### Simple Harmonic Motion Dominic Stone Lab Partner

Experiment 1: Simple Harmonic Motion Dominic Stone Lab Partner: Andrew Lugliani January 9, 2012 Physics 132 Lab Section 13 Theory For this experiment we investigated and learned about simple harmonic motion. To do this we hung and measured different masses on a spring-mass system to calculate the force constant k. Simple harmonic motion is a special type of periodic motion. It is best described as an oscillation motion that causes an object to move back-and-forth in response to a restoring force given by Hooke’s Law: 1) F=-kx Where k is the force constant.

Then using two different procedures, we calculate the value of the force constant k of a spring in our oscillating system. We observed the period of oscillation and use this formula: 2) T=2(m/k) Then we reduced the equation to solve for the value of k by: 3) k=4^2/slope “Slope” represents the slope of the graph in procedure B. k is then the measure of the stiffness of the spring. We can then compare k to that of a vertically stretched spring with various masses M. By using the following equation: 4) Mg=kx Where x is the distance of the stretch in the spring.

To find the value of the constant k we take the data from procedure A and graph it. Using this graph, we use equation: 5) k=g/slope We can compare the two values of the constant k. Both values should be exact since we used the same spring in both procedures. Here simple harmonic motion is used to calculate the restoring force of the spring-mass system. Procedure Part A: First, we set up the experiment by suspending the spring from the support mount and measured the distance from the lower end of the spring to the floor.

After, we hung 100 grams from the spring and measured the new distance created from the stretch of the spring. We then repeated this step for masses 200 to 1000 grams, by increasing the weight by 200 grams each time. Then we took this data and plotted them on a graph with suspended weight Mg versus elongation x. After plotting this data we were then able to evaluate the force constant k from the slope of the graph. Part B: First, we suspend 100 grams from the spring and let it lay at rest.

When the spring was naturally set in its equilibrium position, we slightly pulled down the weight and recorded the time it took for the weight to complete 10 oscillations and calculated the average period of each oscillation. We then repeated this process for masses 100 to 1000 grams by increasing the weight by 100 grams each time. After we completed this process we plotted a graph of T^2 verses suspended mass m with the data. When then found the intercept at T^2=0 to see how it would compare with the value of negative one-third the mass of the spring.

We then also determined the spring constant k by calculating the slope of the graph and compared it with the spring constant k in part B. Data Part A: Mg(Kg/s^2)| X(m)| 1. 96| 0. 39| 3. 92| 0. 63| 5. 88| 0. 86| 7. 84| 1. 11| 9. 8| 1. 36| Part B: M(Kg)| T (s)| T(s)| T^2(s^2)| 0. 1| 8. 24| 0. 824| 0. 679| 0. 2| 9. 87| 0. 987| 0. 974| 0. 3| 12. 74| 1. 274| 1. 623| 0. 4| 14. 57| 1. 457| 2. 123| 0. 5| 16. 23| 1. 623| 2. 634| 0. 6| 17. 49| 1. 749| 3. 059| 0. 7| 19. 21| 1. 921| 3. 69| 0. 8| 20. 26| 2. 026| 4. 105| 0. 9| 21. 69| 2. 169| 4. 705| 1| 22. 89| 2. 289| 5. 24| Data Analysis