A smooth hemisphere of mass 4m and radius R can slide horizontally on a frictionless surface. A small mass m slides down from its top. If hemisphere speed is v when angular coordinate is θ, and particle’s relative angular speed is nv/(R cosθ), find n.
Physics · Mixed Chapter
JEE Main Practice

Question

A smooth hemisphere of mass 4m and radius R can slide horizontally on a frictionless surface. A small mass m slides down from its top. If hemisphere speed is v when angular coordinate is θ, and particle’s relative angular speed is nv/(R cosθ), find n.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Horizontal external force on combined sphere+particle system is zero, so horizontal momentum remains zero. Step 2 — Let hemisphere horizontal velocity be v. Particle relative horizontal velocity is Rω cosθ (opposite direction with this convention). Its absolute horizontal velocity is v − Rω cosθ. Step 3 — Momentum: 4mv + m(v−Rω cosθ) = 0 ⇒ 5v = Rω cosθ. Step 4 — ω = 5v/(R cosθ), so n = 5.