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Chapter 1: Problem 3

A pendulum clock is taken to a repair shop. Its pendulum is replaced by a shorter one that oscillates with a smaller period than the original. What effect, if any, does this have on how the clock runs?

Short Answer

Expert verified
Answer: Replacing a pendulum in a clock with a shorter one will cause the new pendulum to have a smaller period and oscillate faster. As a result, the clock will tick more frequently and run faster than before, leading to inaccurate timekeeping.

Step by step solution

01

Understand the pendulum motion in a pendulum clock

A pendulum clock works on the principle of a simple pendulum. A simple pendulum consists of a weight (called the bob) attached to a string or rod, which oscillates back and forth. In a pendulum clock, each oscillation of the pendulum represents one tick, and a certain number of ticks account for one second.
02

Formula for the period of a simple pendulum

The formula for the period (T) of a simple pendulum is given by: T = 2π √(L/g), where L is the length of the pendulum and g is the acceleration due to gravity (approximately 9.81 m/s²).
03

Shorter pendulum and its effect on the period

Since the pendulum has been replaced by a shorter one in the clock, we need to analyze the effect of this change on the period of oscillation. According to the formula, as L (length) decreases, the period T also decreases. Therefore, the new pendulum with a shorter length will have a smaller period, meaning it will oscillate faster.
04

Effect on the clock's timekeeping

Since the new pendulum has a smaller period, it will oscillate faster, causing the clock to tick more frequently. As a result, the clock will now run faster than it did with the original pendulum, leading to inaccurate timekeeping.

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