An insulated chamber at a height h above the earth's surface and maintained at $$300C$$ has a clock fitted with an uncompensated pendulum. The maker of the clock for the chamber mistakenly designs it to maintain correct time at $$200C$$ 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 $$300C$$. The coefficient of linear expansion of the material of pendulum is (earths$$ radius is $$R)

Question

An insulated chamber at a height h above the earth's surface and maintained at $$300C$$ has a clock fitted with an uncompensated pendulum. The maker of the clock for the chamber mistakenly designs it to maintain correct time at $$200C$$ 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 $$300C$$. The coefficient of linear expansion of the material of pendulum is (earths$$ radius is $$R)

Infinity Learn · Accepted Answer

Clock is designed to indicate correct time at $$200C$$ at height ‘h’. It will indicate correct time at $$300C$$ on the ground in in this case the time period is same as the earlier.∴ $$2$$$$π$$Lgh $$=$$ $$2$$$$π$$L'gshere L' $$=$$ $$L(1+$$α$$10)$$, gs  $$=$$ $$GMR2and$$ gh $$=$$ $$GM(R+h)2$$ ⇒ Lgn $$=$$ Lgs⇒ $$L(R+h)2$$ $$=$$ $$L(1+$$α$$10)R2$$⇒ $$1+$$ hR $$=$$ (1+$$α$$10)12$$⇒ $$1+hR$$ $$=$$ $$1+5$$α $$(by$$ binomial $$expansion)⇒ α $$=$$ $$h5R$$

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