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Chapter 4NEWTON’S LAWS, FORCES, & FREE-BODY DIAGRAMMING

HONORS PHYSICS, 2014-2015

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In-Class Problem Solving

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EXAMPLE 1 (4.6): A freight train has a mass of 1.5 x 107 kg. If the locomotive can exert a constant pull of 7.5 x 105 N, how long does it take to increase the speed of the train from rest to 80 km/h?

EXAMPLE 2 (4.14): The force exerted by the wind on the sails of a sailboat is 390 N north. The water exerts a force of 180 N east. If the boat (including the crew) has a mass of 270 kg, what are the magnitude and direction of the acceleration?

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Free-Body Diagramming:

Consider all of the forces acting on a crate being pushed across a horizontal floor.

How can we calculate the acceleration of the box?

Free-Body Diagramming: 1-D Elevator Example

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Friction FBD Example:

Suppose you shove a box across a surface. What is the acceleration of the box as it slows to a stop?

CASE A: FLAT SURFACE

CASE B: SLOPED SURFACE

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BOX SLIDING DOWN A RAMP EXAMPLE

What is the acceleration of a box sliding down an inclined ramp a) when neglecting frictional forces, and b) when considering frictional forces?

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Name: _____________________________

Honors Physics FBD ActivityEnd-of-the-semester fun with inclined surfaces!!!

Purpose:

This seemingly simple lab activity will shed some light on the frictional force.

Supplies Needed:

Spring scale, quarter, text book, weight, protractor, and boards (for inclination)

Directions for Part ONE:

(read thoroughly before beginning)

Before beginning, wipe down your book cover with a dampened paper towel to remove any surface oils, dirt, or adhesives. Dry the cover thoroughly.

Placing a coin on your textbook cover, begin inclining the cover upright slowly. Have someone measure the angle (try to be as accurate as possible!) as you continue inclining the cover. I highly recommend you slowly raise the cover vertically, and then pause for a moment every degree, being as the coin might finally reach the verge of slipping.

Once the coin slowly starts moving down the slope at what appears to be a constant velocity, record this angle. Do this multiple times, being as our measurement tools aren’t tremendously precise. I recommend putting the coin back at the same initial book cover location. Once your entire group has come to a conclusion regarding the proper angle (you can use an average, or take the most consistent result), record this value in the space below.

Angle Measurements (use 2 sig figs): ___________________________

Average angle (2 sig figs): ___________

Equation Derivation for μk:

Derive an expression for the coefficient of kinetic friction by drawing a FBD of the coin moving at constant speed down the slope. First, draw a diagram of the situation depicting the textbook, the coin, and the following physical necessities for Newtonian Mechanics: FW, FN, FF, and θ. You should NOT use any numbers in your equation derivation, and please keep your angle as theta (don’t plug it in yet! This expression works for ANY angle, not just this situation!!). Translate your picture into a rotated coordinate system, and remember that the coin moves at constant speed down the incline as you derive your expression for the coefficient of friction between the coin and your book. As a hint, your mathematical expression for μk should be a trigonometric function!!!

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What seems particularly interesting for the equation you derived from a physical standpoint?

Compare your coefficient for kinetic friction with that of i) steel on ice and ii) rubber on concrete(see your book or your equation sheet). At what angle would expect a steel object to begin sliding down an icy incline?

Compare your angle value from the previous page (the weighted value given to two significant figures) to some of your classmates recorded values. Explain why there might be some differences (hopefully minor).

Directions for part TWO:

Once you have μk, now place the weight on your slope at some arbitrary angle. Use the provided board(s) to incline your book cover. Be sure to measure the angle made with the horizontal as accurately as possible! Attach your spring scale to the weight, and slowly continue applying force until the weight begins to slide at a constant velocity in the uphill direction. Have someone carefully assess the spring scale force value (in Newtons), and write this value down. Do multiple trials to check for precision and accuracy. Finally, draw a FBD for this motion, and solve for the Normal force provided by your book surface and the Frictional force opposing the motion of your weight. Of course, you will need to use the friction coefficient from the first part of this lab to do your calculations!!

Show FBD and All Calculations HERE:

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Frictional Force: ____________________ Normal Force: ________________________

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Name: _______________________________

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Honors Physics In-Class FBD WorksheetConceptual Practice

1) Your physics book sits motionless on a perfectly flat tabletop. Draw a free body diagram of this situation and label all forces acting on the book.

ΣFx: ________________ ΣFy: ________________

2) Mr. Cunnings is suspended motionless from our ceiling by two vertical ropes. Draw a free body diagram of this situation and label all forces acting on poor Mr. Cunnings.

ΣFx: ________________ ΣFy: ________________

3) An egg is freely falling from a nest in a tree. Neglecting air resistance, draw a free body diagram of all of the forces acting on the egg as it falls.

ΣFx: ________________ ΣFy: ________________

4) A flying squirrel glides from a tree toward the ground at constant velocity (ignore flapping of wings). Take air resistance into account, and label all forces acting on the squirrel in a free body diagram.

ΣFx: ________________ ΣFy: ________________

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5) Your physics book sits motionless on a perfectly flat tabletop. You apply a horizontal force that slides the text book with a rightward acceleration. Draw a free body diagram of this situation and label all forces acting on the book.

ΣFx: ________________ ΣFy: ________________

6) Your physics book sits motionless on a perfectly flat tabletop. You apply a horizontal force that slides the text book with a rightward constant velocity. Draw a free body diagram of this situation and label all forces acting on the book.

ΣFx: ________________ ΣFy: ________________

7) A college student rests a backpack on her shoulder. The backpack is suspended motionless by one single strap from one shoulder. Draw a free body diagram of this situation and label all forces acting on the backpack.

ΣFx: ________________ ΣFy: ________________

8) Mr. Cunnings goes skydiving (again). He jumps out of the plane and is accelerating toward Earth’s surface. Draw a free body diagram of the vertical forces acting on him.

ΣFx: ________________ ΣFy: ________________

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9) During Mr. Cunnings’ skydive, he reaches his “terminal velocity” and is now falling with constant vertical velocity. Draw a free body diagram of the vertical forces acting on him.

ΣFx: ________________ ΣFy: ________________

10) A force is applied to the right to drag a sled across a flat surface of loosely packed snow with rightward acceleration. Draw a free body diagram of all forces acting on the sled.

ΣFx: ________________ ΣFy: ________________11) A force is applied to drag a sled up a hill inclined 23

degrees with rightward acceleration parallel to the hill slope. Draw a free body diagram (original AND translated) of all forces acting on the sled.

ΣFx: ________________ ΣFy: ________________

12) A force is applied to drag a sled up a hill inclined 15 degrees with a rightward acceleration. The applied force is administered at an angle that’s 35 degrees northerly with respect to the hill slope. Draw a free body diagram (original AND translated) of all forces acting on the sled.

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ΣFx: ________________ ΣFy: _______________13) A block of mass m on an inclined plane is joined to a mass

n by a cord over a pulley, as shown below. Draw translated free body diagrams for both blocks.

ΣFx: ________________ ΣFy: ________________

14) Two blocks of mass m1 and m2 are connected by a light string passing over a pulley. The blocks are at rest on inclined frictionless surfaces, and the effects of the pulley are negligible. Given the picture below, draw translated free body diagrams for both blocks! For this problem, you will need to use subscripts for all forces acting on each of the two blocks (e.g. FN1).

ΣFx: ________________ ΣFy: ________________

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15) A lantern of mass m is suspended by a string that is joined to two other strings, as shown in the diagram below.

a. Draw a free body diagram for the forces acting on the lantern.

ΣFx: ________________ ΣFy: ________________

b. Draw a free body diagram for the forces acting on the knot.

ΣFx: ________________ ΣFy: ________________

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Name: ___________________________ Date: ________ Score: ___________

Honors PhysicsNewton’s Laws: Free Body DiagrammingEquilibrium Under the Action of Concurrent Forces

Definitions:

Concurrent Forces: Forces whose lines of action all pass through a common point. For concurrent forces, we will treat all objects (e.g. boxes, humans, footballs) as point objects, which greatly simplifies the problems.

Equilibrium: An object is said to be in equilibrium under the action of concurrent forces provided it is not accelerating. For all of the problems on this worksheet, the objects are in equilibrium, meaning that they are either not moving OR that they are moving with a constant nonzero velocity.

Weight (Fw): The force with which gravity pulls downward on an object.

The Tensile Force (FT): The force acting on a string, cable or chain (or any structural member, for that matter). The tensile force will always tend to stretch these objects. The scalar magnitude of the tensile force (e.g. 22.4 N) is called the tension.

The Friction Force (Ff): The force that opposes the sliding of an object across an adjacent surface that it is in contact with. The friction force will be parallel to the surface and apposite to the direction of the motion (or impending motion) of an object.

The Normal Force (FN): The force on an object that is being supported by a surface. The normal force will always be perpendicular to the surface supporting the object.

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Directions:

Complete the following problems. Give your final answers with the indicated number of significant digits.

1) In the figure shown, the tension in the horizontal cord is 30.0 N. Find the weight of the hanging object. Use three significant figures in your final answer.

2) A rope extends between two poles. A 90.0 N boy hangs from the rope, as shown in the diagram. Find the tension in the two parts of the rope. Use two significant figures in your final answers.

FT1: _______________ FT2: _____________

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3) A 50.0 N box is slid straight across the floor at a constant speed by a force of 25 N, as shown in the figure. Use two significant figures for all answers.

a. How large a friction force impedes the motion of the box?

b. How large is the normal force?

c. Find the coefficient of kinetic friction between the box and the floor

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4) Find the tensions in all of the ropes shown in the figure, assuming that the supported object weighs 600 N (use two significant figures for all answers in this problem).

Answers: FT1: ______ FT2: ____ FT3: ____ FT4: ____ FT5: _____

5) Each of the objects shown in the figure is in equilibrium. Find the normal force, FN, in each of the three cases. Use 1 significant figure for a and c, and 2 significant figures for b.

a)

b)

c)

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6) Considering the same three cases in number 5, find the coefficient of kinetic friction if the object is moving with constant speed. Round all of your answers to two significant figures. Use two significant figures for all answers.

a)

b)

c)

7) Pulled by the 8.0 N block shown in the diagram, the 20 N block slides to the right at a constant velocity. Find μk between the block and the table. Assume the pulley to be frictionless, and use two significant figures in your answer.

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Name: ____________________________ Date: ________ Score: __________

FBD Applied Problems (no diagram provided)

1) Rudolph, the red-nosed reindeer, pulls on a rope attached to a 20.0kg sleigh (Let’s assume Santa isn’t in the sleigh yet, nor are the presents for all of the children of the world). Rudolph pulls with a force of 90.0N at an angle of 30.0 degrees to the horizontal. The coefficient of kinetic friction between the sleigh and the snowy surface is 0.200. a. Draw a free-body diagram of this situation

b. Draw a vector diagram of this situation

c. Find the acceleration of the empty sleigh.

\

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1) Suppose that a box of mass m hangs by a string in an elevator that’s accelerating upward, as shown in the image above. Using a free body diagram (drawn to the side of the image), derive an expression for FT for this situation.

a. How would your expression change if the elevator WAS NOT ACCELERATING?

b. How would your expression change if the elevator were accelerating DOWNWARD?

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2) Now let’s use some numbers. If the box has a mass m = 5 kg and g = 9.8 m/s2,a. What is the tension in the rope if the elevator is MOTIONLESS?

b. What is the tension in the rope if the elevator is moving at constant velocity?

c. What is the tension in the rope if the elevator is accelerating upwards at 2.0 m/s2?

d. What is the tension in the rope if the elevator is accelerating downwards at -1.5 m/s2?

e. What if the elevator cable snapped and the elevator was freely falling down the elevator shaft…what would the tension in the cable be?

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3) Reconsider the elevator situation described in problem 2. In other words, the mass of the box is still 5kg. Suppose that the rope connected to the hanging mass can only withstand tensions less than 75 Newtons. Tensions greater than 75 Newtons would cause the rope to snap. What maximal upward acceleration could the rope withstand before it snaps?

4) Another important equation we’ll encounter that involves springs is F = kx, where F = force, x = the spring stretch/compression distance, and k is the “spring constant.” Note: This equation –Hooke’s Law – is sometimes written as F = -kx, being as springs provide a RESTORING FORCE…for now, however, let’s keep things simple. A box of mass m is hung by a spring from the ceiling of an elevator. When the elevator is at rest, the length of the spring is L = 1m.

a. As the elevator accelerates upward, the length of the spring will be _______ 1 meteri. Greater than

ii. Equal toiii. Less thaniv. It’s impossible to know based on the information given

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b. Now let’s use some numbers in this situation. A single mass m1 = 5 kg hangs from a spring in a motionless elevator. The spring is extended x = 12 cm from its unstretched length. What is the spring constant, in N/m?

c. If the elevator accelerates upward at 2.5 m/s2 what will the length of the spring be when compared to its unstretched length? Give your answer in CENTIMETERS.

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Name: __________________________________

Honors PhysicsFBD Retired Quiz 1

1) Homer Simpson falls off a power plant tower, and, while freely falling to Earth, two of his hairs get snagged around a utility wire. One of the snagged hairs (label this hair 1) is oriented 30 degrees to the left of the vertical, and the other hair (label this hair 2) is oriented 15 degrees above the right horizontal. Homer’s weight is given as 1000N. Assume that both hairs are weightless, and use three significant figures in ALL calculations for this question.

a. What is the tension in hair 1 and in hair 2?

b. What is Homer’s weight in lbs? Note that 1 kg = 2.2 lbs, and use g = 10 m/s2 for this problem.

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2) Calvin and Hobbes are in a wagon traveling down a hill at a constant velocity. Assume that the wagon’s total mass (including Calvin and his stuffed tiger Hobbes) is mw . The hill is sloped at an angle θ with respect to the horizontal. Please take the “downhill” direction to be to the right.

a. Derive an expression for the frictional force acting on their wagon?

b. Derive an expression for the normal force acting on their wagon?

c. Derive an expression for the coefficient of kinetic friction between the wagon wheels and the hill.

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Honors Physics In-Class Examples

When two objects of unequal masses are hung vertically over a frictionless pulley of negligible mass, we call this system an “Atwood Machine.” Given the picture, determine the acceleration of the two objects and the tension in the lightweight cord by deriving equations for each.

a. Assume that m1 = 1kg and m2 = 2kg. What is the acceleration of the system?

b. Assume that m1 = 1kg and m2 = 5kg. What is the acceleration of the system?

c. Assume that m1 = 1kg and m2 = 10kg. What is the acceleration of the system?

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A ball of mass m1 and a block of mass m2 are attached by a lightweight cord that passes over a frictionless pulley of negligible mass (see diagram below). The block rests on a frictionless incline of angle theta. Derive expressions for the magnitude of the acceleration of the two objects and the tension in the cord.

For all parts below, assume the angle is 20 degrees.

a. If m1 = 5kg and m2 = 3kg, what is the acceleration of the system?

b. If m1 = 5kg and m2 = 3kg, what is the tension in the cord?

c. If m1 = 5kg and m2 = 1kg, what is the acceleration of the system?

d. If m1 = 5kg and m2 = 1kg, what is the tension in the cord?

e. If m1 = 5kg and m2 = 10kg, what is the acceleration of the system?

f. If m1 = 5kg and m2 = 10kg, what is the tension in the cord?

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In-Class Practice Problems

EXAMPLE 1 (4.19): Two blocks are fastened to the ceiling of an elevator, as in figure P4.19 on page 110. The elevator accelerates upward at 2.00 m/s2. Find the tension in each rope.

EXAMPLE 2 (4.23): A 5.0kg bucket of water is raised from a well by a rope. If the upward acceleration of the bucket is 3.0 m/s2, find the force exerted by the rope on the bucket.

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EXAMPLE 3 (4.25): A 2000-kg car is slowed down uniformly from 20.0 m/s to 5.00 m/s in 4.00 seconds. a) What average force acted on the car during that time, and b) how far did the car travel during that time?

EXAMPLE 4 (4.28): An object of mass 2.0 kg starts from rest and slides down an inclined plane 80.0 cm long in 0.50 seconds. What net force is acting on the object along the incline?

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EXAMPLE 5 (4.32): a) An elevator of mass m moving upward has two forces acting on it: the upward force of tension in the cable and the downward force due to gravity. When the elevator is accelerating upward, what is greater, the tension or the weight force? b) When the elevator is moving with constant velocity upward, which is greater, the tension or the weight force? c) When the elevator is moving upward, but the acceleration is downward, which is greater, the tension or the weight force? d) Let the elevator have a mass of 1500 kg and an upward acceleration of 2.5 m/s2. Find the tension force. Is your answer consistent with your answer in part A? e) The elevator in part d now moves with a constant upward velocity of 10 m/s. Find T. Is your answer consistent with part b? f) Having initially moved upward with constant velocity, the elevator begins to accelerate downward at 1.50 m/s2. Find the tension force. Is your answer consistent with part c?

EXAMPLE 6 (4.45): Objects with masses of m1 = 10.0 kg and m2 = 5.00 kg are connected by a light string that passes over a frictionless pulley as in picture P4.30 on page 111. If, when the system starts from rest, m2 falls 1.00 m in 1.20 seconds, determine the coefficient of kinetic friction between m1 and the table.

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EXAMPLE 7 (4.46): A car is traveling at 50.0 km/h on a flat highway. A) If the coefficient of kinetic friction between road and tires on a rainy day is 0.100, what is the minimum distance in which the car will stop? B) What is the stopping distance when the surface is dry and the coefficient of kinetic friction is 0.600?

EXAMPLE 8 (4.48): Objects of masses m1 = 4.00 kg and m2 = 9.00 kg are connected by a light string that passes over a frictionless pulley as in P4.48 on page 113. The object m1 is held at rest on the floor, and m2 rests on a fixed incline of θ = 40 degrees. The objects are released from rest, and m2 slides 1.00 m down the incline in 4.00 seconds. Determine a) the acceleration of each object, b) the tension in the string, and c) the coefficient of kinetic friction between m2 and the incline.

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EXAMPLE 9 (4.57): A boy coasts down a hill on a sled, reaching a level surface at the bottom with a speed of 7.0 m/s. If the coefficient of friction between the sled’s runners and the snow is 0.0500 and the boy and the sled together weigh 600.0 N, how far does the sled travel on the level surface before coming to rest?

EXAMPLE 10 (4.59): A box rests on the back of a truck. The coefficient of static friction between the box and the bed of the truck is 0.300. a) When the truck accelerates forward, what force accelerates the box? b) Find the maximum acceleration the truck can have before the box slides.

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EXAMPLE 11 (4.65): Two boxes of fruit on a frictionless horizontal surface are connected by a light string as in Figure P4.65 on page 115, where m1 = 10.0 kg and m2 = 20.0 kg. A force of 50.0 N is applied to the 20.0 kg box. a) Determine the acceleration of each box and the tension in the string. b) Repeat the problem for the case where the coefficient of kinetic friction between each box and the surface is 0.10.

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EXAMPLE 12 (4.69): Three blocks of masses 10.0 kg, 5.00 kg, and 3.00 kg are connected by light strings that pass over frictionless pulleys as shown in Figure P4.69 on page 115. The acceleration of the 5.00 kg block is 2.00 m/s2 to the left, and the surfaces are rough. Find a) the tension in each string, and b) The coefficient of kinetic friction between blocks and surfaces. Assume the same coefficient for kinetic friction applies for all blocks.

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Discussion & Challenge Questions

Relevant Topics• Forces and Free Body Diagrams (FBDs)

1. The Free Body Diagram2. Force Inventory

• Friction1. Kinetic Friction2. Static Friction

Key concepts • Free Body Diagrams

• Force Inventory

◦ Normal Forces (no formula)

◦ Tension Forces (no formula)

◦ Spring Forces ( )

◦ Gravitational Forces

(near Earth surface)

(in general)

• Friction

◦ Kinetic Friction ( )

◦ Static Friction

Has a maximum possible value of

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EQUATIONS

Kinematics

Uniform Circular Motion

Dynamics

F = mg (near Earth’s surface)Fgravity = (in general)

(where G = 6.67 x 10-11)Fspring = -kx

Frictionf = kN (kinetic)f sN (static)

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Exit Ramp

On a trip to the Colorado Rockies, you notice that when the freeway goes steeply down a hill, there are emergency exits every few miles. These emergency exits are straight ramps which leave the freeway and are sloped uphill. They are designed to stop runaway trucks and cars that lose their brakes on downhill stretches of the freeway even if the road is covered with ice. You are curious, so you stop at the next emergency exit to take some measurements. You determine that the exit rises at an angle of 10.0o from the horizontal and is 100.0m long. What is the maximum speed of a truck that you are sure will be stopped by this road, even if the frictional force of the road surface is negligible?

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Hanging Sculpture

You are part of a team to help design the atrium of a new building. Your boss, the manager of the project, wants to suspend a 10.0-kg sculpture high over the room by hanging it from the ceiling using thin, clear fishing line (string) so that it will be difficult to see how the sculpture is held up. The only place to fasten the fishing line is to a wooden beam which runs around the edge of the room at the ceiling. The fishing line that she wants to use is known to be able to support a vertically hanging mass of 10.0 kg, so she suggests attaching two lines to the sculpture to be safe. Each line would come from the opposite side of the ceiling to attach to the hanging sculpture. Her initial design has one line making an angle of 20.0o with the ceiling and the other line making an angle of 40.0o with the ceiling. She knows you are taking physics, so she asks you if her design can work.

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Friction Intro

(a) A block of mass M = 5.0 kg slides on a horizontal table. The kinetic coefficient of friction between the block and the table is µ = 0.38. If the initial speed of the block is 8.0 m/s, how many seconds does it slide before stopping?

(b) A block of mass M = 15 kg is pulled across a horizontal table by a string. The kinetic coefficient of friction between the block and the table is µ = 0.69. If the speed of the block is constant at 2.0 m/s, what must the tension in the string be?

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Friction by Time

A block of mass M is released from rest on an incline of length L which makes an angle θ with the horizontal. The block reaches the end of the incline in time T. Show how the coefficient of kinetic friction can be determined from the measurement of time T.

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Emergency Stop

Your friend has been hired to design the interior of a special executive express elevator for a new office building. This elevator has all the latest safety features and will stop with an acceleration of g/3 in the case of an emergency. The management would like a decorative lamp hanging from the unusually high ceiling of the elevator. He designs a lamp which has three sections which hang one directly below the other. Each section is attached to the previous one by a single thin wire, which also carries the electric current. The lamp is also attached to the ceiling by a single wire. Each section of the lamp weighs 7.0 N. Because the idea is to make each section appear that it is floating on air without support, he wants to use the thinnest wire possible. Unfortunately the thinner the wire, the weaker it is. Since he knows that you have taken a course in physics, he asks you to calculate the force on each wire in case of an emergency stop.

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Name: ___________________________________ Deriving Equations: Pendulum Period

Possible Quantities that might contribute to your expression: Ɵ, L, t, m, g, π

Collect data in a neat table format. In your groups, decide which data you’ll need to collect. I strongly recommend varying different quantities to see if they factor into the equation!!!

“Play around” with the data and see if you can come up with an expression for T, which represents the period (in seconds) for a pendulum. As a hint, you might need a square or a square root somewhere in your expression!

Your best guess at the equation: ________________________________________________