Question

A pendulum clock is running fast. To correct its time, we should

• A. reduce the mass of the bob.
• B. reduce the amplitude of oscillation.
• C. increase the length of the pendulum.
• D. reduce the length of the pendulum.

Hint:

Remember this straightforward mnemonic rule for pendulum clock maintenance:
Running FAST $\rightarrow$ Needs to slow down $\rightarrow$ INCREASE length ($\uparrow l$).
Running SLOW $\rightarrow$ Needs to speed up $\rightarrow$ DECREASE length ($\downarrow l$).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
A pendulum clock that is "running fast" completes its oscillations too quickly. This means its time period ($T$) is shorter than it should be, causing the clock face to get ahead of real time. To correct the clock and slow it down, we need to increase its time period.

Step 2: Key Formula or Approach:
The periodic time of oscillation $T$ for a simple pendulum swinging at small angles is given by the formula: $$T = 2\pi\sqrt{\frac{l}{g}}$$ Where $l$ represents the effective length of the pendulum suspension cord and $g$ is the local acceleration due to gravity. This formula shows that the time period depends strictly on the length of the pendulum: $$T \propto \sqrt{l}$$ Importantly, the time period is completely independent of both the mass of the hanging bob ($m$) and the angular amplitude of oscillation ($\theta$).

Step 3: Detailed Explanation:
Let's evaluate how to alter the parameters to correct the fast clock: To correct a clock that is running fast, we must slow down its rate of oscillation, which requires increasing the time period ($T$). Since $T \propto \sqrt{l}$, increasing the time period requires increasing the effective length ($l$) of the pendulum wire. Modifying mass or changing amplitude has no effect on the time period according to the core governing formula, which eliminates options (A) and (B). Therefore, the correct modification is to increase the length of the pendulum, which matches the instruction in option (C).

Step 4: Final Answer:
To correct the time, we should increase the length of the pendulum, corresponding to option (C).