Relating the Linear and Angular Variables — Question 8

Question 8

Distance from the axis is not distance from the origin

A body rotates about the fixed $z$ axis with $\vec\omega=2\hat z\,\mathrm{rad/s}$ and $\vec\alpha=-\hat z\,\mathrm{rad/s^2}$. A point has position $(3,4,5)\,\mathrm m$.

1. Find its velocity and acceleration vectors.

2. Explain why its height does not affect its speed.

Question 8 - Solution

Distance from the axis is not distance from the origin

(a) Derivation and calculation

Compute the cross products in laboratory coordinates: $$\vec v=2\hat z\times(3\hat x+4\hat y+5\hat z) =\boxed{(-8,6,0)\ \mathrm{m/s}}.$$ $$\vec a_t=-\hat z\times\vec r=(4,-3,0),\qquad \vec a_r=\vec\omega\times\vec v=(-12,-16,0).$$ $$\boxed{\vec a=\vec a_t+\vec a_r=(-8,-19,0)\ \mathrm{m/s^2}}.$$

(b) Result and interpretation

The perpendicular radius and speed are $$r_\perp=\sqrt{x^2+y^2}=5\ \mathrm m,\qquad \boxed{v=\omega r_\perp=10\ \mathrm{m/s}}.$$ The height disappears because $\hat z\times5\hat z=0$; distance from the origin is not the relevant radius.

Physical check

The velocity norm $\sqrt{64+36}=10$ m/s agrees with the perpendicular-radius calculation.