web viewwe suspended a 46 cm spring from the top of a steel rod. the rod screwed into a heavy metal...

A. Student Formal Lab Report Mallok P.January 8, 2010 Physics 12 – SP4U1

The behavior of springs in Static and Dynamic situations Investigating Simple Harmonic Motion and Hooke’s Law

Abstract

A series of tests were made to experimentally determine the k-value for an ideal spring, the period of oscillation for the same spring and the Kinetic coefficient of friction for a tabletop. The k-value was found to be 16. We found the period of oscillation of the same spring to be 0.72 seconds. This value was found to be independent of amplitude of the oscillation. The Kinetic coefficient of friction for the tabletop was found to be 5.2 x 10-4. These values are a result of the application of Hooke’s Law to ideal springs, Simple Harmonic motion and The Law of conservation of Energy.

Introduction

We investigated the Law’s and properties of an oscillating spring system. We wanted to determine the spring constant from a given spring with an unknown k-value in the first part of the experiment. We utilized Hooke’s Law and our knowledge of simple harmonic motion as a basis for our calculations and observations.

Hooke’s Law states that the magnitude of the force exerted by the spring is directly proportional to the distance the spring has moved from equilibrium. It is this directly proportional relationship or Constant that we were searching for in this experiment. Hooke’s law can be expressed mathematically as Fs = -kx. We used this formula in our

calculations. The spring constant or value assigned to the variable k (k-value) was the subject we needed to derive experimentally. We sought to prove, using several different methods, that the k-value is indeed a constant value for the equation.

Fs = -kxWhere Fs is known as the

spring force and x is the distance of compression or expansion experienced by the spring. We also determined experimentally the period of the same spring. In the second part of the experiment we employed what we had learned about Simple Harmonic Motion in our observations. Simple Harmonic Motion (SHM) is defined as a periodic vibratory motion in which the force (and the acceleration) is directly proportional to the displacement. 1 We also know that the period is not dependant on the magnitude and we can expect to see a constant period in our results regardless of the magnitude in each test. We used the formula

T = 2π√m/kto check mathematically the test results from the second part of the experiment. This equation could verify the accuracy of the data of which we used to derive the period of oscillation.

This experiment increased our understanding of simple harmonic

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motion and the physical applications of Hooke’s Law. It also allowed us to see many of the various external influences that can affect the system.

In the third part of the experiment we wanted to derive the Kinetic friction constant from a tabletop. We utilized the spring constant we derived in the first part of the experiment and applied the law of conservation of energy. We used the formula

1/2kx2i - ff Δd = 1/2kx2

f

to calculated the kinetic coefficient of friction (μk) of a tabletop. The law of conservation of energy states that for an isolated system, energy can be converted into different forms, but cannot be created or destroyed. Using this knowledge and test data we were able to conclude the μk of a tabletop.

This exercise increased our ability to derive useful information and conclusions from experimental data and improved our calculation skill set.

Procedure

We suspended a 46 cm spring from the top of a steel rod. The rod screwed into a heavy metal base and attached a clamp to the top of it. We attached the spring to the clamp so that the spring could hang freely and placed the whole apparatus on the table top near the edge of the table so any weights attached to the table would not impact the tabletop when any attached masses were released. We attached a meter stick with 1 cm increments to the rod directly behind the spring so we could clearly record the various positions of the spring throughout the experiment. We then

recorded the normal position of the spring and proceeded to attach 5 different weights of varying masses to the spring and record the extended position of the spring after each mass was attached.(See table 1.0)

After the measurements were recorded we placed a Newton Meter between the spring and the clamp and pulled down on the spring recording the expanded distance of the spring at intervals of 0.5, 1.0, 1.5 and 2.0 Newton’s. (See table 1.1)

In the second part of the experiment we lifted the apparatus onto a stool to allow for greater accuracy in observing the period of oscillation. We removed the Newton meter and re-attached the spring to the clamp. The meter stick remained in place attached to the steel rod throughout the experiment. We then attached a 147.5g mass to the bottom of the spring and recorded the normal of the system. We stretched the mass to various expanded distances and released the mass. With a stopwatch two people recorded the time of one oscillation. We took the average of the time each person recorded and repeated the test three times for each position, taking the average for the times recorded as our final time. We repeated these steps for four different positions. (See table 2.0)

For the third part of the experiment we used a different set-up. We laid a meter stick on a table and secured the same spring to the meter stick. We attached four 200g masses together to achieve a large mass for the experiment. We recorded the normal position of the system. We stretched the mass extending it from the normal position and the recorded

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the distance. We then released the mass and recorded both the time it

took for the mass to stop and the distance at which it stopped. We did this for four different positions.(See table 3.0)

AnalysisTable 1.0 Normal 46cm

Table 1.1 Normal 49cm

0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.5

1

1.5

2

2.5

f(x) = 16.6666666666667 x

Force vs. Position

Position (M)

Forc

e (

N)

Table 1.2

Table 2.0

Table 3.0

We used the equationsFs = mg & Fs = -k Δx

and the data in Table 1.0 to calculate the spring constant in part 1. We averaged the results to give us a final k-value of 16.

We graphed the values obtained from the Newton Meter data set displayed in Table 1.1 to get the results in Table 1.2. The slope of the graph in table 1.2 indicates a k-value of 17. These results are consistent with an ideal spring and are supported by Hooke’s Law. The results of part A and part B in the first experiment are reasonably close in value.

We calculated the average oscillation in part 2 of the lab to be 0.72 seconds based on the experimental data as seen in table 2.0. We used the equation

T=2π√m/kWhere T is the period in seconds, m is the mass in kilograms, and k is the spring constant in Newton’s per meter. We also used the spring constant we calculated in part 1 to arrive at a theoretical period of 0.60 seconds. These values express what we would expect to see in simple harmonic motion. The two periods are different and this could be caused by error introduced into the experiment.

We calculated the the Kinetic coefficient of friction to be 5.2x10-4

using the formula 1/2kx2

i - ff Δd = 1/2kx2f

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Weight KG

Spring Extension

Displacement M

K-Value

0.1 52cm 0.06 16.350.2 58cm 0.12 16.350.05 49cm 0.03 16.350.1475 55cm 0.09 16.080.295 64cm 0.18 16.08

Position mm

Displacement M

Force N

1. 52 0.03 0.52. 55 0.06 1.03. 58 0.09 1.54. 61 0.12 2.0

Mass (KG)

Initial (cm)

Final (cm)

Displacement (M)

Time (s)

0.1475 57 75 0.18 0.600.1475 57 80 0.23 0.800.1475 57 85 0.28 0.750.1475 57 90 0.33 0.76

Mass (KG)

Normal (M)

Start (m)

Stop (M)

Time(s)

0.8 0.435 0.85 0.72 0.30.8 0.435 0.90 0.66 0.40.8 0.435 0.95 0.62 0.40.8 0.435 1.00 0.51 0.4


The k-value was the value derived in part 1 and the remainder of the data was from table 3.0

Discussion / Error

Based on what we know about Hooke’s Law and simple harmonic motion the lab was quite successful in achieving results that fall in line with the theory. Our calculations of the k-value were consistent for both the value based on the first set of data and the value based on the Newton meter experiment. Both of these experiments gave us the data set we were expecting and the results were in line with Hooke’s Law.

The oscillation experiment had a greater difference in the data table and the average period from the data set did vary slightly with the results from the formula calculations. This was mainly due to this section of the experiment having the greatest possibility for error. The results were still within expectations and did adhere to what would be expected.

The results for the calculation of the Kinetic coefficient of friction appear questionable although we didn’t have a similar μk to compare the results do not seem consistent with other kinetic friction values we’ve used in calculations in the past.

Although we endeavor to execute every possible step with the most possible accuracy there remains a large contingency for error. The greatest possibility for error is in the recording or interpreting of data. The scale that was used for most of the experiment was a relatively large scale. We used the side of the meter stick with only the centimeters

marked and divided into quarters instead of the side with the millimeters marked. When recording the normal for most of the spring experiment we received multiple opinions of the actual measurement depending on where each person was positioned relative to the meter stick. Every attempt to mitigate error was made by accepting the averages of the group. The same thing occurred during the timing of the oscillations with several people operating stopwatches with varying results. Again multiple trials and averages were used to try and minimize error. The speed at which most of the oscillations occurred meant that several attempts were made before accurate stop times could be taken as most of the reactions lagged behind the actual end of the oscillations.

Another component of error was the equipment itself. The spring itself, while it falls under an “ideal” spring was deformed at one end. The Newton meter had the ability to calibrate it or “zero it”. As such there could easily be inaccurate readings coming from the meter because its zero could be altered. The stopwatches used were digital programs on various cell phones. While they displayed hundredth of a second accuracy the program itself could lag while stopping and starting the device. They were also awkward to use with one hand and keep and eye on the spring at the same time.

During the third and final part of the lab the spring was laid horizontally to calculate the kinetic co-efficient of friction for the table top as such it was hard to keep the spring in alignment so that force was not being wasted by the deformation of

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the spring. This caused a problem and introduced error into the final part of the lab.

Conclusion

We calculated the spring constant for the given ideal spring to be 16. This value is consistent with Hooke’s Law and the second set of data generated by the Newton meter test. The accuracy of the test data suggests that the set up for the experiment was acceptable. More accurate results could be obtained using a more accurate measuring device or a meter stick marked in millimeter increments. A calibrated Newton meter would also improve accuracy.

In the second section of the experiment we calculated the period of oscillation to be 0.72 seconds. This is in line with the expectations we had and compares well with the results from the calculations.

I believe that we were not fully versed on the various components that make up the simple harmonic motion. I feel we skewed the results on this section of the lab because I remember getting the same period for several of the test and group members dismissing the results because the amplitude of the test was greater than the previous trial. Had we researched more on simple harmonic motion we would have discovered before the lab that period was not dependent on amplitude and we would have accepted more of the results. I see clearly where extensive knowledge of your expectations would allow you to receive correct data and sense quicker when the data is out of line so as to re-evaluate your procedure. I think this

would be a very important part of future labs.

We calculated the Kinetic coefficient of friction to be 5.2 x 10 -4.

While the calculation base and theory behind these results seem sound there is a lot of room for improvement on this test scenario. The formula used in the calculation was based on group discussions and input from the instructor. I believe this number to be too small and I feel it does not reflect what the results should be. There was varying results throughout the group. There was also error in the recording of the data. The data set and the way in which we set up the experiment to get the data was flawed. In future experiments I would use a compressed spring scenario. By compressing the spring as opposed to stretching it, I believe more accurate data could be generated. We would also employ accurate stopwatches and a lighter weight. We had significant problems getting usable results with the method we used for the third part of the lab. I would not repeat the same methodology for the third part of the lab should we chose to re-examine the results in part three of the lab. As the results of the third part of the lab do not coincide with what we would expect to see further experimentation to verify or discount the results would be necessary.

References

1.Hirsch, A., Martindale, D., Stewart, C., Barry, M.2003.Nelson Physics 12 3rd ed.Toronto: Nelson

Hirsch and others(2003:212)

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web viewwe suspended a 46 cm spring from the top of a steel rod. the rod screwed into a heavy metal...

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