Are Angular Quantities Vectors? — Question 5

Question 5

An inclined rotation axis requires a unit direction

A rotor turns right-handed about an axis pointing along $(2,-1,2)$, at angular speed $9\,\mathrm{rad/s}$. Its speed increases at $3\,\mathrm{rad/s^2}$ without changing axis direction.

1. Find angular velocity and angular acceleration in Cartesian components.

2. Explain why multiplying $(2,-1,2)$ directly by 9 gives the wrong angular speed.

Question 5 - Solution

An inclined rotation axis requires a unit direction

(a) Derivation and calculation

Normalize the specified direction: $$|\vec d|=\sqrt{2^2+(-1)^2+2^2}=3,\qquad\hat n=\frac13(2,-1,2).$$ $$\boxed{\vec\omega=9\hat n=(6,-3,6)\ \mathrm{rad/s}},\qquad \boxed{\vec\alpha=3\hat n=(2,-1,2)\ \mathrm{rad/s^2}}.$$

(b) Result and interpretation

Multiplication by the unnormalized direction gives $$|9\vec d|=9|\vec d|=27\ \mathrm{rad/s}\ne9\ \mathrm{rad/s}.$$ The required check is $$|\vec\omega|=\sqrt{36+9+36}=9,\qquad|\vec\alpha|=\sqrt{4+1+4}=3.$$

Physical check

Both vectors are parallel because the axis direction is fixed while the speed increases.