rolling without slipping

Question

For pure rolling, which relation holds between linear velocity v and angular velocity ω (radius R)?

RankGuru · Accepted Answer

1. **Definition of Pure Rolling**: The condition where the point of contact between the object and the surface has zero instantaneous velocity relative to the surface.
2. **Relative Velocity Logic**: The velocity of the point of contact $C$ is $\vec{v}_c = \vec{v}_{cm} + \vec{v}_{rot}$. 
3. **Components**: For a forward-moving wheel, $v_{cm}$ is to the right and $v_{rot} = \omega R$ is to the left at the bottom.
4. **No-Slip Condition**: Setting $v_{cm} - \omega R = 0$ gives $v = \omega R$.
5. **Result**: **$v = \omega R$** is the fundamental constraint for pure rolling.

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