Simple Harmonic Motion

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  • Topic: Simple harmonic motion, Oscillation, Phase
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  • Published : May 8, 2013
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Simple Harmonic Motion
Ethan Albers
Case Western Reserve University, Department of Physics
Cleveland, OH 44106
Abstract:
In this lab, my partner and I observed oscillations that were translational and rotational. The two forms we studied must have a form of a restoring force that is proportional to the displacement of the object from its point of equilibrium. This produces the harmonic motion which this lab wants. At small and big amplitudes we measured/observed the translational oscillation of the spring. To go with this also, we measured it when it had the spring had added mass and when it didn’t have added mass. Additional to this, we were able to measure the rotational oscillation of a Torsion pendulum that was rotating on its central axis. With this data we created a sine curve to display the oscillating effect which was made possible by using the translational oscillation. After that, my partner and I created a histogram that displayed the different lengths of the period of oscillation. This histogram used the Torsion pendulum to make the graph. In both of these mini labs, they displayed the principle of the oscillating effect that is produced by a restoring force. Spring Mass Oscillator:

Introduction and Theory:
The way translational harmonic motion is illustrated is by the oscillation with the spring. The compression and the extension of the spring while it oscillates, shows there is a force that is applied relative to the equilibrium position, x0, and then at a different position x.

(1)
In this equation, K represents the spring constant of the given spring. Additionally, in the lab we added a mass, m, to the system. So we can use an equation that will illustrate the harmonic motion:

(2)
(3)
(4)
In the equation we have the amplitude which is represented by A, and then the frequency of the spring which is, then the phase angle which is, and then lastly x0 which is the equilibrium position. The way we figure out is by determining the starting point and its relative position to x0. We know that at equilibrium the value of =. So with this information we can write the equation that connects frequency with period of the spring, T, by the equation of:

(5)

This is an example of the spring that has an attached mass with a motion sensor Experimental Procedure:
To begin with this in our lab, we tested the motion detector to make sure it was working. We did this by using a notebook to go all around it basically. Based on this we then made the appropriate adjustment to the spring so that it would be more accurate for our data. After that, we used the hanger and added it to the spring and placed it a couple of cm down to analyze a smaller oscillation. Then after using Logger Pro we were able to collect the results and produce a sine graph. Going off of the sine graph we made, we were able to estimate the parameters that are used in the equation two in order to be able to reproduce an accurate graph of the oscillation. We did this multiple times until we were able to come up with an accurate representation of the original sine graph. In order to account for the error analysis, we performed three more trials with having the hanger at small amplitude. After that we used a large amplitude oscillation and also just the hanger, and finally we attached the 50 g to the hanging mass and ran another trial of the oscillation lab. Results and Analysis:

The duplication of the graph of the sine curve and the original one are found at the end of this lab report as graph 1. The parameters have estimation based off of the sine curve and the values that are from this lab sine curve are as followed: Parameter| Estimate| Experimental Result|

A| .11| .05186|
ω| 10.5| 10.86|
φ| 39 | 5.356|
X0| .75| .7790|

Then the values of the small amplitude oscillation test runs have the results below: Parameter| Trial 1| Trial 2| Trial 3...
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