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

Options

A. `h/(19 R)`
B. `(5R)/h`
C. `h/(5R)`
D. `h/(20R)`

Answer

C

Explanation

Clock is designed to indicate correct time at `20^(@)C` at height `'h'`. It will indicate correct time at `30^(@)C` on the ground if in this case the time period is same as the earlier.
`:. 2pisqrt((L)/(g_h)) = 2pisqrt((L')/(g_s))`
here, `L' = L(1+alpha 10), g_(s)=(GM)/(R^2)`
and `g_(h) = (GM)/((R+h)^(2)) rArr L/(g_n) = L/(g_s)`
`rArr L(R+h)^(2) = L(1+alpha 10)R^(2)`
`rArr 1+h/R = (1+alpha 10)^(1//2)`
`rArr 1+h/R = 1+5 alpha` (by binomial expansion)
`rArr alpha = h/(5R)`.