Solids and liquids both expands on heating. The density of substance decreases on expanding according to the relation `rho_(2) = (rho_(1))/(1 + gamma(T_(2)- T_(1)))`, where , `rho_(1) rarr "density at" T_(1) , rho_(2) rarr "density at" T_(2) , gamma rarr` coefficient of volume expansion of substances. When a solid is submerged in a liquid , liquid exerts an upward force on solid which is equal to the weight of liquid displaced by submerged part of solid. Solid will float or sink depends on relative densities of solid and liquid . A cubical block of solid floats in a liquid with half ot its volume submerged in liquid as shown in figure (at temperature `T`) `alpha_(S) rarr` Coefficient of linear expansion of solid
`gamma_(L) rarr "Coefficient of volume expansion of liquid"`
`rho_(S) rarr "Density of solid at temperature" T`
`rho_(L) rarr" Density of liquid at temperature" T`

Assume block does not expand on heating . The temperature at which the block just begins to sink in liquid is

Options

A. `T + (1)/(gamma_(L))`
B. `T + (1)/((2gamma_(L)))`
C. `T + (2)/((gamma_(L)))`
D. `T + (gamma_(L))/(2)`

Answer

A

Explanation

If volume change in solid is zero. Let at `T` , solid sinks `implies rho_(0)(V)/(2)g = Vrho_(S)g` (initially)
Finally `rho_(T) Vg = Vrho_(S)g = (rho_(0))/(2)`
`rho_(T) = (rho_(0))/((1+gamma_(L)DeltaT)) = (rho_(0))/(2) implies 2 = 1 + gamma_(L)DeltaT = DeltaT = (1)/(rho_(L))`
`therefore` Temperature = `T + (1)/(gamma_(L))`