Rotation with Constant Angular Acceleration — Question 5

Question 5

A delayed rotor catches an already moving reference

Rotor A turns steadily at $4\,\mathrm{rad/s}$. Rotor B shares its initial angle but waits $1\,\mathrm s$, then starts from rest with $\alpha_B=4\,\mathrm{rad/s^2}$. Compare unwrapped angles.

1. Find the first catch after B starts.

2. Compare their angular velocities at that time and reject any algebraic root outside B's accelerating interval.

Question 5 - Solution

A delayed rotor catches an already moving reference

(a) Derivation and calculation

Use the delayed time origin only on its valid interval: $$\theta_A=4t,\qquad\theta_B=\tfrac12(4)(t-1)^2=2(t-1)^2,\quad t\ge1.$$ $$2(t-1)^2=4t\Rightarrow t^2-4t+1=0 \Rightarrow\boxed{t_c=2+\sqrt3=3.732\ \mathrm s}.$$ The root $2-\sqrt3=0.268$ s is outside $t\ge1$.

(b) Result and interpretation

At the accepted crossing, $$\boxed{\omega_B(t_c)=4(t_c-1)=4(1+\sqrt3)=10.93\ \mathrm{rad/s}},\qquad \omega_A=4\ \mathrm{rad/s}.$$ B overtakes A in unwrapped angle. Equality modulo $2\pi$ would be a different alignment condition.

Physical check

The relative angular velocity at the catch is positive, $4\sqrt3$ rad/s.