A uniform solid sphere of radius R rotates about a diameter with angular velocity ω₁. During rotation it loses mass uniformly while remaining a uniform sphere of the same density and shape. When its radius becomes R/2, its angular velocity is xω₁. Find x.
Physics · Rotational Motion · angular momentum in pure rotation and its conservation
JEE Mains 2025 4 April Evening Shift

Question

A uniform solid sphere of radius R rotates about a diameter with angular velocity ω₁. During rotation it loses mass uniformly while remaining a uniform sphere of the same density and shape. When its radius becomes R/2, its angular velocity is xω₁. Find x.

Physics · Rotational Motion · angular momentum in pure rotation and its conservation

JEE Mains 2025 4 April Evening Shift

G1 Physics · Accepted Answer

With no external torque, angular momentum is conserved: I₁ω₁ = I₂ω₂. For a uniform solid sphere, I = (2/5)MR². Because density remains constant, M ∝ R³. Therefore, I ∝ R³R² = R⁵. When radius changes from R to R/2: I₂/I₁ = (1/2)⁵ = 1/32. Thus, I₁ω₁ = (I₁/32)ω₂ ⇒ ω₂ = 32ω₁. Hence x = 32.