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### 132-10W10_SHM_

Course: PHYS 132, Spring 2012
School: Cal Poly
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Word Count: 974

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EXPERIMENTS PHYSICS 132 10-1 Experiment 10 Simple Harmonic Motion of a Mass Suspended from a Spring In this experiment you investigate the behavior of a simple physical system consisting of a mass hanging on the end of a spring. Although mechanically simple, this system is important because it exhibits repetitive motion. The mass oscillates! The frequency of oscillation depends on physical properties of the...

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EXPERIMENTS PHYSICS 132 10-1 Experiment 10 Simple Harmonic Motion of a Mass Suspended from a Spring In this experiment you investigate the behavior of a simple physical system consisting of a mass hanging on the end of a spring. Although mechanically simple, this system is important because it exhibits repetitive motion. The mass oscillates! The frequency of oscillation depends on physical properties of the system; the masses in motion and the spring stiffness. You determine whether Hookes Law describing an ideal spring applies to your system and determine the spring constant of your spring by both static and dynamic means. Preliminaries. Part A. Static Determination of Spring Constant. A spring with spring constant k hangs vertically msp k with a mass m attached to the lower end, as in Figure 1. Figure 1. Hanging a mass on a spring An external force F applied to a spring stretches the spring a distance x. Hookes Law states that, for an ideal spring, the applied force and stretch are linearly related F = kx (eq. 1) where k is the spring constant, a measure of spring stiffness. The stretched spring exerts an (equal and opposite) elastic restoring force, Fs = -kx, in the opposite direction. If the mass is stationary, in equilibrium, the elastic restoring force must balance the masss weight, mg = kx (eq.2) where g is the acceleration due to gravity. Part B. Dynamic Determination of Spring Constant. If the mass in the system of Figure 1 is pulled away from its equilibrium position and then released, it will oscillate up and down about its equilibrium position. Motion in which the restoring force is proportional to the displacement but oppositely directed is called simple harmonic motion. In this experiment, the mass of the spring msp is comparable to that of the hanging mass m. This spring mass is usually ignored in model mass-spring systems discussed in textbooks. Realistically, the spring oscillates along with the hanging mass in a complicated way. Only the lowest part of the spring oscillates with the amplitude of the attached mass, while coils near the top oscillate through only a very small amplitude. For this reason, the effective mass of the system is increased by some fraction f of the spring mass msp of the spring. (The fraction f is theoretically determined to be one third (f=1/3) for an ideal spring.) The period of oscillation T is theoretically related to mass-spring system parameters by ( m + fmsp ) T = 2 (eq. 3) k Squaring eq. 3 gives (eq.4) T 2 = (4 2 / k ) m + (4 2 / k ) fmsp Eq. 4 is the equation of a straight line on a graph of squared period vs. hanging mass in standard slopeintercept form. In this experiment, the period is measured for different hanging masses. The graph of squared period vs. hanging mass is constructed, and the 10-2 PHYSICS EXPERIMENTS 132 spring constant and mass fraction determined from the graph. Procedure. Questions (Answer clearly and completely). Part Static A. Determination of Spring Constant. Clamp a long vertical pole to the edge of the lab table. Connect a horizontal bar near the top of the pole and attach one end (the narrower one) of a spring to the end of the horizontal bar. The spring should hang with its narrower end on top so that, as it stretches, it becomes uniform in width. Hang at least five different weights on the spring Measure the amount of spring for each. Make certain not to damage the spring by stretching it beyond its elastic limit. Make a graph with the spring stretch on the vertical axis and the hanging mass on the horizontal axis. Determine the spring constant from the graph. Part B. Dynamic Determination of Spring Constant. Measure and record the mass of the spring msp. Attach 100g total mass to the unattached end of the spring, using the weights mounted on the weight hanger. Record the total hanging mass m. Do not forget to add the hanger mass! Pull the mass several centimeters from equilibrium and release. Start the timer and time five oscillations. Divide by five to determine the period of oscillation. Repeat several times to insure consistency and determine precision of data. Record. Add mass in 50g increments up to about 400g (do not exceed 400g) and determine the period for each hanging mass. Be very careful that the spring is not stretched excessively, where it will be permanently deformed and not able to snap back! Graph the data with the period on the vertical axis and hanging mass on the horizontal axis. Make another graph with period squared on the vertical axis and hanging mass on the horizontal axis. Calculate the spring constant and the effective mass fraction from the appropriate graph. 1. Does your spring obey Hookes Law? 2. Equation 4 predicts a straight line if period squared (on the vertical) is plotted against hanging mass (on the horizontal). What is the equation for the slope of the line in terms of mass-spring parameters (k, fmsp). What is the equation for the vertical axis intercept in terms of mass-spring parameters (k, fmsp). Derive an equation for the horizontal axis intercept in terms of mass-spring parameters (k, fmsp). 3. Is the suggested technique for determining the oscillation period a good one? Would it be better to time a single oscillation? Would it be better to time a hundred oscillations and divide by a hundred? 4. What value do you determine for the spring constant statically? What value do you determine for the spring constant dynamically? These are supposed to be the same. What is the percent difference between these values? How do you account for any differences? 5. What value do you determine experimentally for the effective mass fraction of the spring? What is the percent error between this value and the theoretical result of 1/3? 6. How would the period squared vs mass graph be different from the one you plotted if the spring were stiffer?
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