Rolling as Translation and Rotation Combined — Question 1

Question 1

The top point outruns the wheel center

A wheel of radius $0.40\,\mathrm m$ rolls right without slipping on stationary ground at $v_C=3\,\mathrm{m/s}$. Take $+x$ right, $+y$ up and positive angular velocity counterclockwise.

1. Find angular velocity and velocities of the top and bottom points.

2. Find the velocity of the rightmost rim point and its direction.

Question 1 - Solution

The top point outruns the wheel center

(a) Derivation and calculation

For planar rigid motion, $$\vec v_P=\vec v_C+\vec\omega\times\vec r_{P/C}.$$ The bottom contact condition is $0=V+\omega R$, hence $$\boxed{\omega=-V/R=-3/.40=-7.50\ \mathrm{rad/s}}.$$ At the top and bottom, $$\boxed{\vec v_t=(V-\omega R)\hat x=6\hat x\ \mathrm{m/s}},\qquad\boxed{\vec v_b=\vec0}.$$

(b) Result and interpretation

At the rightmost point, $\vec r=R\hat x$: $$\vec v_r=3\hat x+(-7.5\hat z)\times(.40\hat x) =\boxed{(3\hat x-3\hat y)\ \mathrm{m/s}}.$$ Therefore $$\boxed{|\vec v_r|=3\sqrt2=4.24\ \mathrm{m/s}},\qquad \boxed{\tan^{-1}(v_y/v_x)=-45^\circ}.$$

Physical check

Translation and rotation reinforce at the top, cancel at contact, and are perpendicular at the right rim.