An Angular Simple Harmonic Oscillator — Question 7

Question 7

An unanchored torsional pair has a free overall rotation

Two coaxial rotors of inertias $I_1$ and $I_2$ are joined by a torsional spring of constant $\kappa$. Neither rotor is attached to a fixed angular reference. Let $q=\theta_2-\theta_1$ be the spring twist. External torque is zero.

1. Derive the twist oscillation frequency.

2. If total angular momentum is zero and $I_2=3I_1$, find the ratio of angular amplitudes. How does nonzero total angular momentum alter the motion?

Question 7 - Solution

An unanchored torsional pair has a free overall rotation

(a) Derivation and calculation

$$I_1\ddot\theta_1=\kappa q,\qquad I_2\ddot\theta_2=-\kappa q,\qquad q=\theta_2-\theta_1.$$$$\ddot q=-\kappa\left(\frac1{I_1}+\frac1{I_2}\right)q.$$$$\boxed{\Omega=\sqrt{\kappa/I_{\mathrm{red}}}},\qquad I_{\mathrm{red}}=\frac{I_1I_2}{I_1+I_2}.$$

(b) Result and interpretation

$$L=I_1\dot\theta_1+I_2\dot\theta_2=\text{constant},\qquad\dot\Theta=\frac L{I_1+I_2}.$$$$\delta\theta_1=-\frac{I_2q}{I_1+I_2},\quad\delta\theta_2=\frac{I_1q}{I_1+I_2}.$$$$I_2=3I_1\Rightarrow\boxed{A_1/A_2=3}.$$The oscillatory parts are opposite in phase; common rotation may be superposed.

Physical check

Internal spring torques cannot change the total angular momentum.