A cylinder vessel containing a liquid of density `rho` is closed by a smooth piston of mass `m` as shown. If atmospheric pressure is `P_0`, radius of cylinder is `r` and height of liquid is `h`. Find
(a) Pressure of liquid just below the piston
(b) force at bottom of cylinder.
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Explanation

(a) Forces acting on piston
(i) weight of piston, `mg` (downward)
(ii) force due to air above it, `P_0 A` (downward)
(iii) force due to liquid below it, `PA` (upward)
If the piston is in equilibrium
`PA = P_0 A + mg`
`P = P_0 + (mg)/(A) = P_0 + (mg)/(pi r^2)`
(b) Pressure at bottom of cylinder
`P' = P_0 + (mg)/(pi r^2) + rho gh`
Force at bottom of cylinder
`F = P' pi r^2 = (P_0 + (mg)/(pi r^2) + rho gh)pi r^(2)`.