Rotation with Constant Angular Acceleration — Question 8

Question 8

Calibration from two separated angle readings

A rotor starts at $\theta_0=0$ and has unknown $\omega_0$ and constant $\alpha$. Measurements give $\theta(2)=8\,\mathrm{rad}$ and $\theta(4)=24\,\mathrm{rad}$.

1. Infer both kinematic parameters.

2. Find when angular speed first reaches $10\,\mathrm{rad/s}$ and the angle then.

Question 8 - Solution

Calibration from two separated angle readings

(a) Derivation and calculation

Insert both readings into $\theta=\omega_0t+\tfrac12\alpha t^2$: $$2\omega_0+2\alpha=8,\qquad4\omega_0+8\alpha=24.$$ Subtract twice the first equation from the second: $$4\alpha=8\Rightarrow\boxed{\alpha=2\ \mathrm{rad/s^2}},\qquad \boxed{\omega_0=2\ \mathrm{rad/s}}.$$

(b) Result and interpretation

For $t\ge0$, the rate is positive and increasing: $$\omega(t)=2+2t=10\Rightarrow\boxed{t=4\ \mathrm s}.$$ $$\boxed{\theta(4)=2(4)+\tfrac12(2)(4)^2=24\ \mathrm{rad}}.$$

Physical check

The inferred parameters reproduce both measured angles exactly.