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)gH`
D. `(5)/(3)rho_(0)gH`

Answer

D

Explanation

Pressure at bottom
`=g xx int_(z=0)^(z=H)(1)/(pi r^(2))xx pi r^(2)xx dz xx rho_(0)(2-(z^(2))/(H^(2)))`
`=g xx int_(z=0)^(z=H)rho_(0)(2-(z^(2))/(H^(2)))dz=g xx[rho(2z-(z^(3))/(3H^(2)))]_(z=0)^(z=H)`
`=g rho_(0)(2H-(H^(3))/(3H^(2)))=g rho_(0)(2H-(H)/(3))`
`=g rho_(0)((5H)/(3))=(5)/(3)rho_(0)gh`