Question

A cubical block of density ρ is floating on the surface of water. Out of its height L, fraction x is submerged in water. The vessel is in an elevator accelerating upward with acceleration a . What is the fraction immersed?

Answer 1

When a fluid is subjected to constant vertical acceleration, its free surface remains horizontal as the net effective gravity is acting vertically. But the magnitude of pressure at a point in the fluid increases or decreases depending upon the direction of acceleration.

Through the principle of floating when the block gets submerged in water

`Vpg = V^'Pwg`

Where V' = Volume of water displaced by block

V' = Base area of block × Height of block inside the water

`V^' = L^2x`

Let V be the volume of the block and Pb by the density of the block

Hence, `V = L^2 xx Pw`

`(Pb)/(Pw) = x/L`  ......(1)

`x =  (Pb)/(Pw) L`

As the block lifts upward, the acceleration = (g + a)

Weight of L³ Pb block = `m(g + a) = V c Pb(g + a) = L^3 Pb(g + a)`

Let x1 be the part of block which is submerged.

We know that the buoyant force will be equal and opposite to the weight of the block.

`L^2 Pb(g + a) = x1L^2 Pw(g + a)`

So, `(Pb)/(Pw) = (x1)/L`

So, `x1 = (Pb)/(Pw) L`  ......(2)

From (1) and (2) we can infer that `(Pb)/(Pw) L` is the part of the block which is submerged in water. This is independent of the acceleration of the block (upwards, downwards or at rest).