Are Angular Quantities Vectors? — Question 10

Question 10

Instantaneous angular rates can add about different axes

At an instant a moving carrier has angular velocity $3\hat z\,\mathrm{rad/s}$. A rotor turns relative to it at $4\hat x\,\mathrm{rad/s}$, where $\hat x$ denotes the relative axis expressed in laboratory coordinates at that instant.

1. Find the rotor's instantaneous angular-velocity vector and magnitude.

2. Why does this not license replacing a long sequence of finite rotations by the sum of two angle vectors?

Question 10 - Solution

Instantaneous angular rates can add about different axes

(a) Derivation and calculation

Instantaneous angular velocities add in a common frame: $$\boxed{\vec\omega=\vec\omega_c+\vec\omega_r=4\hat x+3\hat z\ \mathrm{rad/s}}.$$ $$\boxed{|\vec\omega|=\sqrt{4^2+3^2}=5\ \mathrm{rad/s}}.$$

(b) Result and interpretation

For an infinitesimal interval, $$d\vec r=(\vec\omega\times\vec r)\,dt,\qquad \vec\omega(t)=\vec\omega_c(t)+\vec\omega_r(t).$$ The carrier changes the laboratory direction of the relative axis. Finite rotations must retain that ordered history; in general $R_2R_1\ne R_1R_2$. A local vector sum does not imply commuting finite angle vectors.

Physical check

The magnitude 5 is below the scalar sum 7 because the two instantaneous axes are perpendicular.