What is the instantaneous centre of rotation (ICR) and how is it located?

In one line

The ICR is the point, in or outside the body, that has zero velocity at a given instant, so the body seems to rotate purely about it at that instant.

Explained simply

For a fraction of a second, any moving flat body looks as if it is spinning about one fixed pin. That imaginary pin is the instantaneous centre. A moment later it may be somewhere else, so it is 'instantaneous'.

Technically, in general plane motion, the velocity of any point P is \( v_P = \omega \cdot IP \), perpendicular to the line IP, where I is the ICR and \( \omega \) the body's angular velocity.

How it works

1. Find two points A and B on the body whose velocity directions are known.
2. Draw lines through A and B perpendicular to their velocities.
3. Their intersection is the ICR, I.
4. \( \omega = v_A / IA \); then \( v_B = \omega \cdot IB \), direction perpendicular to IB.
5. Use geometry (sine rule or right triangles) to get IA and IB.
6. Special cases: rolling without slip: I is the contact point; if the perpendiculars are parallel (velocities parallel and equal), I is at infinity and the body is momentarily translating (\( \omega = 0 \)).

Key formula

\( \omega = \dfrac{v_A}{IA} = \dfrac{v_B}{IB} \); v in m/s, distances in m, \( \omega \) in rad/s.

Quick example

A 5 m rod AB slides with A on the floor and B on a wall, at 30° to the floor; A moves away from the wall at 2 m/s. Perpendiculars meet at I, where IA = \( 5\sin30^\circ = 2.5 \) m and IB = \( 5\cos30^\circ = 4.33 \) m. \( \omega = 2/2.5 = 0.8 \) rad/s; \( v_B = 0.8 \times 4.33 = 3.46 \) m/s downward.

Common mistakes

• Drawing lines along the velocities instead of perpendicular to them.
• Using ICR to find accelerations; the ICR generally has non-zero acceleration, so the method is only for velocities.
• Using the rod length instead of the distance from I.

How MU asks it

Theory 3 to 4 marks: 'Define the ICR and explain how to locate it'. Numerical 6 to 8 marks: sliding rods, ladders, rolling wheels, two-link mechanisms (syllabus limits to 2 links, no relative velocity method). Draw the configuration to scale-like proportions, mark I, show the geometry for IA and IB.