Math 2.0

Mechanics of particles and systems Definition

Instant centre of rotation

Builds on Angular velocityVelocity

Definition

Let J=(0−110)J = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} be the rotation of ℝ2\R^2 by π/2\pi/2. Let a rigid body move in the plane by x(t)=R(t)X+b(t)x(t) = R(t) X + b(t), where X∈ℝ2X \in \R^2 labels a point of the body. Here R(t)∈SO(2)R(t) \in SO(2) and b(t)∈ℝ2b(t) \in \R^2 are differentiable in time tt. At time tt the velocity of the body point located at x∈ℝ2x \in \R^2 is v(x)=ω J (x−b(t))+b˙(t),v(x) = \omega\, J\, (x - b(t)) + \dot b(t), where ω\omega is the angular velocity, defined by R˙(t)R(t)−1=ωJ\dot R(t) R(t)^{-1} = \omega J. If ω≠0\omega \ne 0, the instant centre of rotation at time tt is the unique point P∈ℝ2P \in \R^2 with v(P)=0v(P) = 0.

Remarks

Solving v(P)=0v(P) = 0 gives P=b(t)+ω−1Jb˙(t)P = b(t) + \omega^{-1} J \dot b(t), using J2=−IJ^2 = -I. Then v(x)=ωJ(x−P)v(x) = \omega J (x - P) for every xx: each point moves perpendicular to the segment joining it to PP, with speed ∣ω∣ ∣x−P∣\abs{\omega}\, \abs{x - P}.

Example: a wheel of radius 11 rolling to the right along the line x2=0x_2 = 0 at unit speed has hub b(t)=(t,1)b(t) = (t, 1) and ω=−1\omega = -1. Then P=(t,1)−J(1,0)=(t,1)−(0,1)=(t,0)P = (t, 1) - J (1, 0) = (t, 1) - (0, 1) = (t, 0), the contact point with the ground.

Non-example: if R(t)R(t) is constant, so ω=0\omega = 0, and b˙≠0\dot b \ne 0, then v(x)=b˙v(x) = \dot b for all xx and no point is at rest; the definition does not apply. The instant centre is also called the instantaneous centre of rotation or the pole of the motion.

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Cross product, concept, no evidence levelCross productconceptDerivative, concept, no evidence levelDerivativeconceptEuclidean vector, concept, no evidence levelEuclidean vectorconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionVelocity, concept, no evidence levelVelocityconceptSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectAngular velocity, concept, no evidence levelAngular velocityconceptInstant centre of rotation, definition, no evidence levelInstant centre of rot…definition
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  1. Cross product concept
  2. Derivative concept
  3. Euclidean vector concept
  4. Identity matrix object
  5. Matrix definition
  6. Vector space definition
  7. Velocity concept
  8. Square matrix definition
  9. Transpose definition
  10. Linear map definition
  11. Orthogonal matrix definition
  12. Multilinear map definition
  13. Orthogonal group object
  14. Alternating multilinear map definition
  15. Determinant definition
  16. Special orthogonal group object
  17. Angular velocity concept
  18. Instant centre of rotation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Instant_centre_of_rotation
About this page
TagRJ6G8HJJM0
URIm2://RJ6G8HJJM0
Kinddefinition
WritingW0 stub
MSC70B10, 53A17
DefinesInstant centre of rotation
UsesAngular velocity, Velocity
Changed3 November 2026

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