Rotation with Constant Angular Acceleration — Question 3

Question 3

Equal-time angle increments reveal a constant acceleration

Over three consecutive $1.0\,\mathrm s$ intervals a rotor advances by $3$, $5$ and $7\,\mathrm{rad}$, with no reversal. Assume constant angular acceleration throughout.

1. Infer the initial angular velocity and acceleration.

2. Predict the next interval displacement and state what a measured $8\,\mathrm{rad}$ would imply.

Question 3 - Solution

Equal-time angle increments reveal a constant acceleration

(a) Derivation and calculation

The increment in interval $n$ of duration $h$ is $$\Delta\theta_n=\omega_0h+\alpha h^2(n-\tfrac12),\qquad \Delta\theta_{n+1}-\Delta\theta_n=\alpha h^2.$$ $$\boxed{\alpha=\frac{5-3}{1^2}=2\ \mathrm{rad/s^2}},\qquad \boxed{\omega_0=\frac31-\frac{2(1)}2=2\ \mathrm{rad/s}}.$$

(b) Result and interpretation

The next predicted displacement is $$\boxed{\Delta\theta_4=2+2(4-\tfrac12)=9\ \mathrm{rad}}.$$ An exact reading of 8 gives a residual $8-9=-1$ rad and violates the common constant-acceleration model; it cannot fit all the stated exact increments.

Physical check

The observed successive differences are both 2 rad, providing two consistent estimates of acceleration.