Kinetic Energy of Rotation — Question 9

Question 9

An energy measurement bounds how far mass can lie from an axis

A rotor has total mass $2\,\mathrm{kg}$, angular speed $10\,\mathrm{rad/s}$ and kinetic energy $18\,\mathrm J$. All its mass lies within a maximum perpendicular distance $R$ from the fixed axis.

1. Infer its inertia and a lower bound on $R$.

2. Can all its mass lie within $0.40\,\mathrm m$? Explain the limiting configuration.

Question 9 - Solution

An energy measurement bounds how far mass can lie from an axis

(a) Derivation and calculation

Infer inertia from the measured energy: $$\boxed{I=\frac{2K}{\omega^2}=\frac{36}{100}=0.36\ \mathrm{kg\,m^2}}.$$ Since $r\le R$ for all mass elements, $$I=\int r^2\,dm\le R^2\int dm=MR^2 \Rightarrow\boxed{R\ge\sqrt{I/M}=\sqrt{0.18}=0.4243\ \mathrm m}.$$

(b) Result and interpretation

The proposed smaller radius permits at most $$I_{\max}(0.40)=2(0.40)^2=0.32\ \mathrm{kg\,m^2}<0.36\ \mathrm{kg\,m^2}.$$ It is impossible. Equality in $I\le MR^2$ requires every mass element at $r=R$, as for an ideal thin hoop.

Physical check

Any mass moved inward reduces inertia at fixed total mass and maximum radius.