Simple Harmonic Motion

If we replace l with h in equation 44 we can write

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Unformatted text preview: . If we replace L with h in equation (44), we can write the period as 8 Figure 8 A diagram of a physical pendulum with the corresponding forces acting on the pendulum. http://upload.wikimedia.org/wikipedia/commons/thumb/3/ 39/Physical-Pendulum-Labeled-Diagram.png (48) 2 As with the simple pendulum, I is the rotational inertia of the pendulum about the pivot point. However, I is NOT simply mh2 (it depends on the shape of the physical pendulum), but it is still proportional to m. A physical pendulum will not swing if it pivots at its center of mass. Formally, this corresponds to putting h=0 into equation (48). That equation then predicts that the period goes to infinity, which implies that such a pendulum will never complete one swing. Corresponding to any physical pendulum that oscillates about a given pivot point with period T is a simple pendulum of length Lo with the same period T. We can find Lo from equation (47). The point along the physical pendulum at distance Lo from the pivot point is called the center of oscillation of the physical pendulum. 2.5.3 Measuring the Acceleration of Gravity using a Physical Pendulum We can use a physical pendulum to measure the free-fall acceleration g at a particular location on Earth’s surface. To analyze a simple case, take the pendulum to be a uniform rod of length L, suspended from one end. For such a pendulum, h in equation (48), the distance between the pivot point and the center of mass is 1 2 (49) Recall that in, Table 1 from the angular motion lab the rotational inertia of a long thin rod about a perpendicular axis through its center of mass is 1 12 (50) From the parallel-axis theorem, (51) we find that the rotational inertia about a perpendicular axis through one end of the rod is 1 12 1 2 1 3 (52) (53) Placing equations (49) and (53) into equation (48) and solving for g we find that (54) 3 Thus by measuring L, and the period T or frequency, we can find the value of g at the pendulum’s location. (If precise measurements are to be made, a number of refinements are needed, such as swinging the pendulum in an evacuated chamber.) 9 2.6 Pendul...
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This document was uploaded on 03/20/2014 for the course PHYS 215 at Lafayette.

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