A cube of coefficient of linear expansion `alpha` is floating in a bath containing a liquid of coefficient of volume expansion `gamma_(l)`. When the temperature is raised by `DeltaT`, the depth upto which the cube is submerged in the liquid remains the same. Find relation between `alpha` and `gamma_(l)`

Answer

`gamma_(l) = 2alpha_(S)`

Explanation

When the temperature is increased , volume of the cube will increase while density of liquid will decrease. The depth upto which the cube is submerged in the liquid remains the same, hence the upthrust will not change.
`F= F'`
`therefore V_(l)rho_(L)g = V_(l)'rho_(L)'g`
(`V_(i)` = volume immersed
`therefore (Ah_(i))(rho_(L))(g) = A(1 + 2alpha_(S) DeltaT)(h_(t)) ((rho_(L))/(1 + gamma_(t)DeltaT))g`
Solving this equation , we get
`gamma_(l) = 2alpha_(S)`