A spherical shell of mass m encloses viscous fluid of mass 21m and radius R, rolling without slip at speed V₀. Fluid angular velocity increases linearly with radius from 0 to ω₀ = V₀/R. Its rotational kinetic energy is written as A m V₀² in the source. Find A.
Physics · Mixed Chapter
JEE Main Practice

Question

A spherical shell of mass m encloses viscous fluid of mass 21m and radius R, rolling without slip at speed V₀. Fluid angular velocity increases linearly with radius from 0 to ω₀ = V₀/R. Its rotational kinetic energy is written as A m V₀² in the source. Find A.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Fluid is distributed uniformly in volume inside radius R, and angular speed depends on distance s from sphere centre: ω(s) = (V₀/R)(s/R). Step 2 — A thin spherical layer has mass dm = (21m)(3s²/R³)ds and its angular rotational moment about a diameter is dI = (2/3)s² dm. Step 3 — Rotational kinetic energy dK = (1/2)dI ω(s)² = (1/3)s²ω(s)² dm. Step 4 — Integrate s⁶ from 0 to R: K = (1/3)(21m)(3/R³)(V₀²/R⁴) × R⁷/7 = 3mV₀². Step 5 — Therefore A = 3.