Seconds pendulum that gains 2 minutes per day A Second’s pendulum (period ≈ 2 seconds) gains 2 minutes in one day. To make it keep correct time, how should its effective length be adjusted?

Introduction / Context:
Pendulum clocks depend on the simple pendulum relation between period and length. Timekeeping errors arise if the period deviates from its calibrated value. Understanding how to correct the period through length adjustment is a classic dynamics application.

Given Data / Assumptions:

• Second’s pendulum nominal period T0 ≈ 2 s.
• Clock gains 2 minutes per day → it runs fast → period is too short.
• Small oscillations; air resistance and amplitude effects neglected.

Concept / Approach:

The ideal pendulum period is T = 2π * √(L/g). Period increases with length. If the clock gains time, the pendulum beats too quickly; therefore, increase the length to slow it down. Changing the bob mass does not change T (for small oscillations), since mass cancels out of the period expression.

Step-by-Step Solution:

Verification / Alternative check:

Quantitatively, if the error is small, ΔT/T ≈ (1/2) ΔL/L. A positive ΔT requires positive ΔL, confirming the qualitative reasoning.

Why Other Options Are Wrong:

(a) Decreasing length makes the period shorter, worsening the gain. (c) and (d) changing mass does not affect period for small swings. (e) is unnecessary because a correct mechanical adjustment exists.

Common Pitfalls:

Confusing amplitude effects or believing bob weight affects period; ignoring that friction/escapement mainly influences amplitude, not the ideal period.

Final Answer:

Must be increased