Kinetic Energy of Rotation — Question 2

Question 2

Equal masses and radii do not fix rotational energy

A thin hoop and a uniform disk each have mass $4\,\mathrm{kg}$ and radius $0.50\,\mathrm m$. Both rotate about their symmetry axes at $6\,\mathrm{rad/s}$. Use $I_{\mathrm{hoop}}=MR^2$ and $I_{\mathrm{disk}}=MR^2/2$.

1. Compare their kinetic energies.

2. How fast must the disk turn to have the same energy as the hoop at the stated speed?

Question 2 - Solution

Equal masses and radii do not fix rotational energy

(a) Derivation and calculation

Insert the common mass and radius in each inertia law: $$I_h=4(0.50)^2=1.00,\qquad I_d=\tfrac12(4)(0.50)^2=0.50\quad\mathrm{kg\,m^2}.$$ $$\boxed{K_h=\tfrac12(1)(6)^2=18\ \mathrm J},\qquad \boxed{K_d=\tfrac12(0.50)(6)^2=9\ \mathrm J}.$$

(b) Result and interpretation

For equal energy with the hoop’s rate held fixed, $$\tfrac12I_d\omega_d^2=\tfrac12I_h(6)^2 \Rightarrow\boxed{|\omega_d|=6\sqrt{I_h/I_d}=6\sqrt2=8.485\ \mathrm{rad/s}}.$$ The disk must rotate faster because more of its mass is nearer the axis.

Physical check

The hoop has twice the inertia and twice the energy at the same angular speed.