Consider a vessel filled with a liquid upto height H. The bottom of the vessel lies in the XY-plane passing through the origin. The density of the liquid varies with Z-axis as `rho(z)=rho_(0)[2-((z)/(H))^(2)]`. If `p_(1)` and `p_(2)` are the pressures at the bottom surface and top surface of the liquid, the magnitude of `(p_(1)-p_(2))` is

Options

A. `rho_(0) gH`
B. `(8)/(5) rho_(0) gH`
C. `(3)/(2) rho_(0) g H`
D. `(5)/(3) rho_(0) g H`

Answer

D

Explanation

Given that `P(Z) = rho_(0) (2- (Z/H)^2 )`
Let a small column of liquid thickness be dz.
Pressure due to column,
`dP_1 = rho_(0) ((2- (Z)/(H))2 ) gdz + P_2`
`P_1 = int_(0)^(pi) rho_(0) (2- (Z/H)^2 )gdz + P_2`

`rho_(0) g int(0)^(pi) ( 2 - (Z^2)/(H^2))dz+ P_2`
`= rho_(0) g (2H - (1)/(3H^2 )(H^3) )+P_2`
`P_1 = rho_(0) g (2H - (H)/(3) ) + P_2`
`P_1 - P_2 = rho_(0) g (5H)/(3) = (5)/(3) rho_(0) g H`