HOOKE'S LAW AND SIMPLE HARMONIC MOTION OBJECT

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1 5 M19 M19.1 HOOKE'S LAW AND SIMPLE HARMONIC MOTION OBJECT The object of this experiment is to determine whether a vertical mass-spring system obeys Hooke's Law and to study simple harmonic motion. THEORY References: Sections 13.1 and 13.3, College Physics, Serway and Vuille The restoring force, F, of an ideal spring is said to obey Hooke s law: F = k x (1) where F is the restoring force exerted by the stretched or compressed spring; k is the spring constant of the spring; x is the displacement of the system from its unstrained length. In the case of the vertical spiral spring and pan used in this experiment, when a mass is added to the pan and the system is lowered to its new equilibrium position, the spring stretches until the restoring force due to the additional stretch of the spring equals the weight of the added mass. The displacement, x, is the displacement of the system from its position before any mass was added, i.e. the displacement from the original equilibrium position. If the mass at the end of the spring is pulled down and released, the system oscillates. For a system in which the restoring force obeys Hooke s Law, the oscillatory motion is called simple harmonic motion. It can be shown that for a body moving in simple harmonic motion, T, the period of the motion (the time for one complete oscillation) is given by EXPERIMENT m T = 2π (2) k Equipment: spring-pan assembly, masses, photogate timer, electronic balance The apparatus consists of a hanging spring, with a pan suspended from it on which masses may be placed. The position of the pan can be read on a millimetre scale printed on a reflective metal plate mounted on the post supporting the spring-pan assembly. Procedure: 1. Record the mass of the pan. 2. Set the spring displacement scale to read zero with no mass on the pan. Add a mass of 50 grams and record the scale reading. The pointer on the pan should be aligned with its reflected image to eliminate any error due to parallax when taking readings. Record the displacement scale readings for load masses of 100 to 350 grams in 50 g increments. Now take off the masses one at a time, confirming that the displacement scale readings are the same as previously recorded within experimental error. Checkpoint 1 ask the TA to review your data. 3. With the load mass of 350 grams, start the mass-spring system oscillating in the vertical direction (try to avoid any pendulum-like swing) and use the photogate to measure the time for 10 complete oscillations (i.e. 10 periods), following the procedure provided during your lab period. 4. Repeat the above measurement (time for 10 periods) for masses of 300, 250, 200, and 150 grams. Checkpoint 2 ask the TA to review your data.

2 ANALYSIS 5. Calculate the force exerted by the spring (= m load g) and the error in the spring force. 6. Plot a graph of the force exerted by the spring versus the displacement x. 7. Determine the spring constant, k, of the spring from the graph. 6 Checkpoint 3 ask the TA to review your work for steps 5 to 7. In equation (2) the mass m is the total mass that is undergoing oscillations. M19.2 Thus we will use m = m pan + m load (3) 8. Referring to equation (2), plot the measured period versus an appropriate function of m to show that your mass-spring system is undergoing simple harmonic motion. 9. Using the results from your graph of period as a function of oscillating mass, perform the necessary calculations to attempt to verify equation (2). Be sure to clearly state what you are calculating and why. CONCLUSION Checkpoint 4 ask the TA to review your work for steps 8 and Do your results confirm that your mass-spring system obeyed Hooke s Law? Explain. 11. Explain clearly in your own words whether you feel justified in concluding that your mass-spring system is an example of a system that can undergo simple harmonic motion? SOURCES OF ERROR Checkpoint 5 ask the TA to join your discussion of the Conclusion. 12. Consider the effects of any assumptions made regarding the spring. 13. Consider the effects of additional forces on the pan. 14. Was the motion of the pan purely vertical? If not, consider the effects of the observed motion. 15. The mass of the pan was added to the load mass to obtain the total mass that was oscillating. Was any other mass ignored in this calculation? If so, specify and discuss. Checkpoint 6 ask the TA to join your discussion of the Sources of Error.

3 M19 7 HOOKE'S LAW AND SIMPLE HARMONIC MOTION DATA & RESULTS mass of pan, m pan = _ ± kg Load Mass m load (kg) Spring Force (m load g) (N) Force, Displacement, and Oscillation Data for Spring Scale Reading x (cm) (± cm) T 10, time for 10 periods (s ± s) Avg. Period, T (=T 10 /10) (s) (± s) Oscillating Mass (m load + m pan ) (kg) ± ± ± ± ± ± ± ± ±

4 Spring Force versus Displacement Spring Force (N) Displacement from Equilibrium, x (cm)

5 Measured Period versus Period (s) ( )

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