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**Unformatted text preview: **Chapter 28 Oscillations: The Simple Pendulum, Energy in Simple Harmonic Motion 197 28 Oscillations: The Simple Pendulum, Energy in Simple Harmonic Motion Starting with the pendulum bob at its highest position on one side, the period of oscillations is the time it takes for the bob to swing all the way to its highest position on the other side and back again. Dont forget that part about and back again. By definition, a simple pendulum consists of a particle of mass m suspended by a massless unstretchable string of length L in a region of space in which there is a uniform constant gravitational field, e.g. near the surface of the earth. The suspended particle is called the pendulum bob. Here we discuss the motion of the bob. While the results to be revealed here are most precise for the case of a point particle, they are good as long as the length of the pendulum (from the fixed top end of the string to the center of mass of the bob) is large compared to a characteristic dimension (such as the diameter if the bob is a sphere or the edge length if it is a cube) of the bob. (Using a pendulum bob whose diameter is 10% of the length of the pendulum (as opposed to a point particle) introduces a 0 . 05% error. You have to make the diameter of the bob 45% of the pendulum length to get the error up to 1%.) If you pull the pendulum bob to one side and release it, you find that it swings back and forth. It oscillates. At this point, you dont know whether or not the bob undergoes simple harmonic motion, but you certainly know that it oscillates. To find out if it undergoes simple harmonic motion, all we have to do is to determine whether its acceleration is a negative constant times its position. Because the bob moves on an arc rather than a line, it is easier to analyze the motion using angular variables. The bob moves on the lower part of a vertical circle that is centered at the fixed upper end of the string. Well position ourselves such that we are viewing the circle, face on, and adopt a coordinate system, based on our point of view, which has the reference direction straight m L Chapter 28 Oscillations: The Simple Pendulum, Energy in Simple Harmonic Motion 198 downward, and for which positive angles are measured counterclockwise from the reference direction. Referring to the diagram above, we now draw a pseudo free-body diagram (the kind...

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