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: 31

Answer

It should be shortened by 0.5 mm.

Work Step by Step

In a day, the clock should count off 86400 seconds. Instead, it counts off only 86400-21 = 86379 seconds. Therefore, the current period is too long by a factor of 86400/86379, and the pendulum should be shortened to make it swing more quickly. $$T_{new}=\frac{86379}{86400}T_{old}$$ The period of a pendulum is given by $T=2 \pi \sqrt{\frac{\mathcal{l}}{g}}$. $$2 \pi \sqrt{\frac{\mathcal{l}_{new}}{g}}=\frac{86379}{86400}2 \pi \sqrt{\frac{\mathcal{l}_{old}}{g}}$$ $$\mathcal{l}_{new}=(\frac{86379}{86400})^2\mathcal{l}_{old}$$ $$\mathcal{l}_{new}=(\frac{86379}{86400})^2(0.9930m)=0.9925m$$ The pendulum should be shortened by 0.0005m = 0.5 mm.
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