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A piece of metal floats on mercury. The ...

A piece of metal floats on mercury. The coefficient of volume expansion of metal and mercury are `gamma_1 and gamma_2`, respectively. if the temperature of both mercury and metal are increased by an amount `Delta T`, by what factor does the frection of the volume of the metal submerged in mercury changes ?

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The correct Answer is:
A, B, D
`f_i = (rho_s)/(rho_l) rArr f_i = (rho_s^')/(rho_l^')`
`:. (f_i^')/(f_i) = ((rho_s^')/rho_s)((rho _l)/rho_l^')=((1 + gamma_l Delta T)/(1 + gamma_s Delta T)) = ((1 + gamma_2 Delta T)/(1 + gamma_1 Delta T))`
Now, `1/(1 + gamma_1 Delta T) ~~ (1 - gamma_1 Delta T)`
`:. (f_i^')/(f_i) = (1 + gamma_2 Delta T)(1 - gamma_1 Delta T)`
Neglecting `gamma_1 gamma_2(Delta T)^2` term, we get
`(f_i^')/(f_i) ~~ (gamma_2 - gamma_1) Delta T`
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