Physics: Principles with Applications (7th Edition)

Published by Pearson
ISBN 10: 0-32162-592-7
ISBN 13: 978-0-32162-592-2

Chapter 11 - Oscillations and Waves - Problems - Page 324: 34

Answer

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Work Step by Step

Suppose the pendulum starts at maximum displacement. The equation of motion is known. $$\theta=\theta_o cos \omega t=\theta_o cos 2 \pi f t $$ The initial angle $\theta_o $ is given as 12 degrees, but we may convert that to radians, and find an expression for the angular displacement in radians at any time t. $$\theta=(12^{\circ})\frac{\pi \;rad}{180^{\circ}} cos 2 \pi (2.5\;Hz) t $$ $$\theta=\frac{\pi}{15} cos 5 \pi t $$ Now, evaluate this at the 3 times that are stated. a. $$\theta(0.25\;s)= \frac{\pi}{15} cos 5 \pi (0.25\;s)=-0.15\;rad$$ b. $$\theta(1.60\;s)= \frac{\pi}{15} cos 5 \pi (1.60\;s)=\frac{\pi}{15}\;rad$$ This makes sense, because the time is exactly 4 periods, so the pendulum has returned to its starting point. c. $$\theta(500\;s)= \frac{\pi}{15} cos 5 \pi (500\;s)=\frac{\pi}{15}\;rad$$ This makes sense, because the time is exactly 1250 periods, so the pendulum has returned to its starting point.
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