Question

A uniform metal rod is used as a bar pendulum. If the room temperature rises by ${10}^{\circ }C$, and the coefficient of linear expansion of the metal of the rod is $2\times {10}^{-6}per{}^{\circ }C$, the period of the pendulum will have percentage increase of

• A. $-2\times {10}^{-3}$
• B. $-1\times {10}^{-3}$
• C. $2\times {10}^{-3}$
• D. $1\times {10}^{-3}$

Answer 1

The correct option is D $1\times {10}^{-3}$
the fractional increment in time with Temperature change
$\frac{\Delta t}{t}=\frac{1}{2}\alpha \Delta T$ eq(1)
where
$\frac{\Delta t}{t}$ = fractional increment
$\alpha$ = coefficient of linear expansion
$\Delta T$= change in temperature
$\Delta T$ = ${10}^{o}C$
$\alpha$ = $2\times {10}^{-6}pe{r}^{o}C$

by eq(1)
$\frac{\Delta t}{t}$ = $\frac{1}{2}\times 2\times {10}^{-6}\times \left(10\right)$
$\frac{\Delta t}{t}$ = ${10}^{-5}$
percentage change of period
%$\frac{\Delta t}{t}$ = $\frac{\Delta t}{t}\times 100$
= ${10}^{-5}\times 100$
= ${10}^{-3}$ = $1\times {10}^{-3}$
so D is correct.