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Chapter 10: Problem 12

\(\underline{Q C}\) A grandfather clock is controlled by a swinging brass pendulum that is \(1.3 \mathrm{~m}\) long at a temperature of \(20^{\circ} \mathrm{C}\). (a) What is the length of the pendulum rod when the temperature drops to \(0.0^{\circ} \mathrm{C}\) ? (b) If a pendulum's period is given by \(T=2 \pi \sqrt{L / g}\), where \(L\) is its length, does the change in length of the rod cause the clock to run fast or slow?

Short Answer

Expert verified
a) The length of the pendulum rod at 0.0掳C is slightly less than 1.3 m due to thermal contraction. b) The clock runs faster due to the decreased period of the pendulum.

Step by step solution

01

- Calculate Change in Length due to Lower Temperature

We'll calculate the change in length of the pendulum rod due to the changed temperature. The formula for linear thermal expansion we need to use is \(\Delta L = L_o \cdot \alpha \cdot \Delta T\), where \(\Delta L\) is the change in length, \(L_o\) is the initial length (1.3 m in our case), \(\alpha\) is the coefficient of linear expansion for brass (\(\approx 1.9 \cdot 10^{-5}K^{-1}\)), and \(\Delta T\) is the change in temperature (20 - 0 = 20掳C).
02

- Calculate New Length of the Pendulum Rod

To find the new length of the pendulum rod at 0掳C, we subtract the change in length from the original length, \(L = L_o - \Delta L\).
03

- Determine Pendulum's Period Change

Next, we analyze the change in pendulum鈥檚 period due to the change in length using the formula \(T=2 \pi \sqrt{L / g}\), where \(T\) is the period, \(L\) is the length and \(g\) is the acceleration due to gravity. Since the period depends on the square root of the length, when the length decreases, the period also decreases.
04

- Interpretation of the Clock Speed

In the last step, we conclude that when the period of the pendulum decreases, the clock will run faster than normal because the pendulum controls the motion of the clock. Therefore, the change in length of the rod will cause the clock to run fast.

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