First slide

Thermal expansion

Question

An insulated chamber at a height h above the earth's surface and maintained at $30^{0} C$ has a clock fitted with an uncompensated pendulum. The maker of the clock for the chamber mistakenly designs it to maintain correct time at $20^{0} C$ 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 $30^{0} C$ . The coefficient of linear expansion of the material of pendulum is (earths radius is R)

Moderate

A

$\frac{h}{19 R}$
B

$\frac{5 R}{h}$
C

$\frac{h}{5 R}$
D

$\frac{h}{20 R}$

Solution

Clock is designed to indicate correct time at $20^{0} C$ at height ‘h’. It will indicate correct time at $30^{0} C$ on the ground in in this case the time period is same as the earlier.
$∴ 2 π \sqrt{\frac{L}{g_{h}}} = 2 π \sqrt{\frac{L^{'}}{g_{s}}}$
here $L^{'} = L (1 + α 10), g_{s_{}} = \frac{GM}{R^{2}}$
and $g_{h} = \frac{GM}{{(R + h)}^{2}} \Rightarrow \frac{L}{g_{n}} = \frac{L}{g_{s}}$
$\Rightarrow L {(R + h)}^{2} = L (1 + α 10) R^{2}$
$\Rightarrow 1 + \frac{h}{R} = {(1 + α 10)}^{\frac{1}{2}}$
$\Rightarrow 1 + \frac{h}{R} = 1 + 5 α (by binomial expansion)$
$\Rightarrow α = \frac{h}{5 R}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App