Instantaneous Centre of Rotation

Think of a bicycle wheel moving in a smooth motion over a road. At any given moment, as the wheel is rotating and moving ahead, there is a special point, the point that touches the road, that is stationary at that moment. The whole wheel seems to rotate around this point at that time. This point is referred to as the Instantaneous Centre of Rotation (ICR). The idea of ICR enables us to reduce complicated motion to an instantaneous pure rotation around one point. It is a fundamental concept in kinematic and rotational dynamics studies, particularly for planar rigid bodies in motion.

Definition

The point in or outside a moving body about which all other points in the body are rotating at a given instant, and which has zero instantaneous velocity.

It's referred to as "instantaneous" since it is the centre of rotation only at that specific moment. The position of the ICR could also change as the object moves on.

Formula

If a body rotates about an ICR with angular velocity ω, then the linear velocity v of any point on the body is:

Formula

where:

v – velocity of the point
ω – angular velocity
r – distance between the point and the ICR

Pure Rotation and General Motion

There are two modes of motion for rigid bodies:

Pure Rotation – Every point in the body traces out circles about a fixed axis. The ICR is stationary.

General Plane Motion – Both translation and rotation. The ICR is able to move and might not be within the body itself.

ICR in Rolling Without Slipping

It is a special case of a wheel rolling without slipping.

The contact point with the ground has zero velocity, so it is the ICR. The centre of the wheel advances while the remaining wheel revolves about the ICR. This concept makes rolling motion analysis simpler, which is prevalent in vehicles, gears, and rotating equipment.

Velocity at the top of the wheel = 2v

Centre point velocity = v (same as translational speed)

Velocity at the bottom point = 0 (ICR).

Points to Remember

ICR is not necessarily within the body. It can be external.
It's only useful for velocity, not acceleration or forces.
ICR is applied only to rigid bodies and to planar motion.

Advantages of ICR

Simplifies velocity analysis of complex motion.
Provides a geometric insight into motion.
Reduces the need for complex calculus.

Disadvantages of ICR

It is applicable to velocity but not to acceleration or force analysis.
The ICR changes with time, particularly in non-uniform or complex motion.
For 3D motion, the idea becomes less practical and more difficult to visualise.

Examples & Applications

Swinging Door: The hinge is a fixed ICR around which the door swings.
Human Joints: When a person moves or walks, their knees and elbows are shifting ICRs.
Engineering: Examining mechanisms such as linkages, cams, and wheels.
Vehicle Dynamics: The study of tire motion and steering systems.
Biomechanics: Modelling human joint motion for sports science and medical diagnostics.
Robotics: Planning motion in robotic arms and rotating components.

Summing Up

The Instantaneous Centre of Rotation is a great physics and engineering tool for the analysis of rigid body motion. In transforming complex motion into easy, instantaneous rotation about a point, the ICR provides both a geometric and algebraic benefit. From spinning wheels to robot joints, knowing the ICR helps us to solve practical problems of motion better. Though limited to velocity analysis and 2D motion, its usefulness across many domains makes it an essential concept in mechanics.

Frequently Asked Questions

Q1. In rolling without slipping, where is the ICR?

It lies at the point of contact between the rolling object and the surface, because that point has zero velocity at that instant.

Q2. What is the difference between the centre of mass (COM) and ICR?

The centre of mass is a fixed point representing the average position of the mass distribution in the object. The ICR is a point about which instantaneous rotation occurs (not always within the object) around which it seems to turn at the instant. COM can or cannot be located at the ICR.

Q3. Is the ICR an actual physical point?

Not necessarily. The ICR is usually a geometric or virtual point. It can be inside the body, on the body, or even outside the body.