A pendulum clock loses 12 seconds a day if the temperature is 40∘C and goes fast by 4 seconds a day if the temperature is 20∘C. Find the temperature at which the clock will show correct time, and the coefficient of linear expansion of the metal of the pendulum shaft.
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Let T be the temperature at which the clock is correct.
Time lost per day =
12 × a × (rise in temperature) × 86400 s, (see hint)
⇒ 12 = 12 × a × (40−T) × 86400. ...........(1)
Time gained per day -
12 × a × (rise in temperature) × 86400 s,
⇒ 4 = 12 × a × (T−20) × 86400. ...........(2)
On adding (1) and (2), we get
64 = 86400 × a × (40−20)
⇒ a = 1.85 × 10−5/∘C.
On dividing (1) and (2), we get
12(T−20) = 4(40−T)
⇒ T = 25∘C
⇒ clock shows correct time at 25∘C only!