt and impede the group's progress. Usually nothing can be accomplished until the conflict is resolved. If that is the case, the parties need to discuss the problem, using the listening techniques that have been discussed. Ignoring or Ridiculing Others Subgroups or factions can form in groups with one or more people excluded. Sometimes the people who are outside of the "in" group will be the subject to criticism or ridicule. Knowing how to work with people we're not necessarily comfortable with is an ability that will serve you well in the work world. Each group member must make every effort to work with every other group member. The Group Member Who Does Not Do His/Her Share of the Work A group member may be unwilling to cooperate with others, may not complete assigned tasks, or may not come to meetings. You should be talk directly with the person to tell him/her the effect his/her actions are having on the group.


47

Teamwork Activity Note: you should complete this Activity yourself before the Practical. During the Practical all members of the Team will discuss their rankings and attempt to come to a consensus about the three most important characteristics, and the three characteristics that are most disruptive. Listed below are 12 characteristics of work teams. Please go through those characteristics and pick three that you feel are essential for good team performance. Rank them in this way: 1--most important 2--second most important 3--third most important Then go through the remaining items on the list and mark the three that you feel most interfere with team performance. Rank them: 4--most disruptive 5--second most disruptive 6--third most disruptive _____ 1. Competitiveness among members _____ 2. Everyone sticks closely to the point _____ 3. The team avoids conflict _____ 4. Members rotate the leadership position _____ 5. Each member gives and receives feedback _____ 6. A detailed plan is suggested for each team meeting _____ 7. Each team member is assertive _____ 8. Informal sub-teams form _____ 9. Members freely express negative feelings _____ 10. The overall goals of the team are explicitly set _____ 11. Information is freely shared among team members _____ 12. Each person's ideas are taken into consideration and assessed


48

Teamwork Homework This Assignment should be done by you individually, and turned in via turnitin.com. It should have a maximum length of three pages (750 words). There are two objectives to this assignment. One is to give you practice in researching, writing about, and presenting on a real-world problem, situation, or topic that is more complex than what you have done in class so far (the task). The second is to give you an opportunity to observe team dynamics, including your own role in them, so that you can work more productively within a team (the process). To that end, please structure your analysis around the following questions.

Individual Behavior Analysis 1. Describe a specific instance in which you or another team member (but please don’t identify the person by name) did something that furthered progress on either the task or the process. 2. Describe a specific instance in which you or another team member (but please don’t identify the person by name) did something that hindered progress on either the task or the process. 3. Describe one or two ways in which you demonstrated leadership on the team, or ways in which you intend to demonstrate leadership. 4. Describe a specific instance in which you supported another team member who was in the leadership position. 5. Describe one or two ways in which you are working to make yourself a more effective team member.

Team Dynamics Analysis 6. Describe one or two situations in which the team functioned effectively. To what do do you attribute that positive interaction? 7. Describe one or two situations in which the team functioned poorly. To what do you attribute those difficulties?


49

8. How does the team make decisions? 9. How does the team assign tasks? 10. Have specific team members taken on specific roles (again, no names, please)? 11. Are there mechanisms in place for team members to alternate roles? If not, how could that happen? 12. Are there any problems within the team that you want to discuss with your instructor or TA? (All conversations will be strictly confidential.) This Module was written by David M. Harrison, Dept. of Physics, Univ. of Toronto in August 2007. It is largely based on materials developed by Lori Breslow, Director, MIT Teaching and Learning Laboratory and Senior Lecturer, MIT Sloan School of Management. We thank Dr. Breslow for supplying the originals or her materials. Last revision: August 24, 2007

50




• Kinetic, potential, elastic, and total energy • Work

The Activities

Activity 1

Two balls are launched with equal initial speeds along tracks as shown. Friction and air resistance are negligible.

A. Predict which ball reaches the end of its track first. B. You may check your prediction with a Flash animation at:

http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/RacingBalls/RacingBalls.html

A similar situation for skiers instead of balls is at:

http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/RacingSkiers/RacingSkiers.html

Of course, with animations one may program the wrong answer. A video of a real apparatus is at:

http://www.physics.umd.edu/lecdem/services/demos/demosc2/c2-11.mpg

Was your prediction correct?

C. Qualitatively explain the results of the race using conservation of energy D. Qualitatively explain the results of the race using forces and accelerations.


51

Activity 2

In Mechanics Module 3 Activity 11 you may have whirled a ball on a string in a vertical circle and noted that the speed of the ball at the bottom of the circle is greater than the speed at the top. At that time we did not explore why this is so very carefully.

1. Qualitatively explain the difference in the speeds using conservation of energy. 2. Qualitatively explain the difference in the speeds using forces and acceleration.

Activity 3

Three balls are the same height h above the ground and are fired with the same initial speeds v0. Ball A is fired straight up, ball B is fired horizontally, and ball C is fired straight down. Air resistance is negligible.

A. Rank the speeds, from the largest to the smallest, of the three balls when they hit the ground. Explain.

B. Rank the time, from the largest to the smallest, it takes the three balls to hit the ground. Explain.

Activity 4

Joe is standing on the ground, Peter is standing on a 10 m high cliff, and Amanda is at the bottom of a 20 m deep pit, as shown. All three are using coordinate systems with the vertical axis directed up. Joe’s coordinate system has the zero of the vertical axis at ground level. Peter’s coordinate system has the zero of the vertical axis at the height of the cliff. Amanda’s coordinate system has the zero of the vertical axis at the bottom of the pit.


52

A ball of mass m is initially at rest at ground level, Position A above the pit.

A. What is the gravitational potential energy of the ball for Joe, for Peter, and for Amanda?

B. The ball is then raised to the height of the cliff, Position B, and is held at rest. What is the gravitational potential energy of the ball at Position B for Joe, for Peter, and for Amanda?

C. The ball is then released from rest and strikes the ground at the bottom of the pit, Position C. What is the gravitational potential energy of the ball at Position C for Joe, for Peter, and for Amanda?

D. What is the speed of the ball at Position C for Joe, for Peter, and for Amanda?

The following is used in Activities 5 - 8 A horizontal spring has an equilibrium position x0. When the mass m is at position x0 as shown the spring exerts no force on it. When the spring is either stretched or compressed, the position of the mass is x and the force the spring exerts on the mass is:

F = -k(x – x0) 4.1 We assume an ideal spring and negligible air resistance. If the mass is oscillating, the mechanical energy is conserved and equal to:

20

2 )(21

21 xxkmv −+ 4.2

Activity 5

In the above figure we have chosen a coordinate system that points from the right to the left.

A. For values of x > x0 does the force point in the +x or the –x direction? B. For values of 0 < x < x0 does the force point in the +x or the –x direction? C. For values of x < 0 does the force point in the +x or the –x direction?


53

Activity 6

The term 20 )(

21 xxk − is the elastic potential energy of the spring. Explain in your own

words where the 21 term comes from. There are at least two ways that you may wish to

think about this.

1. In a graph of the force versus the distance, what physical quantity is given by the area under the graph?

2. What is ∫ dxF ?

Activity 7

Now the same spring is suspended vertically. You may wish to note that the coordinate axis is now labeled y, while in the introduction above it was labeled x. The position labeled x0 to the right is the same equilibrium position of the spring as before.

A. If the mass is at position x0, the equilibrium position of the spring, draw the free body diagram of the forces acting on the mass.

B. At this position what is the net force acting on the mass? C. If the mass is at some position y0, the net force on the mass

is zero. Draw the free body diagram of the forces acting on the mass.

D. What is the expression for y0 in terms of m, g, k, and x0? E. What is the total vertical force acting on the mass as a

function of k, y, and y0? F. What is the total mechanical energy when the mass is

oscillating?

Activity 8

Suspend the supplied mass from the supplied spring. Place the Motion Sensor under the mass with the transducer pointing up so it tracks the position of the mass. Later we will learn how to describe the distance as a function of time. Here we will begin to explore oscillatory motion and look at the total mechanical energy of the system.


54

Collect distance-time data for the mass when it is vertically oscillating. Recall that the Motion Sensor can only measure distances greater than 0.15 m. This means that for the coordinate system of Activity 7, if the Motion Sensor is at y = 0 the bottom of the mass must always have a value of y > 0.15m.

• Set the vertical position of the mass-spring position so that when the mass is oscillating the minimum distance from the Motion Sensor is close to but greater than 0.15 m. Try to have the mass moving only up and down.

• Set the Motion Sensor for the wide beam. On some units this is indicated by an icon of a person.

• After starting the MotionSensor.vi software, set the sample rate to about 110 samples per second.

• Collect data for just a few oscillations. Here are some tips for analyzing your data:

• It is likely that there will be noise in your values of the distance. These propagate to even greater noise in the displacement, velocity and acceleration. Use the cursors in the main Distance-Sample plot to select a reasonably clean set of data encompassing at least a bit more than half of one complete oscillation. The velocity-time graph is often particularly useful in determining the “region of interest” that you wish to keep. Sometimes the data will be so noisy that it is a good idea to take another set.

• The acceleration plot will be particularly noisy. By default this plot displays all of the values. You can adjust the minimum and maximum values of the plot to show the main features of the data without showing any noisy values by double-clicking on the minimum or maximum value of the vertical axis, putting in a new value, and pressing Return on the keyboard.

Now the Activities:

A. From your data, what is the value of y0? B. When the mass is at y0, is its speed a maximum or a minimum? C. What is the value of the speed when the mass is at y0? Try to account for the noise

in the plot by assigning an error to the value. D. When the mass is at y0, is its acceleration a maximum or a minimum? E. When the mass is at y0 what is the value of its acceleration? Try to account for the

noise by assigning an error to the value. F. From your data what is the maximum amplitude of the oscillation? When the

mass is at this position, is its speed a maximum or a minimum? G. What is the value of the speed when the mass is at its maximum amplitude? What

is the error in this value?


55

H. When the mass is at its maximum amplitude is its acceleration a maximum or a minimum?

I. What is the value and error of the acceleration? What is the value for the spring constant k and the error in this value?

J. When the mass is at y0 what is the total mechanical energy? When the mass is it the maximum amplitude of the oscillation what is the total mechanical energy? Is mechanical energy conserved within errors? Explain.

Activity 9 For an ideal spring-mass system we may write Newton’s 2nd Law as:

makxF =−= (9.1)

If one knows integral calculus then Eqn. 9.1 can be integrated to show that the mechanical energy Emech is conserved, where:

22

21

21 kxmvEmech += (9.2)

In fact, integral calculus was invented by Newton (and independently by Leibniz) to do just this sort of Physics calculation. Here you will use only the calculus of derivatives to show the relation between these equations. Let us assume that the elastic potential energy somehow depends on the spring constant k and how much the string is stretched from its equilibrium position x. Then is has a form:

fe xkd (9.3)

where d, e, and f are numbers. Then the mechanical energy is:

femech xkdmvE += 2

21 (9.4)

You will determine the values of d, e, and f.


56

A. If the mechanical energy is conserved what must be the value of dt

dEmech ? Explain.

B. Calculate dt

dEmech from Eqn. 9.4, and set the result equal to your answer to Part A.

Compare to Eqn. 9.1 to determine the values of d, e, and f. Is your result consistent with Eqn. 9.2?

We see, then, that integration and differentiation are two sides of the same coin. Mathematicians call this the fundamental theorem of calculus.

Activity 10

In Mechanics Module 2 Activity 9 you determined the angle with the horizontal, θ , that the track must make if the cart is to roll down it with constant speed. You will need to use your data from Module 2 Activity 9 for this Activity. As the cart rolls down the track is mechanical energy conserved? If your answer is Yes, explain. If your answer is No, where did the energy go? How much mechanical energy is lost if the cart travels 1 m down the track?

Activity 11

Work is a word that is used both in Physics and in everyday life. Although the meanings of the word in these two contexts are similar, they are not identical. In 5 minutes or less think of as many uses as you can of the word work in your everyday life that do not correspond to the Physics definition, and illustrate each by using it in a complete sentence.

Activity 12

The relation between the s component of a force acting on an object, Fs, and the potential energy U is:

dsdUFs −= (11.1)

Note the minus sign. On the next page are some other equations from the textbook with minus signs:


57

tvvxx )(21

00 +=−

20 2

1 gttvy y −=

1221 onon FF −=

kxF −=

UW Δ−= Are any of these minus signs conceptually the same as the one that appears in Eqn. 11.1? Explain.

Activity 13

As you may already know Leibniz, a contemporary and bitter rival of Newton, believed that the “vis viva” mv2 was the most crucial concept for understanding mechanics. Newton believed that the momentum vm r was the most important quantity.

A. Now, we usually use one-half of the vis viva, ½ mv2, and call it the kinetic energy. In your own words describe how the factor of one-half arises in the definition of kinetic energy.

B. Who do you think was right about the most important quantity: Newton or Leibniz? Why?

Activity 14 A popular toy is part of a hollow rubber sphere that pops when inverted and dropped. It is often called a “popper.” Cock the supplied popper a few times. You may find you need to curl the edges down a bit to get it to stay cocked. You will be using two Force Sensors in parallel to measure the force needed to cock the popper. The program that reads the two Force Sensors and displays their sum is named ForceSensorsDoubled. For more details see the Force Sensor manual at: http://faraday.physics.utoronto.ca/Practicals/Equipment/ForceSensor.pdf


58

You will use the supplied “hook” and piece of Plexiglas with a hole in it as shown. Be sure to center the popper on the hole in the Plexiglas, or the popper can get pulled through the hole when it is cocked.

A. Use the doubled Force Sensors to estimate the force needed to cock the popper. Estimate the total work you need to do to cock it. Where did the energy go?

B. When placed on the floor some of the poppers uncock themselves almost immediately and fly straight up into the air. Others need to be dropped onto the floor from a height of a few centimeters. You do not want to launch it from the tabletop: it will often hit the ceiling. Determine whether your popper launches itself spontaneously or needs to be dropped. Repeat a few times, and note how high above the floor it was dropped from, if applicable, and how high it flies vertically up into the air.

C. Describe all of the energy transformations that occur as you cock the popper and then have it fly up into the air. Is the total energy conserved throughout all of these transformations? Do some rough calculations to justify your answer.

This Student Guide was written by David M. Harrison, Dept. of Physics, Univ. of Toronto, in the Fall of 2008. Last revision: October 30, 2008. Activity 3 is from Randall Knight, Student Workbook that accompanies the 1st edition of Physics for Scientists and Engineers (Pearson Addison-Wesley, 2004), Section 10.3, Activity 10. Activity 14 is similar to Priscilla Laws et al., Workshop Physics Activity Guide (John Wiley, 2004), Unit 11, 11.7.

59




• Angular Momentum • Rotational Motion • Torque • Moment of Inertia • Rotational Dynamics

Activity 1

The figure on the next page shows four cases of objects in uniform circular motion: A, B, C, and D. The motion of all four masses are in the horizontal plane. Recall that for an object of mass m moving with speed v in a circle of radius r the angular momentum L = mvr.

A. Rank in order from the largest to the smallest the angular momentum L of the four cases. Explain your reasoning.

B. Recall that for an object in circular motion with radius r and speed v, the angular

velocity ω is defined as:rv

≡ω . Call the angular velocity of case A ωA. Express

the angular velocity of cases B, C, and D in terms of ωA. C. Now write the angular momentum of the four cases in terms of m, r, and ωA. Does

this form make it easier or more difficult to rank the angular momenta as in Part A? Explain.


60

Activity 2

The four masses of Activity 1 have been connected to a massless frame that is rotating in the horizontal plane about the central pivot point, as shown on the next page. The “spokes” of the frame are rigid.

A. Will the motions of the masses tend to distort the frame in the horizontal plane? B. What is the total angular momentum of the four masses? C. What is the moment of inertia of the combined system of four masses?


61

Activity 3

Whirl the ball on a string in a horizontal circle, being careful not to hit anybody with it. Try to maintain the ball at constant speed. You will find it useful to run the string through the supplied drinking straw and hold the straw in your hand, keeping the string taut with your other hand. If you make the axis of rotation directly over your head you will be much less likely to hit yourself in the head with the ball.

A. Was the linear momentum vmp rr= of the ball conserved as the ball moved in

uniform circular motion? Explain. B. Was the kinetic energy ½ mv2 of the ball conserved as the ball moved in uniform

circular motion? C. Was the angular momentum of the ball, L = mvr = mr2ω, conserved as the ball

moved in uniform circular motion? D. Whirl the ball again in a horizontal circle. Reduce the radius r of the circle to

about ½ r by pulling on the string hanging below your hand. What happened to the speed of the ball? Did you notice anything about the pull on your hand by the string? If yes, what?

E. Was the kinetic energy conserved as the radius of the circle was being reduced? Explain.

F. Was the angular momentum conserved as the radius of the circle was being reduced? Explain.

G. If the speed of the ball was v when the radius of the circle was r, what was the speed when the radius of the circle was ½ r? If the magnitude of the angular velocity of the ball was ω when the radius of the circle was r, what was its magnitude when the radius of the circle was ½ r?


62

Activity 4

A Flash animation illustrating some of the points of this Activity and the next one is available at: http://www.upscale.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/RollingDisc/RollingDisc.html. A full screen version which is easier for a group of people to see is at: http://www.upscale.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/RollingDisc/RollingDisc.swf A bicycle wheel of radius R rolls to the right without slipping. The velocity of the axle of the wheel relative to an observer standing on the road is vr . At the moment shown in the figure Point A is in contact with the road, and Point B is at the top of the wheel.

A. At the moment shown what is the instantaneous velocity of Point A relative to an observer standing on the road?

B. For the person riding on the bicycle, about what point is the wheel rotating? What are the velocities of Points A and B and the axel at the moment shown in the figure? What is the angular velocity ωB of the wheel?

C. For an observer standing on the road, about what point is the wheel rotating? For this observer what is the angular velocity ωR of the wheel? What is the instantaneous velocity of Point B for this observer at the moment shown in the figure?


63

Activity 5

To keep them from slipping off the tracks, train and streetcar wheels have a flange, as shown. The radius of the part of the wheel in contact with the rail is R1, and the radius of the flange is R2. The wheel is rolling to the right without slipping. The velocity of the axle of the wheel for an observer who is stationary relative to the track is vr . At the moment shown in the figure Point A is in contact with the track and Point C is at the bottom of the flange. At the moment shown what is the velocity of Point C for an observer who is stationary relative to the track? Does this answer surprise you? Explain.

Activity 6

A. Here is a figure of a yoyo that is in free fall: the

string is not attached to anything and is not shown in the figure. Draw a free body diagram of the forces acting on the yoyo. Assume that air resistance in negligible.


64

B. Here is a cross section of a yoyo that is falling with the end of the string fixed to a support. In Part A, you could reasonably assume that the yoyo is a point particle. The free body diagram for this case must treat the yoyo as an extended body, and where the forces are exerted on it is important. Draw an extended free body diagram of the forces acting on the yoyo.

C. If both yoyos are released at the same time from the same height do they both fall at the same rate? Which moves fastest? Confirm your prediction by dropping the yoyo with and without you holding the string; catch the yoyo of Part A so it doesn’t get damaged by colliding with the floor or tabletop.

D. Explain the results of Part C qualitatively using Newton’s Laws.

E. For the yoyo of Part B, can the force exerted on the yoyo by the string ever do work on it? Explain the result of Part C qualitatively using conservation of energy.

Activity 7

A. A uniform meter stick of

length L = 1.0 m has a 0.20 kg mass suspended by a string from its left side, and rests on a pivot that is 0.25 m from the left side. If the meter stick is balanced what is its mass?

B. What is the force exerted on the meter stick by the pivot? C. Evaluate the total torque exerted on the meter stick about the pivot point. Repeat

for the total torque evaluated about the left side of the meter stick, where the 0.20 kg mass is attached. Repeat for the total torque evaluated about the far right side of the meter stick. Can you generalize these results to a statement about evaluating the torque for a body that is in equilibrium? Explain.


65

D. The meter stick has been cut in half. The right half is attached to a massless frame that is free to rotate about the pivot. The 0.20 kg mass is suspended from the left side of the frame 0.25 m from the pivot. Are the half meter stick, frame and mass balanced?. What is the mass of the half meter stick? Is your answer consistent with that mass of the full meter stick of Part A? Explain.

E. The full meter stick and 0.20 kg mass is tilted, held at rest, and gently released. What will be its motion? Explain.

Activity 8

A. A yoyo sits on the tabletop, and is gently pulled to the right by the horizontal string, which is wound about the axel as shown in the cross-section view. The pull is gentle enough that the yoyo does not slip. Predict the motion of the yoyo. Using the supplied yoyo, confirm your prediction.


66

B. A yoyo sits on the tabletop, and is gently pulled to the right by the horizontal string, which is wound about the axel as shown in the cross-section view: note that now the string is attached to the bottom of the axel. The pull is gentle enough that the yoyo does not slip. Predict the motion of the yoyo. Using the supplied yoyo, confirm your prediction.

C. Explain the results of Parts A and B. Assume that the radius of the axel of the yoyo is r, and the radius of the yoyo itself is R. What are the total torques acting on the yoyo about its axis of rotation for Parts A and B?

D. Suppose that in the arrangement of Part B the string is not horizontal, but instead pulls the yoyo to the right and up. As the angle of the string is increased predict what will happen. Test your prediction. Can you explain?

Activity 9

A dumbbell consists of two masses m and 2m separated by a distance d by a massless rod. The dumbbell rests on a frictionless horizontal table, and a force F is pulling mass m to the right. In the figure we are looking down at the dumbbell from above

A. Are there one or more forces that can be applied to the dumbbell that will cause it to move with only translational motion, without any rotation? If yes what is/are those forces, magnitude, direction, and applied to what part of the dumbbell? If not, explain.

B. In addition to the force shown in the figure are there one or more forces that can be applied to the dumbbell that will cause it to move with only constant translational speed, without any rotation or acceleration? If yes what is/are those forces, magnitude, direction, and applied to what part of the dumbbell? If not, explain.


67

Activity 10

Two uniform cylinders are made of the same material, have the same thickness, and are rotating about their axes of symmetry. Cylinder B has twice the radius of Cylinder A. Recall that the moment of inertia of a cylinder of mass M and radius R that is rotating about its axis of symmetry is

2

21 MRI = .

a) If the two cylinders are to have the same angular momentum, what must be the

relation between their angular speeds ωA and ωB? b) If the two cylinders are to have the same rotational kinetic energy, what must be

the relation between their angular speeds ωA and ωB?

Activity 11

You are driving a screw into a piece of wood. In addition to the screw’s rotation it moves down into the wood. What is the relation between the direction of the angular momentum vector of the turning screw and the direction it is moving into the wood?


68

Activity 12

A gyroscope is spinning as shown.

A. What is the direction of the angular momentum vector of the gyroscope?

B. What is the direction of the torque being exerted on the gyroscope? C. From your answer to Part B what can you conclude about the magnitude of the

vertical component of the angular momentum as the motion proceeds? D. If the gyroscope were not spinning, what would the torque you found in Part B

cause the gyroscope to do? E. If the spinning gyroscope did what you predicted in Part D what would happen to

the vertical component of the angular momentum? Is this possible? F. Now predict the direction of precession of the gyroscope. Check your prediction

with the supplied gyroscope. G. Now the gyroscope is rotating in the

opposite direction. Repeat Parts A – F. This guide was written in May 2008 by David M. Harrison, Dept. of Physics, Univ. of Toronto Last revision: November 6, 2008.

69

PHY131 Practicals Manual Oscillations Module

Oscillations Module Student Guide


Periodic and simple harmonic motion. Simple harmonic motion as one component of uniform circular motion. Pendulums when the small angle approximation is not valid. Double pendulums, spring-mass systems with an oscillating support point, and

chaotic systems.

The Activities

Activity 1

Here is the URL of a Flash animation of two systems executing harmonic motion. http://www.upscale.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/SHM/TwoSHM.html The above link is to a fixed size animation which works nicely if only one person it viewing it. For use in the Practical itself a version which can be resized to be larger so that the entire Team can see it is better. Here is a link to such a version: http://www.upscale.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/SHM/TwoSHM.swf Open the animation. In your own words describe which characteristics of these two systems are the same. In your own words describe what is different for these two systems.

Activity 2

Here is the URL of a Flash animation comparing uniform circular motion to simple harmonic motion: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/Circular2SHM/Circular2SHM.html The above link is to a fixed size animation which works nicely if only one person it viewing it. For use in the Practical itself a version which can be resized to be larger so that the entire Team can see it is better. Here is a link to such a version: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/ClassMechanics/Circular2SHM/Circular2SHM.swf


70

Open and run the animation.

A. For an object in uniform circular motion, the angle θ as a function of time changes according to: θ = ω t where ω is called the angular velocity. What are the units of ω?

B. For an object in uniform circular motion, the y coordinate of a point on the object changes as a function of time according to: y = r sin(ω t) What is the meaning of r? What are its units?

C. An object executing Simple Harmonic Motion is described by: y = r sin(ω t) Note that this is the same equation as in Part B. However in this case ω is called the angular frequency. In this case what are the units of ω? In your own words explain why two somewhat different names are used for the same symbol ω. For Simple Harmonic Motion what name is usually used for r? What are the units of r?

Activity 3

A. Here is a plot of the position versus

time for a particle. Does the motion appear to be periodic? Does the motion appear to be Simple Harmonic? What is the period of the motion? What is the frequency of the motion?


71

B. Here is another position versus time plot. For values of the time between 0 and 8 s it is identical to Part A. Does this motion appear to be periodic? For the particle of Part A, if you only have data for times between 0 and 8 s can you tell the difference between that motion and the motion shown here?

C. Here is another plot of the position versus time for a particle. Does the motion appear to be periodic? Does the motion appear to be Simple Harmonic? What is the period of the motion? What is the frequency of the motion? What is the angular frequency of the motion?

D. Does it make sense to talk about the angular frequency of the motion of Part A? Explain.

Activity 4

Mount the supplied spring on the support and hang the supplied mass from it. Position the Motion Sensor under the mass and pointing up at it, as shown. You will connect the Motion Sensor to the U of T DAQ Device. You may have used this same setup in Mechanics Module 5 Activity 8. Recall that the Motion Sensor can only measure distances greater than 0.15 m. This means that for the given coordinate system, if the Motion Sensor is at x = 0 the bottom of the mass must always have a value of x > 0.15m.


72

Set the vertical position of the mass-spring position so that when the mass is oscillating the minimum distance from the Motion Sensor is close to but greater than 0.15 m. Try to have the mass moving only up and down.

Set the Motion Sensor for the wide beam. On some units this is indicated by an icon of a person.

After starting the MotionSensor.vi software, set the sample rate to about 110 samples per second.

Collect data for a few oscillations. Now the Activities:

Does the motion appear to be Simple Harmonic? What is the period, frequency, angular frequency, and amplitude of the motion?

What physical characteristics must the spring-mass system have for the motion to be truly Simple Harmonic and not just approximately so? Do you think this spring-mass system has those characteristics?

Activity 5

In Activity 4 a Motion Sensor is used to measure the position versus time for a mass oscillating on a spring. We took about 110 samples every second and, depending on the particular spring and mass used the period of the oscillation was about 0.5 s. Imagine we have an ideal spring-mass system oscillating with maximum amplitude = 1 m and angular frequency ω = 6 s-1. The motion is:

)6sin()sin( ttAy == ω

A. What is the period of the oscillation? B. Imagine we use a Motion Sensor to measure the position once per second and take

the first measurement at t = 0 s. Remember that in the above formula the argument to the sine function is in radians. What will be the measured values of y for the first few measurements? Is your results reasonable? Explain. You may find it helpful to draw a rough sketch of y versus t for a few periods of the oscillation and locate on the graph the points where you would have measured the values of y.


73

Activity 6

The figure shows the potential energy of a particle oscillating on a spring. Note that the horizontal axis has units of cm.

What is the spring’s equilibrium length? What is the value of the spring constant k?

The turning points of the particle are at 10 cm and 30 cm. What is the maximum amplitude of the particle’s motion? What is the particle’s maximum kinetic energy?

Sketch a graph of the particle’s kinetic energy as a function of x. What is the shape of the sketch?

The particle has a mass of 2.0 kg. What is the particle’s maximum speed? Sketch a graph of the particle’s speed as a function of x. What is the shape of the sketch?

If the total energy of the particle is tripled, what will be the value of the maximum speed of the particle?

If the total energy of the particle of Parts A - D is tripled, what will be its maximum amplitude?

Activity 7

A block of mass M is attached to a spring of spring constant k, and oscillates back and forth with amplitude A on a frictionless surface. At the moment shown the block is at its maximum amplitude and a lump of putty of mass m is dropped from a very small height and sticks to the block. The mass of the spring is negligible. What are the new amplitude and frequency of the block plus putty system?


74

Activity 8

A vertical massless spring with spring constant k has a platform of mass M fixed to it. A mass m is sitting on top of the platform, but is not fixed to it. The two masses are oscillating vertically with amplitude A. There is a vertical position, the equilibrium point, where the total force acting on the two masses as a single system, their weight plus the force exerted by the spring, is zero.

Imagine that the oscillation is so extreme that the mass m just loses contact with the platform. Where is this going to occur and why?

What is the value of A when for which the mass m just loses contact with the platform?

So far, except for Activities 3A and 3B, all the motions we have investigated were at least approximately Simple Harmonic. In Activities 9 and 10 we will explore two pendulums for which the deviations from Simple Harmonic are not negligible. For both we concentrate on the how the period of the pendulum varies with the maximum amplitude. Newton’s Laws for a simple pendulum of length L gives us a differential equation:

0sin2

2

=+ θθLg

dtd (1)

In classes and the textbook, we almost always restrict ourselves to the case where the angle θ is always small, so that we can approximate that:

θθ ≈sin (2) where θ is in radians. Then we can approximate Eqn (1) as:

02

2

=+ θθLg

dtd (3)


75

Except for the differences in symbols, this is the same equation as for a spring mass system:

02

2

=+ xmk

dtxd (4)

You may have studied how to do an approximation of this system in the Numerical Approximation Module. The solution to Eqn (3) is simple harmonic motion with a maximum amplitude θmax and period:

gLT π20 = (5)

The reason why we usually approximate Eqn (1) with Eqn (3) is that an exact analytical solution of Eqn (1) does not exist! With considerable effort and sophisticated mathematics, from Eqn (1) one can determine that the period of the pendulum with a maximum amplitude of θmax is given by:

⎮⌡

⌠−

=max

0 max0 coscos

2θ

θθθ

πdTT (6)

This is not terribly helpful, since the integral involves elliptical integrals which are not analytically solvable either, so one must either numerically approximate to find the value of T or look up the integral in some table. These sorts of integrals are so common that considerable effort has been expended to develop computer algorithms that are fast and accurate. Table 1 gives some values of T/T0.2

Table 1

θmax (rads) T/T0 0.01000 1.000010.10000 1.00063

π/4 1.03997π/2 1.18034

2.000000 1.328903.000000 2.57123

π ∞ There are a few ways to avoid becoming mired in all the mathematics: 2 These values were calculated with Mathematica.


76

1. Use numerical approximation. 2. Build and measure a real pendulum. 3. Do even more mathematics to simplify Eqn (6) into an approximate form that can

be solved with a simple calculator. In Activity 9 we will use numerical approximation, and Activity 10 we will use a real pendulum. If you wish to see the sort of gyrations necessary to get a formula that can be solved with a calculator, see for example F.M.S. Lima and P. Arun, “An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime”, American Journal of Physics 74(10), 892 – 895 (2006), http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000074000010000892000001&idtype=cvips&gifs=yes.

Activity 9

Here are some values of the period of a simple pendulum for a length L = 1.0000 m taking g = 9.8000 m/s2 using Eqn. 5 and the values of Table 1.

Table 2

θmax (rads) T (s) 0.01000 2.007100.10000 2.00835

π/4 2.08732π/2 2.36905

2.000000 2.667233.000000 5.16070

π ∞

A listing of the code for Pendulum.py appears in Appendix 1. We assume that the mass is connected to the support by a massless rigid rod. The algorithm used to approximate this system is very similar to the one used for the spring-mass system investigated in the Numerical Approximation Module. Examine the code and describe in your own words how the algorithm approximates the motion of the pendulum.

The code also estimates the period of the oscillation. Describe in your own words how the estimation is done.

The “master” copy of Pendulum.py is located at: Feynman:Public/Modules/Oscillations


77

Copy the file to your Team’s area on the server. Start the IDLE for VPython program and open the copy of Pendulum.py. Use Run / Run Module or press the F5 key on your keyboard to start the animation. How does the estimated value of the period compare to the value in Table 1? Can you think of a better way to determine the period of the simulation?

Modify the code so that when the animation starts it prints the maximum amplitude and the timestep of the approximation. Test your changes to make sure that they work. You will wish to know that when you change the code and run the animation VPython will over-write the file with the new version. A tip: when you start the animation, two windows are opened, a Python Shell which shows the results of all print statements in the code and a window of the animation. To stop the animation close the animation window, but leave the Python Shell window open. This will be useful for Part E.

Does the numerical approximation agree with the values of Table 2? How does the approximation do for different values of the timestep? You may wish to print the window of the data for the estimated periods for various maximum amplitudes and timesteps and staple it into your lab book.

Activity 10

For a real pendulum, things are not quite so simple. In the figure we show a real pendulum. The distance between the pivot point and the centre of mass is L, the moment of inertia of the object about the pivot is I, and the total mass of the object is m. Often this is called a physical pendulum. Using the usual convention that positive torques cause counter-clockwise rotations, the torque exerted on the pendulum is:

θτ sinmgL−= (7) Newton’s 2nd Law for rotational motion is:

τθαIdt

d 12

2

=≡ (8)

Thus, the equation of motion is:


78

0sin2

2

=+ θθI

mgLdtd (9)

Note that except for differences in the symbols this has the same mathematical structure as Eqn (1) for a simple pendulum. Therefore within symbol changes the solution is also the same. In particular, in the small angle approximation the motion is simple harmonic and the period of oscillation is:

mgLIT π20 = (10)

Similarly, for oscillations where the maximum amplitude is not small Eqn (6) is also true:

⎮⌡

⌠−

=max

0 max0 coscos

2θ

θθθ

πdTT (11)

Therefore the ratios of T/T0 from Table 1 are also true for this case. Here you will side-step trying to solve Eqn (11) by taking data on how the period of a real physical pendulum varies with its maximum amplitude. You will use a Pasco Rotary Motion Sensor, the U of T Data Acquisition Device (DAQ), and the RMS program. The RMS program is based on the SignalExpress software platform from National Instruments. The Rotary Motion Sensor is mounted on a support, and has a rod with a mass on it mounted on it. The rod and mass will be the physical pendulum you will study. Also part of the rotating system are the plastic disc on which the rod is mounted and the axel connected to the Rotary Motion Sensor.

Setup There are two phone plugs on cables connected to the Rotary Motion Sensor. These are plugged into the corresponding terminals on the Data Acquisition Device. On the left side of the Device are two pairs of terminals labeled Digital Channels; each pair has one labeled with a yellow circle and the other with a black circle. The corresponding yellow and black plugs from the sensor are plugged into pair labeled 0, yellow to yellow and black to black. When you start RMS the position of the Rotary Motion Sensor defines the angle to be zero. Make sure that the pendulum is stationary and start RMS.


79

Occasionally the hardware and software gets confused about the resolution of the measurement.

1. Rotate the pendulum by one full rotation. The angle on the graph should read 6.28 radians (= 360º) either a positive or negative number.

2. If the angle is not correct, disconnect the Rotary Motion Sensor from the DAQ the plug it back in.

3. Stop and restart the RMS program and check that one full rotation reads 6.28 radians.

4. Stop RMS.

Trial Run

It is a good idea to do a trial run first. You should do a run with a maximum amplitude radian1max ≈θ .

Make sure the pendulum is stationary. Start RMS. Rotate the pendulum to some about 1 radian. You can read the angle on the graph. Release the pendulum. The graph will show the angle of the pendulum as a function

of time. After about 10 oscillations stop RMS. Right click on the graph. Choose Visible Items / Cursors. Drag the solid cursor to one of the early oscillations. The horizontal line of the cursor

will track the data. Drag the dotted cursor to one of the later oscillations. At the bottom of the graph the Dx field is the time between the two cursors. y1 and

y2 are the angles of the first and second cursors respectively, in radians, and Dy is the change in the angle.

Thus you may calculate the value of the period and the maximum amplitude.

Note: due to a mis-feature of SignalExpress, sometimes when you take another data set it is difficult to use the cursors. To fix, turn the cursors off and then back on using Visible Items / Cursors.

There are some issues that should be considered:

A. The time resolution of the software is 0.05 s. You can confirm this by moving one of the cursors the minimum possible amount and seeing how much the value of the time changes. Considering that the measured value of the time when the pendulum is at maximum amplitude is unlikely to be the exact value when it actually occurred, a reasonable value for the error in the measurement of the time Δt is perhaps ± 0.03 s. Is this a reasonable estimate? If yes, explain why in your own words. If no, what is a more reasonable value?


80

B. If you position the cursors at successive maximum values of the amplitude, then the time between them is the period T of oscillation. What is a reasonable estimate of the uncertainty in the period, ΔT?

C. Say you position the cursors for n oscillations. Call the final value of the time tf and the initial value ti. The software shows the value of tf - ti as Dx at the bottom of the window. The value of period is T = Dx/n. Now what is the uncertainty in the period, ΔT? Is this a better way to determine the period?

D. Since this is not a perfect frictionless system, the maximum amplitude decreases with time. You can see this in your data. Thus the value of θmax is not well defined. What is a reasonable estimate of the error in its value? Why? How does the value depend on the number of oscillations between the cursors?

E. Considering the results of Part C and Part D, how many oscillations should you use to determine the values of the period for a given maximum amplitude? Note that there is a trade-off: increasing the number of oscillations n reduces the error in the period but increases the error in the maximum amplitude. Explain your choice.

Data Collection and Analysis Collect data for the period for a number of values of the maximum amplitude. Use CreateDataSet to create a dataset of values for the period and maximum amplitude, including their errors, and save it into your Team’s area. You will want the values of the maximum amplitude to be the Independent (x) Variable, and the values of the period be the Dependent (y) Variable. You can then use ViewDataSet to view your data. You may wish to print the window and staple it into your lab book. Does the data look reasonable? Does it appear to be consistent with the values of Table 1?

Parameterising the Data You have determined the period of the pendulum for measured values of the maximum amplitude. Here we explore how to use that data to determine the value of the period for maximum amplitudes that have not been measured. The relation between the period and the maximum amplitude is a complicated function f() involving elliptical integrals:

⎮⌡

⌠−

==

max

0max

0max coscos2)(

θ

θθθ

πθ dTfT (12)


81

However, it might be reasonable to approximate the function as a polynomial:

...)4()3()2()1()0( 4max

3max

2maxmax +++++= θθθθ aaaaaT (13)

Can you eliminate the odd coefficients a(1), a(3), etc. from the series using a physical

argument? If yes, what argument can you use? Use the PolynomialFit program to find the best fit the data. Do not include any terms

that are physically unreasonable. You will find that adding some terms to the polynomial will not improve the quality of the fit: your goal is to find the minimum number of terms in the polynomial that provides a good fit to the data. Any fitted parameters a(i) that are zero within errors should not be included in the fit.

From the best fit, what is the value of T0? How well does the best fit do in duplicating the values of Table 1? Does it do a better

job for small angles than for large ones? Activities 9 and 10 were concerned with pendulums whose motion was approximately but not exactly Simple Harmonic. In Activities 11 and 12 we will investigate some periodic systems whose motion is not even close to being Simple Harmonic. Both Activities are only brief looks at the two systems, and are just for your interest. Both systems that are investigated are chaotic. Here are some characteristics that all chaotic systems have:

No analytic formula can even approximate the motion. The motion will never repeat. Ever! If two identical periodic systems are started with almost identical positions and

speeds, soon their motions will be radically different from each other. Chaotic systems are deterministic. If they start with exactly identical initial conditions

their subsequent behavior will be exactly the same. Characteristic 3 is sometimes called the butterfly effect. This is because if history were a chaotic system then the outcome of World War II could have been determined by whether or not a butterfly landed on a particular flower in the Himalaya Mountains in 1848. You may learn more about chaotic systems from: http://www.upscale.utoronto.ca/PVB/Harrison/Chaos/Chaos.html


82

Activity 11

Our first example of a non-harmonic pendulum is the double pendulum. It is an example of a chaotic system. Start the IDLE for VPython program. Use File / Open … to open the doublependulum.py program in the examples directory; this directory is the default one that is opened when IDLE is first started. Run the program. You are welcome to look at the code for this program, but will wish to know that it is written in terms of a sophisticated form of Newton’s Laws called a Lagrangian formulation. You may investigate the double pendulum further with a Java applet by Peter Selinger at Dalhousie University, Halifax Nova Scotia: http://www.mscs.dal.ca/~selinger/lagrange/doublependulum.html From the trace of the trajectory you can see characteristic #2 above illustrated. By clicking on the Restart button you can see characteristic #4 demonstrated.

Activity 12

In Activity 4 you may have used a Motion Sensor to track the position of a mass oscillating up and down on a spring. If the support of the spring is oscillating up and down, for some frequencies and amplitudes of oscillation this system too is chaotic. FEED ME. This Student Guide was written by David M. Harrison, Dept. of Physics, Univ. of Toronto in May 2008 Last revision: November 8, 2008. Activity 3.A is based on Activity 14.1 Part 3 of Randall D. Knight, Student Workbook with Modern Physics (Pearson Addison-Wesley, 2008). Activity 6 is based on Activities 1.43 Parts 10, 11, and 12 of Knight’s Student Workbook. Activities 7 are 8 are from David Harrison and William Ellis, Student Activity Workbook, 3rd ed. (Norton, 2008), Activities 15.8, 15.9.


83

Appendix 1 – Listing of Pendulum.py # Solve the pendulum using numerical approximation # Copyright (c) 2008 David M. Harrison # The next line is an internal revision control id: # $Date: 2008/05/10 10:33:45 $, $Revision: 1.1 $ # Import the visual library from visual import * # The initial angle in radians. theta = pi/2.0 # The initial angular velocity omega = 0 # Set g and the length of the pendulum g = 9.80 L = 1.00 # These four lines control the size of the window of # the animation and the scale. The details of these lines # are not important for our purposes. scene.autoscale = 0 scene.height = 600 scene.width = 600 scene.range = vector(2.0,2.0,2.0) # # Now we build the pendulum which we will animate. # # The support for the pendulum support = cylinder( pos = (0, 0, -0.5), axis = (0,0,1), radius = 0.02) # The "frame" construct groups two or more objects into a single one. # Here we group the cylinder and the sphere into a single object # which is the pendulum. pendulum = frame() cylinder(frame=pendulum, pos=(0,0,0), radius=0.01, length=1, color=color.cyan) sphere(frame=pendulum, pos=(1,0,0), radius=0.1, color=color.red) # Position the pendulum. pendulum.pos = (0,0,0) # Rotate the pendulum about the z axis. Note that VPython measures # angles with respect to the x (horizontal) axis. We are measuring # angles with respect to the vertical (-y axis) so we subtract # pi/2.0 radians from the angle. pendulum.rotate(axis = (0,0,1), angle = theta - pi/2.0) # The time t = 0.


84

# Below we will want to store the old value of the time. # Set it the "impossible" value of -1 initially. t_old = -1. # The time step dt = 0.0005 # The value "1" is equivalent to true. So this causes the while # loop to run forever. while 1: # Set the rate of the animation in frames per second rate(1/dt) # The angular acceleration, i.e. the second derivative of the # angle with respect to time. alpha = -(g/L) * sin(theta) # The new value of the angular velocity omega = omega + alpha * dt # The change in the angle of the pendulum d_theta = omega * dt # A rough and ready way to estimate the period of the oscillation. # It the angle is positive and adding d_theta to it will make # it negative, then it is going through the vertical # from right to left. if(theta > 0 and theta + d_theta < 0) : # If t_old is > 0, then this is not the first cycle of # the oscillation. The difference between t and t_old # is the period within the resolution of the time step dt # and rounding errors. Print the period. if(t_old > 0): print "Estimated Period =", t - t_old, "s" # Store the current value of the time in t_old t_old = t # Rotate the pendulum about the z axis by the change in the angle pendulum.rotate(axis = (0,0,1), angle = d_theta) # Update the value of the angle theta = theta + d_theta # Update the time t = t + dt

85

PHY131 Practicals Manual Numerical Approximation Module

Numerical Approximation Module Student Guide


• Introduction to the Python programming language. • Numerical approximation as an alternative to analytic solutions.

Introducing Python Here we briefly introduce the Python language and some of the programming constructs that will be used in the main part of this Module. The Python programming language is free and open source, with a huge community of developers. Although it is an ideal first language to learn, you may wish to know that it is not a “toy”. It is used extensively by Google, NASA, the Large Hadron Collider just being lit up in Switzerland, Youtube, Air Canada, and many more. Traditionally the first computer program simply prints hello, world. Here is a complete Python program that does this:

print "hello, world" Here is another complete program that also prints hello, world:

what = "world" print "hello,", what

The first line of this program assigns world to a variable named what. The next line then prints hello, followed by whatever the variable named what is set to, world in this case. The Python interpreter executes the lines of this “program” in order. Today we will wish to have Python execute some lines of the program over and over again. We will use a while loop to do this. This loop has the form:

while something_is_true: execute this line of the program then execute this line of the program then execute this next line of the program

After executing the third line after the while statement, it goes back to the while statement: if something is still true then it executes the following lines again, and so on.


86

We have prepared a program named LoopDemo.py which demonstrates this loop. Here is a listing of the program.

Listing of LoopDemo.py

# All lines like this one that begin with a "#" # are comments. All other non-blank lines are # program statements. # Set a variable named "x" to a value of 0 x = 0 while x < 3: print x

# Increase the value of x by one. x = x + 1 # End of the while loop. Go back to # the while statement again.

You may wish to know that the lines following the while statement must be indented as shown. Start the IDLE for VPython program. Use File / Open … to open the file LoopDemo.py which is located in Feynman:Public/Modules/NumerApprox folder. Predict what will happen when this program is run. Check your prediction by running the program: use Run / Run Module or press the F5 key on your keyboard. Sometimes we wish to use a while statement to have the program execute the same lines over and over until it is manually stopped. The LoopDemo2.py file in the same directory does exactly this. A listing of this program is in Appendix 1. Predict what will happen when this program is run. Check your prediction by running it. Also in the Feynman:Public/Modules/NumerApprox folder is the file LoopDemo3.py, and a code listing is in Appendix 2. It differs from LoopDemo2.py in two ways:

1. The first print t statement is removed. 2. Inside the while loop the two statements that increment the value of the time

and prints the value of the time are reversed.


87

Predict what will happen when this version is run. Check your prediction by opening the file and running it.

The Spring-Mass System and Numerical Approximation For a mass m on a spring with spring constant k Newton’s Second Law is:

2

2

dtxdmkx

maF

=−

= (1)

This is a second-order differential equation, and if one knows enough calculus one can solve it to get:

)sin( tamplx ω= (2)

where:

mk

=ω

But if one doesn’t know enough calculus or just doesn’t want to bother with a differential equation, a moderately powerful computer provides a nice alternative. The basic idea is that we will start with the mass at some known position and calculate its acceleration, how fast it is moving and where it will be small timestep Δt later, and keep doing this over and over again. Here is how one may do this numerical approximation:

1. From the mass’ current position x we can calculate the acceleration a of the mass:

xmka −=

2. If the speed of the mass is v, then calculate a new speed vnew = v + a Δt. 3. If the position of the mass is x, calculate a new position xnew = x + vnew Δt. 4. Go back to Step 1 and repeat.

Of course, this method is just an approximation. However given a sufficiently powerful computer to do the calculations we can make the approximation as close to correct as we wish by making the timestep Δt sufficiently small. We have prepared a Visual Python (VPython3) animation which both uses Eqn. 2 and implements the numerical approximation described above. 3 VPython is free, open source, and available for Windoze, Mac, Linux and UNIX from http://www.vpython.org/.


88

The Activity

A. Open the IDLE for VPython program. Use File / Open … to open the file SHM.py which is located in Feynman:Public/Modules/NumerApprox. Use Run / Run Module or press the F5 key on your keyboard to start the animation. The upper yellow sphere uses Eqn. 2, and the lower green sphere uses the numerical approximation. Can you see any differences between the motions of the two spheres? For fun you may wish to know that:

• Holding down the right mouse button and moving the mouse allows you to rotate the view of the animation.

• Holding down both mouse buttons and moving the mouse up or down allows you to zoom in and out on the animation.

B. For your convenience a listing of the SHM.py code is included in Appendix 3 of this document. In the Feynman:Public/Modules/NumerApprox folder the file CodeBig.pdf also lists the code using big fonts; you may wish to print this file and place the pages on the whiteboard using small magnets. Including empty lines there are 90 lines in the file. How many of them are program statements?

C. Some lines of the code are used only for the animation of the yellow ball; some lines are only for the animation of the green ball; some lines are shared for the animations of both balls; still other lines are commands to control the animation speed, set up the calculation loop, or set the “stage” for the animation. Circle or use a highlighter on all the lines in the code that are used only for the animation of the yellow sphere and label them with Y; if a yellow highlighter is available it would be a good choice for this.

D. Preferably using a different color pen or highlighter, circle or highlight all the lines in the code that are used for the animation of both spheres and label them with B.

E. Describe in your own words how the program animates the motion of the yellow ball.

F. From the parameter values set in the code calculate the period T of the oscillation. Does your calculated value match the actual period you see in the animations?

G. About 60% down the code listing the maximum amplitude of the motion ampl is calculated. Did you circle this in Part C? If not, should you have? Is the calculation correct? (Hint: think about conservation of energy.)

H. Preferably using a third color pen or highlighter circle or highlight all the lines in the code that are used only for the animation of the green sphere and label them with G; a green highlighter would be ideal if available. Circle or highlight all the lines that control the animation speed, set up the calculation loop, or set the “stage” for the animation, and label them with C; a fourth color pen or highlighter would be nice if possible. Follow the code for all the lines that are used for the


89

animation of the green sphere. Does it surprise you that nowhere in these lines of code does a trig function appear? Explain.

I. In the code for the yellow ball, the value of the time is incremented and then the new position of the ball is calculated. Is this correct? What if those two lines were reversed?

At the end of this Module, you will want to staple your “de-constructed” code into your lab book.

For the Keen Here are some things you may wish to do. They are not intended to be part of the Activity of this Practical, but instead some things you may wish to explore on your own. Some systems, particularly chaotic ones, are not analytically solvable: there is no equation that describes the motion. For such systems numerical approximation is the only way that they may be studied. When VPython first starts, using the File / Open … command lists the examples that are shipped with the software. The doublependulum.py program in that directory is an example of a chaotic system which is not analytically solvable but here is solved by numerical approximation. The physics behind this animation is fairly formidable, but the basic idea is the same as the SHM.py code you used here. There are many other interesting examples that are shipped with the software. You may also save a copy of the SHM.py file and try modifying it by changing some of the parameters set in the code. You will want to know that by default every time you run the program VPython first saves the code into the file. Thus you may wish to consider working on a copy of the master file, named perhaps SHM_work.py. One simple change you could make to SHM.py involves efficiency. As written determining the yellow sphere’s position involves calculating the angular velocity sqrt(k/mass) for every iteration of the loop. Calculating the value once before entering the loop and then using the calculated value would mean that the program has to perform many less calculations.


90

Appendix 1 – LoopDemo2.py Code Listing # All lines like this one that begin with a "#" are # comments. All other non-blank lines are program # statements. # Import the visual library. from visual import * # Set the time t = 0 # Set the timestep dt = 1 # Print the current value of the time print t # The next line causes the indented lines that follow # it to be repeatedly executed in the loop. The construct: # 1==1 # means "is one is equal to one?" which is always true. # Thus double equal signs like this mean something different # than a single equal sign, such as is used above to set the # values of the time and the timestep. while 1==1: # Do one calculation every second rate(1) # Increment the value of the time and print the result. # Here the single equal sign means set the value of t to # whatever appears to the right of the equal sign. t = t + dt print t # End of the while loop. Go back to the rate(1) statement # and start over.


91

Appendix 2 – LoopDemo3.py Code Listing # All lines like this one that begin with a "#" are comments. # All other non-blank lines are program statements. # Import the visual library. from visual import * # Set the time t = 0 # Set the timestep dt = 1 # The next line causes the indented lines that follow # it to be repeatedly executed in the loop. The construct: # 1==1 # means "is one is equal to one?" which is always true. # Thus double equal signs like this mean something different # than a single equal sign, such as is used above to set the # values of the time and the timestep. while 1==1: # Do one calculation every second rate(1) # Print the time and then increment its value. # Here the single equal sign means set the value of t to # whatever appears to the right of the equal sign. print t t = t + dt # End of the while loop. Go back to the rate(1) statement and # start over.


92

Appendix 3 – SHM.py Code Listing # All lines like this one that begin with "#" are comments. # All other lines are program statements. # The next line is an internal revision control id: # $Date: 2007/11/08 17:19:19 $ $Revision: 1.2 $ # Copyright (c) 2007 David M. Harrison # Import the visual library. from visual import * # These four lines control the size of the window of # the animation and the scale. The details of these lines # are not important for our purposes. scene.autoscale = 0 scene.height = 400 scene.width = 800 scene.range = vector(60, 60, 60) # Create the green ball that will execute simple harmonic motion # by numerical integration. greenBall = sphere (color = color.green, radius = 2) # yellowBall will execute simple harmonic motion using a sine function. yellowBall = sphere (color = color.yellow, radius = 2) # The initial x position of the balls: this is # the equilibrium position. x = 0 # Position the balls. pos is a built-in of VPython, and # lists the (x,y,z) coordinates. The x axis is horizontal, # y axis is vertical, and the z axis is perpendicular to # the plane of the screen. We place the green ball just # below the center of the scene, at y - -10. # greenBall.pos = (x,-10,0) # yellowBall is above the first ball: it's y coordinate is 10, # just above the center of the scene. yellowBall.pos = (x, 10, 0) # The initial x component of the velocity of the balls: # all other components are zero. vx = 150 # The spring constant k = 9.0 # The mass of the balls mass = 1.0 # The amplitude of yellowBall's motion ampl = sqrt(mass/k) * vx


93

# The time t = 0 # This is the time step dt = 0.005 # This causes the following indented lines # to be executed forever in a loop. while 1 == 1: # Set the rate of the animation rate(1/dt) # The acceleration in the x direction. a = -(k/mass) * x # Update the speed using the acceleration. Note # that we "recycle" the variable vx, replacing the # old value with the new one. vx = vx + a*dt # Update the x position of the ball using the speed. x = x + vx*dt # Position greenBall at the new x position greenBall.pos = (x, -10, 0) # Update the time t = t + dt # Now we calculate simple harmonic motion using # a sine function and position yellowBall using the result # of the calculation x2 = ampl * sin( sqrt(k/mass)* t) yellowBall.pos = (x2,10,0) This Guide was written by David M. Harrison, Dept. of Physics, Univ. of Toronto in November 2007. Last revision: March 4, 2008.

94

PHY131 Practicals Manual Fluids Module

Fluids Module Student Guide

Concepts of this Module • Fluids • Pressure • Buoyancy • Fluid dynamics

The Activities

Activity 1

Open the gas-properties.jar animation which is located at feynman:public/Modules/Fluids. There are many useful ways to use this animation, and we will only draw you attention to a couple of things that you may wish to do; you are encouraged to explore further. Here is a screen shot of the default animation after some Heavy Species molecules have been pumped into the container:


95

A. You will notice that the reading of the Pressure gauge is not constant. Explain why this is so. What would be necessary for the pressure reading to be more constant? How would you present a value for the pressure that also expresses your observed variations?

There are many options for controlling the animation. We shall describe two of them.

1. By default the acceleration due to gravity g is zero. You may introduce a non-zero value of g with the Gravity slider.

2. By clicking on the Measurement Tools button you may turn on the Layer tool. This tool measures the pressure in the gas at a specified height; you may drag the position of the measurement with the mouse. You can also specify the time over which the value of the pressure is averaged.

Here are some suggested explorations.

B. Use the Layer tool with various settings of the Averaging Time. Describe what happens. If this was not part of your answer to Part A, should it have been?

C. With Gravity set to 0, predict how the pressure in the gas varies with height. Check your prediction using the Layer tool. Were you correct? If the pressure varies with height, does it vary is the height, the height squared, one over the height, or what?


96

D. Introduce a non-zero Gravity. Predict how the pressure varies with the height. Check your prediction. Were you correct? If the pressure varies with height, does it vary is the height, the height squared, one over the height, or what?

Activity 2

Cylinder A is being filled to the level shown. As the water is added to the cylinder it flows along the horizontal pipe and up B, C and D, which are all open at their tops.

Rank the heights of the water in A, B, C, and D when A is filled. Check your prediction using the supplied apparatus. Was your prediction correct? If yes, what physical principles did you use to make a correct prediction? If no, explain the actual result.

Activity 3

A rigid rectangular container filled with water is at rest on a table as shown. Two imaginary boundaries divide the water into three layers of equal volume. No material barrier separates the layers.

A. Draw a free body diagram for each layer. The label for each force should include: • A description of the force, and


97

• The object on which the force is exerted, and • The object exerting the force.

B. Rank the magnitude all the vertical forces you have drawn for Part A, from the smallest to the largest. Explain how you determined the ranking.

C. Rank the magnitude of all the horizontal forces you have drawn for Part A, from the smallest to the largest. Explain how you determined the ranking.

Activity 4 A small square hole of area A is cut in the side of the container of Activity 2. The centre of the hole is a height z above the tabletop. Consider the rectangular section of water of area A aligned with the hole, as shown.

A. Draw a free body diagram of all the forces acting on the rectangular section of water.

B. What will happen to the water just inside the hole?

Activity 5 A bucket of water has a spring soldered to the bottom. Attached to the other end of the spring is a cylindrical cork of mass m, height h and area A which is stationary below the surface of the water, as shown. The top of the cork is a depth d below the surface of the water. The spring has a spring constant k and is stretched a distance x from its equilibrium position. The density of the water is ρ. Draw a free body diagram of all the vertical forces acting on the cork. Evaluate the magnitude of those forces. Determine x, the amount that the string is stretched from its equilibrium position.


98

Activity 6

A hydrometer measures the density of a liquid. They are widely used to measure the alcohol content in the brewing of beer, the electrolyte content of battery acid, and more. The device is placed in the liquid whose density is to be measured, and the density is read by the place on the scale where the surface of the liquid touches the stem. Here is a close-up figure of two possible ways that the markings on the scale of the hydrometer can be arranged. Which of these arrangements are correct? Explain.

Activity 7

Please do this Activity with all the apparatus in the supplied dishpan to minimize the water spilled onto the tabletop. You are supplied with a beaker. You should fill it with water nearly to the top. Place the supplied medicine dropper in the water with the squeeze bulb on top. Suck enough water up into the medicine dropper that it just barely floats. You are supplied with an empty 2 liter plastic pop bottle. Fill it to the brim with water. Transfer the filled medicine dropper to the water in the pop bottle. Screw the top tightly on the bottle. Squeeze the bottle. What happens to the medicine dropper? What happens when you quit squeezing the bottle? Explain why squeezing the


99

bottle and increasing the pressure of all the fluids within would cause the observed motion. This is called a Cartesian diver. The supplied toothpicks make it easy to “fish” the medicine dropper out of the bottle. When you are finished with this Activity, carefully empty all the water into the sink.

Activity 8

You may have noticed that the bubbles in a glass of a carbonated beverage (soda, beer, champagne, etc) accelerate as they rise from the bottom. Explain.

Activity 9

A ship is in a canal lock, which is only a little bit larger than the ship itself. The ship is loaded with steel ingots, which are large bars of steel. The crew becomes angry with the captain of the ship and throws the steel ingots overboard into the water of the lock. Does the level of the water in the lock rise, lower, or stay the same? Check your prediction. Place the supplied plastic tank in the dishpan and fill the tank about half-way with water. Place the supplied weight in the bottom of the supplied plastic boat and gently place it in the water. You may mark the height of the water in the tank with a small piece of masking tape. Carefully lift the boat out of the water, place the weight at the bottom of the tank, and put the boat back in the water. When you are finished with this Activity, carefully empty the water into the sink.

Activity 10 A water tank with water of height h has a small hole cut in the side at height z. The water strikes the ground at x. The figure shows the streamline from the top of the water at A to just outside the hole B. Recall that Bernoulli’s equation is:


100

constgyvp =++ ρρ 2

21

If the hole is small, it is reasonable to approximate that the speed of the water at A is zero. Since point A and B are in contact with the outside air, it is reasonable to approximate that the pressure is the same at point A and B, that of atmospheric pressure in the room.

A. What will be the shape of the stream of water emerging from the hole until it strikes the ground?

B. Without using any equations, describe how the speed of the water at B varies with z, How will the distance x depend on z?

C. Use Bernoulli’s equation and your knowledge of projectile motion to derive the answers to Part B. For what value of z will x be a maximum? What approximations are you making? Are those approximations reasonable?

D. You are supplied a tank with small holes cut in it at values of z = 0.75 h, 0.50 h, and 0.25 h, where the height of the water h is indicated by a mark on the tank. Place the tank in the supplied dishpan: place it on one end of the dishpan with the holes pointing towards the other end of the dishpan. Fill the tank with water to the mark. As the water level drops appreciably add water. Is what you see consistent with your results from Parts B and C?

When you are finished with this Activity, carefully empty the water into the sink.

Activity 11

When an object falls through a fluid, either a liquid or a gas, there are three forces that act on it:

1. The downward force due to gravity, GFr

. This is the weight of the object.

2. An upward buoyant force, BFr

. As Archimedes realized over 2,000 years ago, this is equal to the weight of the fluid that the object displaces.

3. An upward drag force, DFr

. In this Activity we will concentrate on the drag force exerted on a sphere falling through a fluid. We assume that the surface of the sphere is perfectly smooth. We will use the following variables in the discussion:

• r: the radius of the ball. • v: the instantaneous speed of the ball. • ρ: the density of the fluid. • η: the kinematic viscosity of the fluid. This is sometimes called liquid friction. It

is measured in units of pressure × time.


101

Here are some values for the density and viscosity of various fluids.

Fluid Density (kg/m3) Viscosity (mPa-s) Superfluid -- 0 Air (20 ºC) 1.2 0.0182

Water (20 ºC) 998 1.00 Olive Oil (88 ºC) 914 43.2 Glycerine (20 ºC) 1260 658

Honey (20 ºC) 1,500 5,000 There are various ways that the fluid can flow around the sphere. If the speed of the ball is small, the flow is “smooth” or “laminar”. In this case it turns out that the drag force is proportional to the speed.

vFD ∝

This was first shown by Stokes in 1851. This will be explored further in Part E. When the ball is going faster, turbulence develops in the fluid behind the ball. In this case the drag force is approximately proportional to the speed.

2vFD ∝ Note that in both of these cases, as the ball’s speed increases the drag force increases. Thus at some point the drag plus buoyant forces approaches the magnitude to the weight of the ball, so asymptotically there is zero net force acting. Then the speed of the ball becomes constant: the value of the speed is called the terminal velocity. For a sky diver falling face down with arms and legs outstretched, the terminal velocity is about 55 m/s. If the sky diver falls feet first, feet together and arms close to the body, the terminal velocity goes up to about 90 m/s. When the sky diver opens the parachute, the drag force goes way up and the terminal velocity falls to about 5 m/s.


102

Whether the fluid flow around the ball is laminar or turbulent turns out to depend only on a single dimensionless number, the Reynolds number Re.

vrηρ

≡Re

Re Type of Flow ≤ 1 Laminar

1 - 1 × 103 Transition 1 × 103 – 1.5 × 105 Turbulent

Note that for constant r, ρ, and η the Reynolds number is proportional to speed. Therefore when a ball is dropped from a large height, Re increases until the terminal velocity is reached. When the Reynolds number reaches ~ 1.5 × 105 the forces on the fluid near the ball become extreme, and both the wake and the layer of fluid right next to the ball become turbulent. This causes a sudden change in the way the luid flows around the ball, and the turbulent wake becomes narrower. When this happens, the drag force drops and the acceleration of the ball increases. This is called the drag crisis. As the speed increases further, the drag force resumes increasing with speed.

A. A ball of radius r is falling through a fluid and at some time has an instantaneous speed v. A second ball of radius 2r is falling through the same fluid. At what instantaneous speed will the second ball have the same flow pattern of fluid around it as the first ball?

B. Here is the URL of a Flash animation of dropping a ball from the CN Tower: http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/FluidDynamics/BallCNTower/BallCNTower.html

The above link is to a fixed size animation which works nicely if only one person it viewing it. For use in the Practical itself a version which can be resized to be larger so that the entire Team can see it is better. Here is a link to such a version:

http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/FluidDynamics/BallCNTower/BallCNTower.swf Open the animation and explore how it works. For the keen some details about this animation and the one you will explore in Part D are in the Appendix.

C. For air, in SI units the Reynolds number is:

vrvr 000,70Re ≅≡ηρ


103

For the billiard ball, 5-pin bowling ball, and 10-pin bowling ball calculate the speed for which the drag crisis occurs. Are these results consistent with what you see in the animation of Part B?

D. Here is the URL of a Flash animation of dropping a ball in a liquid: http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/FluidDynamics/ViscousMotion/ViscousMotion.html

As with the animation of Part B, you may access a resizable version at:

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/FluidDynamics/ViscousMotion/ViscousMotion.swf

Open the animation and explore how it works.

E. For small Reynolds numbers, so the fluid flow is laminar, the drag force is:

vrFD ηπ6=

You may be surprised by the fact that the density of the fluid does not appear in this equation. When the sphere is at terminal velocity the net force is zero:

BGD FFF −=

Therefore:

rFFv BG

alter ηπ6min−

=

For the animation of Part D, set the following values:

• r = 20 mm • η = 5850 mPa-s • ρliquid = 1500 kg/m3 • ρball = 5000 kg/m3

Note the values of the ball weight and buoyant force and their difference. Run the animation and note the terminal velocity. Now set the radius of the ball to 25 mm. Adjust the viscosity of the liquid so the drag force will be the same as the previous case. Adjust the liquid and ball densities so the ball weight minus the buoyant force is about the same as the previous case; you are unlikely to find values of the densities which are exactly the same, but can find ones that make the value almost the same. Does the animation match the theory? In particular is the motion of the ball the same for these two cases?


104

Appendix Although the details of how the animations of Activity 10 work are not important for your learning of fluid dynamics, here we “lift the hood” to discuss the internals of the animations. Except in the limit of laminar flow, the theory of drag forces is not easily solvable. Thus the animation uses a mixture of experimental data and some heuristic formulae that describe the data reasonably well. It turns out to be convenient to describe the drag force in terms of a drag coefficient CD(Re), which is a function only of the Reynolds number. Then the drag force is:

22(Re) vrCF liquidDD ρ=

For laminar flow (Re ≤ 1):

Re6(Re) π

=DC

For larger values of the Reynolds number, experimental data on the dependence of the drag coefficient on the Reynolds number must be used. The data used in the animation is adapted from H. Edward Donley, UMAP Journal 12(1), 47 (1991), http://www.ma.iup.edu/projects/CalcDEMma/drag/drag7.html. To parameterize this data involves some truly ugly equations. We used forms by John Versey and Nigel Goldenfleld, “Simple viscous flows: From boundary layers to the renormalization group”, Rev. Mod. Phys. 39(3), 883 (2007), http://rmp.aps.org/browse. The Donley data and the Versey and Goldenfleld interpolation of it is shown on the next page. The turbulent flow case (1 × 103 < Re < 1.5 × 105 ) corresponds to the part of the above plot where the drag coefficient is approximately constant independent of the Reynolds number. Note that, despite the notation used in the axes of the figure, the values are the natural logarithms of the values. The drag crisis is when the drag coefficient suddenly drops. Once the drag force for a given speed has been determined, then we know the net force acting on the ball and hence its acceleration. We use a numerical approximation to find the motion of the ball. The method is similar to one you may have explored in the Numerical Approximation Module. For a time step dt.


105

1. From the acceleration, calculate the new speed of the ball: vnew = vold + a × dt

2. From the new speed calculate the new position of the ball: ynew = yold + vnew × dt

3. From the new value of the speed calculate the new drag force and then the new acceleration of the ball.

4. Go to 1 and repeat. The above scheme turns out to not be accurate enough for our animation, so an extension of it called a 4th order Runge-Kutta is used. It turns out that for this calculation to be stable we must iterate the Runge-Kutta 10 times for every frame of the animation. Since the animation runs at 12 frames per second, this means that the time step dt is 1/120 = 0.17 s. This Student Guide was written by David M. Harrison, Dept. of Physics, Univ. of Toronto, May 2008. The animation used in Activity 1 is from the Physics Education Technology (PhET) group at the University of Colorado, http://phet.colorado.edu/new/index.php. Activity 2 is based on Lillian McDermott et al., Tutorials in Introductory Physics (Prentice Hall, 20020, ST 219. Activities 5 and 7 are based on David M. Harrison and William Ellis, Student Activity Workbook, 3rd ed. (Norton, 2008), 18.4 and 18.6. The figure for Activity 5 is slightly modified from a figure from Wikipedia, http://en.wikipedia.org/wiki/Hydrometer, retrieved June 19, 2008. Last revision: November 20, 2008.

phy131h1s practicals manual - department of physicsjharlow/teaching/phy131s09/practicals... ·...

Documents