Physics

Problem

The density of a liquid in an open tank varies with depth according to

\rho(y)=\rho_0\bigl(1+ky\bigr)

, where

y

is the depth measured downward from the free surface,

k

is a constant with dimensions

\text{m}^{-1}

and

\rho_0

is a positive constant. A cube of edge

a

is immersed such that its centre is at depth

a

below the surface. Neglect atmospheric pressure. Which of the following gives the magnitude of the net upward force on the cube? a)

\rho_0 g a^{3}\bigl(1+\tfrac{1}{2}k a\bigr)

\rho_0 g a^{3}(1+2 k a)

\rho_0 g a^{3}(1+k a)

\rho_0 g a^{3}