The Angular Momentum of a Rigid Body Rotating About a Fixed Axis — Question 6

Question 6

Energy and axial momentum together determine inertia and speed

A rigid rotor turning about a fixed principal axis has $L_z=6\,\mathrm{kg\,m^2/s}$ and rotational energy $K=9\,\mathrm J$.

1. Infer $I$ and signed angular velocity.

2. If only $K$ were known, what ambiguity would remain?

Question 6 - Solution

Energy and axial momentum together determine inertia and speed

(a) Derivation and calculation

Eliminate angular speed between $L_z=I\omega$ and $K=I\omega^2/2$: $$K=\frac{L_z^2}{2I}\Rightarrow\boxed{I=\frac{L_z^2}{2K}=\frac{6^2}{2(9)}=2.00\ \mathrm{kg\,m^2}}.$$ Then $$\boxed{\omega=L_z/I=6/2=+3.00\ \mathrm{rad/s}}.$$

(b) Result and interpretation

With only the energy measurement, $$I\omega^2=2K=18\ \mathrm J.$$ For every $I>0$, the speeds $$\omega=\pm\sqrt{18/I}$$ give the same energy. Thus neither inertia, speed magnitude, nor rotation sense is determined separately. The signed angular momentum supplies the additional information.

Physical check

The inferred pair gives $I\omega=6$ and $I\omega^2/2=9$, recovering both data.