Kinetic Energy of Rotation — Question 3

Question 3

Equal rim speeds remove radius from a family comparison

Two uniform disks have the same mass $M$ but radii $R$ and $2R$. Each rotates about its fixed symmetry axis with the same rim speed $v$. Use $I=MR^2/2$.

1. Compare their angular speeds and kinetic energies.

2. Would the same conclusion hold if they had equal angular speed instead?

Question 3 - Solution

Equal rim speeds remove radius from a family comparison

(a) Derivation and calculation

The rim constraints and inertias are $$\omega_1=\frac vR,\quad\omega_2=\frac v{2R}=\frac{\omega_1}{2},\qquad I_1=\frac12MR^2,\quad I_2=4I_1.$$ $$K_1=\frac12\left(\frac12MR^2\right)\frac{v^2}{R^2} =\boxed{\frac14Mv^2}=K_2.$$

(b) Result and interpretation

If angular speed is held equal instead, $$\boxed{\frac{K_2}{K_1}=\frac{I_2}{I_1} =\frac{(2R)^2}{R^2}=4}.$$ Equal rim speed and equal angular speed are different experimental constraints.

Physical check

At equal rim speed, the factor of four in inertia cancels the factor of one quarter in squared angular speed.