Question

A clock with a brass pendulum shaft keeps correct time at a certain temperature,
How closely must the temperature be controlled if the clock is not to gain or lose more than $1s$ a day? $\left({\alpha =2\times {10}^{-5}}^{\circ }{C}^{-1}\right)$

• A. $|\theta |<{1.1574}^{\circ }C$.
• B. $|\theta |<{2.1574}^{\circ }C$.
• C. $|\theta |<{13.1574}^{\circ }C$.
• D. $|\theta |<{3.1574}^{\circ }C$.

Answer 1

The correct option is A $|\theta |<{1.1574}^{\circ }C$.

Number of seconds lost or gained per day$=\frac{1}{2}\alpha \theta \times 86400$,
where $\theta =$rise or drop in temperature; $\alpha =$coefficient of linear expansion of shaft.
We want that $|\frac{1}{2}\alpha \theta \times 86400|<1$
$⟹|\theta |<\frac{2}{2\times {10}^{-5}\times 86400}⟹|\theta |<{1.1574}^{\circ }C$
Hence temperature should not increase or decrease by more than ${1.1574}^{\circ }C$. It does not depend upon time period.