Newton's Second Law for Rotation — Question 6

Question 6

Two driven tests separate inertia from bearing resistance

A rotor moving in the positive direction is tested at the same speed. Applied torque $5\,\mathrm{N\,m}$ gives $\alpha=6\,\mathrm{rad/s^2}$; applied torque $3\,\mathrm{N\,m}$ gives $\alpha=2\,\mathrm{rad/s^2}$. Assume constant unknown inertia and the same unknown opposing bearing torque in both tests.

1. Infer the inertia and bearing torque magnitude.

2. Find angular acceleration immediately after the drive is removed at that speed.

Question 6 - Solution

Two driven tests separate inertia from bearing resistance

(a) Derivation and calculation

Both tests share the same opposing magnitude $f$: $$5-f=6I,\qquad3-f=2I.$$ $$2=4I\Rightarrow\boxed{I=0.50\ \mathrm{kg\,m^2}},\qquad \boxed{f=3-2(0.50)=2.0\ \mathrm{N\,m}}.$$

(b) Result and interpretation

Immediately after removing the drive while positive motion continues, $$\boxed{\alpha=\frac{0-f}{I}=-\frac{2.0}{0.50}=-4.0\ \mathrm{rad/s^2}}.$$ This sign applies during positive rotation; any extrapolation through rest requires reconsidering the bearing law.

Physical check

The inferred pair reproduces both applied torques: $6I+f=5$ and $2I+f=3$ N m.