Note that changes as the pendulum moves 88 the term

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Unformatted text preview: o the center of mass of the bob is l , and the angular distance of the pendulum from the vertical is θ . (Note that θ changes as the pendulum moves.) 88 The term “simple pendulum” refers to a point mass hanging from a massless rod, which swings back and forth when it is displaced from its equilibrium position (straight down). We will analyze the motion of the simple pendulum by considering the torque generated by gravity about the pivot point. τ pivot = I α = ￿ × ￿ . rF (5) where I = m l 2 is the bob’s moment of Inertia about the pivot, α is the angular acceleration of the pendulum, and ￿ × ￿ = −m g sin θ . rF Solving equation 5 for α , we obtain α =− g sin(θ ). l If θ is small, we can make the approximation1 that sin(θ ) ≈ θ , which leads to (6) g α ≈ − θ. l (7) There is a direct parallel between Equation 7 and Equation 1, so we can see that in the case of the simple pendulum, ￿ g ω= . (8) l Problem 10.3 The angular frequency of oscillation in simple harmonic motion usually depends on two terms: one related to the restoring force and another related to the inertia of the oscillating mass. In the case of the simple pendulum, does g represent the restoring force or the inertia? How about l ? One interesting feature of this equation is that it shows us that, provided oscillations are small (we can make the approximation shown in Equation 6), the frequency of oscillation of the pendulum does not depend on the mass of the pendulum bob or on the amplitude of the pendulum’s oscillation. Problem 10.4 As the length of a pendulum’s string goes up, does the angular frequency of its oscillation go up or down? Does the period of its oscillation go up or down? 3 5 can be seen most easily by examining the Taylor series expansion of sin θ = θ − θ + θ − . . . and letting 3! 5! θ → 0. Which term goes to zero most slowly? 1 This 89 Problem 10.5 Does ω depend on how far back you pull the pendulum bob initially, provided you can still make the approximation in...
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This note was uploaded on 09/13/2011 for the course ECONOMICS 101 taught by Professor Gerson during the Spring '11 term at University of Michigan.

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