A pendulum clock gains 5 min/day — how should length be changed

A clock gains time means it runs fast — it completes more oscillations per day than required. If it gains 5 minutes in one day (86400 seconds), it ticks through 86400 seconds of time in only $86400 - 300 = 86100$ actual seconds.

The clock’s period $T_1$ is slightly less than the correct period $T_0 = 2$ seconds (for a seconds pendulum). Since the clock is fast, it is completing cycles too quickly — the period is too short.

The time period of a simple pendulum is:

This shows $T \propto \sqrt{L}$. If we want to increase the period (slow the clock down), we need to increase the length.

This is the physical intuition: a longer pendulum swings more slowly, taking more time per oscillation, so the clock runs more slowly.

Let $T_0$ = correct period, $T_1$ = current (too short) period.

The clock gains 5 minutes in 24 hours = 86400 seconds. This means in $T_1$ seconds (one correct day), the clock shows 86400 + 300 = 86700 seconds.

So the ratio:

Wait — let’s think carefully. The clock completes $\dfrac{86700}{T_0}$ oscillations in one day (86400 actual seconds). For correct time, it should complete $\dfrac{86400}{T_0}$ oscillations.

The actual ratio:

Since the clock gains, it does more oscillations → shorter period → $T_1 < T_0$ ✓

Using $T = 2\pi\sqrt{L/g}$:

We need the new length $L_{new}$ such that $T_{new} = T_0$:

The length should be increased by approximately 0.69% of its current length.

To slow the clock down (from gaining to correct):

• Increase the length of the pendulum by about 0.69% of its current length.

In a typical grandfather clock, this is done by lowering the bob (the heavy weight at the bottom of the pendulum) using an adjusting screw.