Are Angular Quantities Vectors? — Question 1

Question 1

Opposite viewing ends describe the same angular velocity

A shaft lies along the laboratory $z$ axis. An observer on the positive $z$ side looking toward the origin sees counterclockwise rotation at $6\,\mathrm{rad/s}$.

1. Write the angular-velocity vector and describe what an observer on the negative $z$ side sees.

2. If the speed decreases at $2\,\mathrm{rad/s^2}$, find the angular-acceleration vector.

Question 1 - Solution

Opposite viewing ends describe the same angular velocity

(a) Derivation and calculation

Use a single laboratory axis for both observers. The right-hand rule gives $$\boxed{\vec\omega=6\hat z\ \mathrm{rad/s}},\qquad |\vec\omega|=6\ \mathrm{rad/s}.$$ The negative-$z$ observer sees clockwise motion because the line of sight is reversed; the laboratory vector remains $+6\hat z$.

(b) Result and interpretation

For fixed direction $\hat z$, $$\vec\alpha=\frac{d\vec\omega}{dt}=\frac{d|\omega|}{dt}\hat z =\boxed{-2\hat z\ \mathrm{rad/s^2}}.$$ $$\frac{d|\omega|}{dt}=\frac{\vec\omega\cdot\vec\alpha}{|\omega|} =\frac{6(-2)}6=-2\ \mathrm{rad/s^2}.$$

Physical check

The negative dot product identifies slowing independently of which end is viewed.