An Angular Simple Harmonic Oscillator — Question 4

Question 4

A tangential impulse launches angular oscillation

A disk on a torsional suspension is initially at its untwisted equilibrium and at rest. Its inertia is $I=.020\,\mathrm{kg\,m^2}$ and stiffness $\kappa=.50\,\mathrm{N\,m/rad}$. A brief tangential impulse $J=.10\,\mathrm{N\,s}$ acts at radius $R=.20\,\mathrm{m}$.

1. Find the initial angular speed and ensuing twist amplitude.

2. Find the maximum restoring torque and verify the oscillator energy using both the initial angular motion and the turning point.

Question 4 - Solution

A tangential impulse launches angular oscillation

(a) Derivation and calculation

For a tangential impulse,$$\Delta L=RJ=(.20)(.10)=.0200\,\mathrm{kg\,m^2/s}.$$$$I\dot\theta_+=RJ\Rightarrow\boxed{\dot\theta_+=1.00\,\mathrm{rad/s}}.$$$$\Omega=\sqrt{\kappa/I}=\sqrt{.50/.020}=5.00\,\mathrm{rad/s},\quad\boxed{\theta_A=\dot\theta_+/\Omega=.200\,\mathrm{rad}}.$$

(b) Result and interpretation

$$\boxed{|\tau|_{\max}=\kappa\theta_A=.100\,\mathrm{N\,m}},\qquad\boxed{E=\tfrac12I\dot\theta_+^2=\tfrac12\kappa\theta_A^2=.0100\,\mathrm J}.$$$$\Delta\mathbf L=\mathbf r\times\mathbf J,\qquad J\parallel\mathbf r\Rightarrow\Delta L=0.$$A radial impulse therefore does not launch torsional oscillation about the axis.

Physical check

The impulse changes angular velocity while angular position remains continuous.