The Angular Momentum of a Rigid Body Rotating About a Fixed Axis — Question 7

Question 7

Equal angular momentum reverses an energy comparison

A hoop and solid disk have equal mass $M$ and radius $R$ and rotate about their central normal axes. Their positive axial angular momenta are equal to $L$.

1. Compare their angular speeds and energies.

2. How would the energy comparison differ if angular speeds were equal instead?

Question 7 - Solution

Equal angular momentum reverses an energy comparison

(a) Derivation and calculation

At the same positive angular momentum, $$\omega_h=\frac{L}{MR^2},\qquad\omega_d=\frac{L}{MR^2/2}=\boxed{2\omega_h}.$$ Since $K=L^2/(2I)$, $$K_h=\frac{L^2}{2MR^2},\qquad K_d=\frac{L^2}{MR^2}=\boxed{2K_h}.$$

(b) Result and interpretation

At equal angular speed instead, $$K_h=\tfrac12MR^2\omega^2,\qquad K_d=\tfrac14MR^2\omega^2,$$ $$\boxed{K_h=2K_d}.$$ The comparison reverses because fixed $L$ makes energy inversely proportional to inertia, whereas fixed $\omega$ makes it directly proportional.

Physical check

Both comparisons follow from $K=L\omega/2$ once the held quantity is specified.