Question

An insulated chamber at a height $h$ above the earth’s surface and maintained at ${30}^{\circ }C$ has a clock fitted with an uncompensated pendulum. The maker of the clock for the chamber mistakenly designs it to maintain correct time at ${20}^{\circ }C$ at that height. It is found that if the chamber were brought to earth’s surface the clock in it would click correct time at ${30}^{\circ }C$. The coefficient of linear expansion of the material of pendulum is (earth’s radius is $R$):

• A. $\frac{R}{5h}$
• B. $\frac{5R}{h}$
• C. $\frac{h}{5R}$
• D. $\frac{h}{20R}$

Answer 1

The correct option is C $\frac{h}{5R}$

Time period at height $H$ = Time period at ground level

$t_G=t_H$

$2\pi \sqrt{\frac{l_0}{g_0}}=2\pi \sqrt{\frac{l^{\prime }}{g^{\prime }}}$

$\Rightarrow \frac{l_0}{g_0}=\frac{l^{\prime }}{g^{\prime }}$

Here, $l^{\prime }=l_o(1+\alpha \Delta T)$
$g^{\prime }=g_0(1-\frac{2h}{R})$ (because $h<<R$)

$\Rightarrow \frac{l_0}{g_0}=\frac{l_o(1+\alpha \Delta T)}{g_0(1-\frac{2h}{R})}$

$\Rightarrow 1=\frac{1+\alpha \Delta T}{1-\frac{2h}{R}}$

$\Rightarrow 1-\frac{2h}{R}=1+\alpha (30-20)$

$\Rightarrow \alpha =\frac{h}{5R}$