The Rotational Variables — Question 3

Question 3

A cubic angle history has two different kinds of stationary behavior

A rotor has unwrapped angle $\theta(t)=t^3-6t^2+9t$ in SI units for $0\le t\le4\,\mathrm s$.

1. Find the times when it stops and determine whether each is a reversal.

2. Find when its angular acceleration vanishes. Is the rotor then stopped?

Question 3 - Solution

A cubic angle history has two different kinds of stationary behavior

(a) Derivation and calculation

Differentiate and factor: $$\omega=\frac{d\theta}{dt}=3t^2-12t+9=3(t-1)(t-3).$$ $$\omega=0\quad\Longrightarrow\quad\boxed{t=1\ \mathrm s,\ 3\ \mathrm s}.$$ The signs on $(0,1),(1,3),(3,4)$ are $+,-,+$, respectively; both zeros are reversals.

(b) Result and interpretation

Differentiate once more: $$\alpha=\frac{d\omega}{dt}=6t-12=0\quad\Longrightarrow\quad\boxed{t=2\ \mathrm s}.$$ $$\omega(2)=12-24+9=\boxed{-3\ \mathrm{rad/s}}.$$ The signed velocity has a minimum there, while the rotor continues clockwise.

Physical check

The stopping times are zeros of the velocity graph; zero acceleration occurs at its horizontal tangent.