In one line
For rolling without slipping, the contact point with the ground is the ICR, so every point's velocity equals \( \omega \) times its distance from the contact point.
Explained simply
A rolling wheel's bottom point touches the road and, for that instant, is not moving relative to the road, otherwise the tyre would skid. So the whole wheel momentarily pivots about that bottom point. That is why the top of a bicycle tyre looks blurred while the bottom looks sharp.
Technically, no slip means \( v_{contact} = 0 \), so I = contact point, and \( v_O = \omega r \) for the centre O.
How it works
- Mark the contact point as I.
- From the centre velocity: \( \omega = v_O / r \).
- For any point P on the wheel, find the distance IP (geometry in the circle).
- \( v_P = \omega \cdot IP \), directed perpendicular to IP.
- Top point: IP = 2r, so \( v_{top} = 2v_O \). End of horizontal diameter: IP = \( r\sqrt2 \), \( v = \sqrt2\,v_O \) at 45°.
- For a rod attached to the wheel, use the velocity of the attachment point as the known velocity for the rod's own ICR.
Key formula
\( v_O = \omega r \), \( v_P = \omega \cdot IP \). r in m, \( \omega \) in rad/s.
Quick example
A wheel of radius 0.5 m rolls with centre speed 10 m/s. \( \omega = 20 \) rad/s. Top point: \( 20 \times 1.0 = 20 \) m/s forward. Point at the front end of the horizontal diameter: \( IP = 0.707 \) m, \( v = 14.14 \) m/s at 45° below the horizontal, forward. Contact point: 0.
Common mistakes
- Taking the centre as the ICR (true only if the wheel spins in place).
- Using \( v = \omega r \) with r from the centre for points other than the centre.
- Applying the method when the wheel slips; then the contact point is not the ICR.
How MU asks it
Common 5 to 6-mark numerical: 'A wheel rolls without slipping ... find velocities of points A, B, C on the rim', or a wheel with a rod attached (2-link limit). Draw the wheel, mark I, write IP for each point and the velocity with direction.