An Angular Simple Harmonic Oscillator — Question 1

Question 1

A static twist and a period identify different parameters

A torsional oscillator obeys $\tau=-\kappa\theta$ with angle in radians. A static external torque of $0.060\,\mathrm{N\,m}$ holds it at $0.20\,\mathrm{rad}$. After release its small-oscillation period is $1.50\,\mathrm{s}$. Neglect damping.

1. Infer the torsion constant and rotational inertia.

2. If released from rest at the held angle, find its maximum angular speed and stored elastic energy.

Question 1 - Solution

A static twist and a period identify different parameters

(a) Derivation and calculation

$$\tau=-\kappa\theta,\qquad\kappa=\frac{.060}{.20}=\boxed{.300\,\mathrm{N\,m/rad}}.$$$$I\ddot\theta+\kappa\theta=0,\quad T=2\pi\sqrt{I/\kappa}.$$$$\boxed{I=\kappa\left(\frac{T}{2\pi}\right)^2=.300\left(\frac{1.50}{2\pi}\right)^2=.01710\,\mathrm{kg\,m^2}}.$$

(b) Result and interpretation

$$\Omega=2\pi/T=4.18879\,\mathrm{rad/s},\quad\boxed{|\dot\theta|_{\max}=\Omega\theta_A=.83776\,\mathrm{rad/s}}.$$$$\boxed{E=\tfrac12\kappa\theta_A^2=\tfrac12(.300)(.20)^2=.00600\,\mathrm J}.$$

Physical check

At maximum angular speed, $I\dot\theta_{\max}^2/2$ equals the initial torsional energy.