A solid sphere with uniform density and radius R is rotating initially with constant angular velocity ω₁ about its diameter. After some time during the rotation it starts losing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius becomes R/2 is xω…

Question

A solid sphere with uniform density and radius R is rotating initially with constant angular velocity ω₁ about its diameter. After some time during the rotation it starts losing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius becomes R/2 is xω₁. The value of x is .

Accepted Answer

32. The sphere keeps its shape and its uniform density, so mass goes as volume: M∝ R³. Moment of inertia about a diameter: I=2/5MR²∝ R³· R²=R⁵. No external torque acts, so angular momentum about the axis is conserved: I₁ω₁=I₂ω₂. (ω₂)/(ω₁)=(I₁)/(I₂)=(R/(R/2))⁵=2⁵. x=32. The fifth power is the point of the question: two powers come from the R² in the formula and three more from the shrinking mass.

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