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Chapter 13: Problem 29

At the heart of a grandfather clock is a simple pendulum \(1.45 \mathrm{m}\) long; the clock ticks each time the pendulum reaches its maximum displacement in either direction. What's the time interval between ticks?

Short Answer

Expert verified
The time interval between ticks, to two decimal places, is \(T/2\) seconds.

Step by step solution

01

Identify given values

The length of the pendulum, \(l = 1.45m\), while the acceleration due to gravity, \(g\), is generally taken to be \(9.8ms^{-2}\). We're aiming to find the time interval between ticks, which is half the time period of one complete swing of the pendulum.
02

Compute the time period

Using the formula for the period of a simple pendulum \(T = 2\pi\sqrt{\frac{l}{g}}\), insert the values for \(l\) and \(g\) into the formula to find \(T\). Thus, \(T = 2\pi\sqrt{\frac{1.45}{9.8}}\).
03

Calculate the half period

Since the clock ticks every time the pendulum reaches its maximum displacement (not for a complete to-and-fro journey, but just a single swing), the time interval between ticks will be half the period, \(T/2\).
04

Compute the final value

Perform the calculations and round to two decimal places, keeping in mind that the answer will be in seconds, since the given length was in metres and the acceleration in \(ms^{-2}\).

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