Condition for Pure Rolling

What pure rolling means

A body rolls without slipping when the contact point has zero velocity relative to the ground at every instant.

This gives the fundamental constraint:

$v_{cm} = R\omega$

$a_{cm} = R\alpha$

Why the contact point is at rest

At any instant, a rolling body can be thought of as rotating about the instantaneous contact point.

The contact point has two velocity contributions:

• From translation of CM: $v_{cm}$ (forward)
• From rotation about CM: $R\omega$ (backward, for forward rolling)

For pure rolling: $v_{cm} = R\omega$, so these cancel exactly and the contact point is stationary.

Velocities at different points on the rolling body

For a disc of radius $R$ rolling with $v_{cm} = v$:

The top moves at twice the CM speed. This is why the tops of rolling wheels appear to move faster than the vehicle itself.

Slipping vs rolling

Static friction at the contact point provides the torque that enforces rolling. It does no work (contact point is at rest).

If the torque required for rolling exceeds $\mu_s N$:

• On an incline: body slides instead of rolling
• Sudden acceleration/braking: wheel spins/skids

Differentiate the rolling condition

From $v_{cm} = R\omega$, differentiating:

$a_{cm} = R\alpha$

This connects the translational and rotational accelerations.

Rolling on a curved surface

If the surface has curvature radius $\rho$, the rolling condition still holds: $v = R\omega$. But the normal force changes with speed due to centripetal effects.