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LAB# 15 SPRING CONSTANT AND SIMPLE HARMONIC MOTION (SHM) Introduction: Any system that has inertia and elasticity will return to its original shape after being slightly deformed and, can execute harmonic motion. All such systems experience restoring forces that push or pull them back to their equilibrium position or configuration. A mass attached to a vertical spring is one such system. In this case the restoring force is proportional to the masses displacement from equilibrium. Such a system executes simple harmonic motion and the restoring force of the spring follows Hooke’s law, assuming of course that the displacements from equilibrium are not too large. Here the distance x is understood to mean the distance the mass has been displaced from its equilibrium position. The quantity k is called the spring constant and is a measure of the strength of the spring. Once the spring is manufactured k is fixed, it depends on the material of the spring, the size and shape of its coils, and other physical properties of the spring. You should note the negative sign in Equation (1). This says that the force due to the spring is always in a direction opposite to the displacement of the mass from its equilibrium position, thus the spring force is a linear restoring force. Our objectives in this experiment are to measure the spring constant, k , for our spring and to compare the theoretical prediction for the period of oscillation of the mass-spring system with direct measurement of the period. Equipment: Coil spring, “Lab#15 SHM” software application, dual range force sensor, motion sensor, platform balance and stand, LabPro interface, and set of masses with hanger. Theory: Applying Newton’s second law to our system, and using Equation (1) for the force on the mass by the spring it can be shown that the system will execute simple harmonic motion with a period given by, , 2 k m T (2) where m is the mass of the object attached to the end of the spring. As it turns out, the mass of the spring itself does affect the motion of the system. A typical estimate for the effective mass of , kx F (1) Figure 1: Equipment for "Spring Constant and Simple Harmonic Motion" experiment showing the force sensor, motion sensor, spring, and hanging masses.
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the spring is to take 1/3 of its mass. This makes our theoretical prediction for the period of motion of our system, , ) 3 / ( 2 k M m T S
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This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.

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