A sphere is just immersed in a liquid. Find the ratio of hydrostatic force acting on bottom half and top half of the sphere.
Physics · Mixed Chapter
JEE Mains

Question

A sphere is just immersed in a liquid. Find the ratio of hydrostatic force acting on bottom half and top half of the sphere.

G1 Physics · Accepted Answer

Step 1: Let the sphere's radius be R, with its top at the liquid's free surface. At polar angle θ measured from the upward vertical, pressure is p=ρgR(1−cosθ). Step 2: Vertical force component on an element is dFᵥ=p cosθ (2πR²sinθ dθ). For the upper hemisphere integrate θ=0 to π/2: Fₜₒₚ=(π/3)ρgR³ (downward). Step 3: By Archimedes' principle the net upward force on the entire sphere is (4π/3)ρgR³. The lower hemisphere's upward component must exceed the upper downward component by this amount. St…