Are Angular Quantities Vectors? — Question 7

Question 7

A proper change of axes preserves the angular vector magnitude

A rotor has $\vec\omega=(3,4,0)\,\mathrm{rad/s}$. New right-handed axes are $\hat e_1=(4/5,-3/5,0)$, $\hat e_2=(3/5,4/5,0)$ and $\hat e_3=\hat z$.

1. Find the angular-velocity components in the new axes.

2. Find the new components of $\vec\alpha=(1,-2,0)\,\mathrm{rad/s^2}$ and decide whether the angular speed is increasing.

Question 7 - Solution

A proper change of axes preserves the angular vector magnitude

(a) Derivation and calculation

Components in an orthonormal basis are scalar projections: $$\omega_1=\vec\omega\cdot\hat e_1=\frac{12-12}{5}=0,\qquad \omega_2=\frac{9+16}{5}=5,\qquad\omega_3=0.$$ $$\boxed{\vec\omega^{\prime}=(0,5,0)\ \mathrm{rad/s}},\qquad |\vec\omega^{\prime}|=|\vec\omega|=5.$$

(b) Result and interpretation

Project acceleration in the same basis: $$\alpha_1=\frac{4+6}{5}=2,\qquad\alpha_2=\frac{3-8}{5}=-1, \qquad\boxed{\vec\alpha^{\prime}=(2,-1,0)\ \mathrm{rad/s^2}}.$$ $$\frac{d|\omega|}{dt}=\frac{\vec\omega\cdot\vec\alpha}{|\omega|} =\frac{3-8}{5}=\boxed{-1\ \mathrm{rad/s^2}}.$$ The rotor slows; the new components give the same dot product $0(2)+5(-1)=-5$.

Physical check

A proper orthonormal change of axes preserves both vector magnitudes and dot products.