Are Angular Quantities Vectors? — Question 9

Question 9

Coaxial rates add after the reference sense is fixed

A turntable rotates at $+3\hat z\,\mathrm{rad/s}$ relative to the lab. A small coaxial rotor turns at $-8\hat z\,\mathrm{rad/s}$ relative to the turntable. The turntable's rate increases at $1\,\mathrm{rad/s^2}$ while the relative rate stays fixed.

1. Find the rotor's laboratory angular velocity and acceleration.

2. Is its laboratory angular speed initially increasing or decreasing, and when does it reverse if the acceleration persists?

Question 9 - Solution

Coaxial rates add after the reference sense is fixed

(a) Derivation and calculation

All rates are expressed about the same laboratory axis: $$\vec\omega_{\mathrm{lab}}=\vec\omega_{\mathrm{table}}+\vec\omega_{\mathrm{rel}} =(3-8)\hat z=\boxed{-5\hat z\ \mathrm{rad/s}}.$$ $$\vec\alpha_{\mathrm{lab}}=(1+0)\hat z=\boxed{\hat z\ \mathrm{rad/s^2}}.$$

(b) Result and interpretation

Under the continued constant acceleration, $$\omega_z(t)=-5+t,\qquad \omega_z(t_r)=0\Rightarrow\boxed{t_r=5\ \mathrm s}.$$ Initially, $$\left.\frac{d|\omega|}{dt}\right|_0=\frac{(-5)(1)}5=-1\ \mathrm{rad/s^2}.$$ The rotor slows to rest, then accelerates in the positive sense.

Physical check

The relative rotor rate stays negative even when the laboratory rate changes sign.