Question

A clock regulated by a second pendulum keeps the correct time. During winter the length of the pendulum decreases by 1 %. How much will the clock gain or lose in one day?

Answer 1

Data: L' = (1 − 0.01) L_s = 0.99 L_s, g = 9.8 m/s²

T = `2π sqrt(L/g)`

∴ At a given place (constant g), T ∝ `sqrtL`

The period of a second pendulum is T_s = 2 s

∴ `(T')/T_s = sqrt((L')/L_s)`

= `sqrt0.99`

= `sqrt(99 xx 10^-2)`

= 9.950 × 10⁻¹

∴ T' = 0.995 T_s = 0.995 × 2 = 1.99 s

Hence the given pendulum clock will gain

2 − 1.99 = 0.01 s every 1.99 s in winter.

∴ Time gained in 24 hours

log 24         1.3802
log 36      + 1.5563
                   2.9365

log 1.99   − 0.2989
                   2.6376
antilog 2.6376  = 434.1

= `(24 xx 3600 xx 0.01)/1.99`

= 434.1 s

= 7 min 14.1 s