page_title: Instant centre of rotation
section_title: Instant center of a wheel rolling without slipping
hierarchical_section_title: Instant centre of rotation / Instant center of a wheel rolling without slipping
context_page_description: The instant center of rotation, also called instantaneous velocity center, or also instantaneous center or instant center, is the point fixed to a body undergoing planar movement that has zero velocity at a particular instant of time. At this instant, the velocity vectors of the trajectories of other points in the body generate a circular field around this point which is identical to what is generated by a pure rotation.
  Planar movement of a body is often described using a plane figure moving in a two-dimensional plane. The instant center is the point in the moving plane around which all other points are rotating at a specific instant of time.
  The continuous movement of a plane has an instant center for every value of the time parameter. This generates a curve called the moving centrode. The points in the fixed plane corresponding to these instant centers form the fixed centrode.
  The generalization of this concept to 3-dimensional space is that of a twist around a screw. The screw has an axis which is a line in 3D space, and the screw also has a finite pitch.
context_section_description: Consider the planar movement of a circular wheel rolling without slipping on a linear road; see sketch 3.  The wheel rotates around its axis M, which translates in a direction parallel to the road.  The point of contact P of the wheel with road does not slip, which means the point P has zero velocity with respect to the road.  Thus, at the instant the point P on the wheel comes in contact with the road it becomes an instant center.
  The set of points of the moving wheel that become instant centers is the circle itself, which defines the moving centrode.  The points in the fixed plane that correspond to these instant centers is the line of the road, which defines the fixed centrode.
  The velocity vector of a point A in the wheel is perpendicular to the segment AP and is proportional to the length of this segment.  In particular, the velocities of points in the wheel are determined by the angular velocity of the wheel in rotation around P.  The velocity vectors of a number of points are illustrated in sketch 3.
  The further a point in the wheel is from the instant center P, the proportionally larger its speed.  Therefore, the point at the top of the wheel moves in the same direction as the center M of the wheel, but twice as fast, since it is twice the distance away from P.  All points that are a distance equal to the radius of the wheel 'r' from point P move at the same speed as the point M but in different directions.  This is shown for a point on the wheel that has the same speed as M but moves in the direction tangent to the circle around P.
caption: Sketch 4: Example relative center of rotation. Two bodies in contact at C, one rotating about A and the other about B must have a relative center of rotation somewhere along the line AB. Since the parts cannot interpenetrate the relative rotation center must also be along the normal direction to the contact and through C. The only possible solution is if the relative center is at D.
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