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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 fraction of the volume of the metal submerged in mercury changes ?

Text Solution

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Let the total volume of metal in air and mercury be V and `V_(s)` respectively, `rho and sigma` be the densities of s . metal and mercury respectively. when the metal floats m equilibrium,
Fraction of the volume submerged,
`f_(s) = (V_(s))/(V) = (rho)/(sigma)" "` .......(1)
When the temperature changes, the fraction of volume submerged changes as densities change.
`(Delta _(s))/(f_(s)) = (f_(s))/(f_(s)) - 1 = (rho^(1))/(sigma^(1)) xx (sigma)/(rho) - 1 " " ` ...... (2)
Densities of liquid and metal decrease as temperature increases
`rho = (rho)/(1 + gamma_(1) Delta T ) , sigma. = (sigma)/(1 + gamma_(2) Delta T)`
Which on substitution in eqn. (2) yields
`(Delta f_(s))/(f_(s)) = (1 + gamma_(2) Delta T)/(1 + gamma_(1) Delta T) - 1 = ((1 + gamma_(2) Delta T ) - (1 + gamma_(1) Delta T ))/((1 + gamma_(1) Delta T))`
`((gamma_(2) - gamma_(1))Delta T )/((1 + gamma_(1) Delta T)) = (gamma_(2) - gamma_(1)) Delta T (1 - gamma_(1) Delta T)`
= `(gamma_(2) - gamma_(1) ) Delta ` T
`[ " As" (1)/(1 + gamma_(1) Delta T) = (1 + gamma_(1) Delta T)^(-1) = (1- gamma_(1) Delta T)`
and `gamma_(1) gamma_(2)` small number, it has been neglected] When the solid is completely submerged,
1 = `f_(s) (1 + gamma Delta T ) " or " Delta T = (1 - f_(s))/(gamma f_(s))`
(b) `f_(s_(1)) = f_(s) (1 + gamma Delta T_(1) ), f_(s_(2)) = f_(s) (1 + gamma Delta T_(2))`
`(f_(s_(1)))/(f_(s_(2))) = ( (1 + gamma Delta T_(1))/(1 + gamma Delta T_(2)) ) . ` ON solving for `gamma` , we have
` gamma = (f_(s_(1)) - f_(s_(2)))/(f_(s_(2)) Delta T_(1) - f_(s_(1)) Delta T_(2))`
(c) As `f_(s) = (rho)/(sigma) and f_(s) = (rho.)/(sigma.)`
`f_(s). = ((rho)/(1 + gamma_(1) Delta T)) ((1 + gamma_(2) Delta T))/(sigma ) f_(s). = f_(s) ((1 + gamma_(2) Delta T )/(1 + gamma_(1) Delta T ))`
If `gamma_(2) gt gamma_(1) , ` the solid sinks.
If `gamma_(2) = gamma_(1)` , no effect on submergence.
If ` gamma_(2) lt gamma_(1)`, the solid lifts up .
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