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A piece of metal floats on mercury. The volume coefficient of expansion of the metal and mercury are gamma_(1) and gamma_(2) respectively. If the temperature of both mercury and metal are increased by DeltaT, by what factor does the fraction of the voilume of the m.etal submerged in mercury change ? |
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Answer» Solution :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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