Rolling as Translation and Rotation Combined — Question 3

Question 3

Two rim measurements reveal translation and spin separately

At one instant a circular wheel of radius $0.30\,\mathrm m$ has horizontal top velocity $5\,\mathrm{m/s}$ and horizontal bottom velocity $1\,\mathrm{m/s}$, both rightward. Its center has no vertical velocity.

1. Infer center velocity and angular velocity.

2. Does it roll without slipping on stationary ground or on a belt moving right at $1\,\mathrm{m/s}$?

Question 3 - Solution

Two rim measurements reveal translation and spin separately

(a) Derivation and calculation

The horizontal velocities at opposite rim points satisfy $$v_t=V-\omega R=5,\qquad v_b=V+\omega R=1.$$ Adding and subtracting, $$\boxed{V=(5+1)/2=3.00\ \mathrm{m/s}},$$ $$\boxed{\omega=(1-5)/(2(.30))=-20/3\ \mathrm{rad/s}}.$$

(b) Result and interpretation

Slip is determined relative to the contacting surface: $$v_{\mathrm{slip}}=v_b-U.$$ For stationary ground, $\boxed{v_{\mathrm{slip}}=1.00\ \mathrm{m/s}}$ rightward. For the stated belt, $$\boxed{v_{\mathrm{slip}}=1-1=0}.$$ The belt contact is instantaneously no-slip; its subsequent behavior also depends on the accelerations and available friction.

Physical check

The arithmetic mean of the opposite-point velocities recovers the center velocity.