Question

If keeping the mass constant a rotating disc’s radius is suddenly reduced to half. Then how much would be its new angular velocity value?

Answer 1

Initial moment of inertia of the disc,
I₁ = \(\frac{1}{2}\) MR₁²
and initial angular velocity = ω₁
After reducing the radius to half (i.e. ,R₂ = R₁/ 2), the new moment of inertia,
\(I_{2}=\frac{1}{2} M R_{2}^{2}=\frac{1}{2} M \cdot\left(\frac{R_{1}}{2}\right)^{2}=\frac{1}{2} M \frac{R_{1}^{2}}{4}\)
or I₂ = \(\frac{1}{4}\) I₁
If ω₂ be the new angular velocity, then according to the law of conservation of angular momentum,
I₂ ω₂ = I₁ ω₁ or \(\frac{I_{1}}{4}\) ω₂ = I₁ ω₁
or ω₂ = 4ω₁
Thus the angular velocity will become four times.