---
title: "Locating the instantaneous centre"
subject: "Engineering Mechanics"
chapter: 5
key: "semester_3/engineering_mechanics:instantaneous-centre-locating-methods"
source: https://www.bctengineer.com/q/semester_3/engineering_mechanics/instantaneous-centre-locating-methods
site: BCT Engineering
---

# Locating the instantaneous centre

Engineering Mechanics · Chapter 5: Kinematics of Rigid Bodies

- Asked in: year not given, 5 marks
- Video: [Instantaneous Center of Zero Velocity (learn to solve any problem step by step)](https://www.youtube.com/watch?v=1aQ9EZGMdDk) — Question Solutions
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/engineering_mechanics/instantaneous-centre-locating-methods
- Whole chapter: https://www.bctengineer.com/md/semester_3/engineering_mechanics/chapter-5

## Question

Define the instantaneous centre of rotation. Explain, with sketches, how it is located when (a) the directions of velocities at two points are known, (b) the velocities at two points are parallel and perpendicular to the line joining them (give both cases), and (c) a body rolls without slipping. State whether the acceleration of the instantaneous centre is zero.

## Short answer

Instantaneous centre is the point with zero velocity in plane motion, about which the body momentarily rotates. It is located by intersecting perpendiculars to known velocities of two points, joining velocity tips when parallel and perpendicular to the line joining points, or at the contact point for rolling without slipping. Its acceleration is generally non-zero.

## Answer

**Instantaneous centre (IC)** of a body in plane motion is the point (on the body or its extension) whose velocity is zero at that instant. At that instant the body appears to rotate about it with angular velocity $\omega$, so the velocity of any point P is $v_P = \omega\,(IP)$, perpendicular to $IP$.

#### Methods of locating the IC

1. **Directions of velocities of two points known (not parallel).** Draw perpendiculars to $v_A$ at A and to $v_B$ at B. Their intersection is the IC (example: ends of a sliding ladder, or crank pin and slider of an engine).

```text
      A ->v_A     perpendicular lines meet at I
      |
      |       B
      |      /|
      +-----I-+ 
```

2. **Velocities parallel at A and B and both perpendicular to AB.** Join the tips of the velocity vectors with a straight line; its intersection with the line AB gives the IC. If the vectors point the same way the IC lies outside AB (on the side of the smaller velocity); if they are opposite it lies between A and B. Because $v_A/IA = v_B/IB$, this follows from similar triangles.

3. **Velocities parallel but not perpendicular to AB, equal in magnitude.** The IC is at infinity: the body is in instantaneous translation ($\omega = 0$).

4. **Rolling without slipping.** The point of contact C with the fixed surface has zero velocity, so the IC is at the contact point. For a wheel of radius $r$: $\omega = v_O/r$, and the top point moves at $2v_O$.

#### Is the acceleration of the IC zero?
No. The IC has zero velocity but in general it has acceleration, so it must not be used to find accelerations as if it were a fixed point. For a rolling wheel, for example, the contact point has an acceleration $\omega^2 r$ directed toward the wheel centre.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The instantaneous centre is:
   A. The centre of mass
   B. The point with zero velocity at that instant
   C. The point with zero acceleration
   D. The geometric centre of the body
   Answer: B. Velocities then are v_P = omega x IP, perpendicular to IP. (Key point: Definition)

2. For velocities at A and B parallel and perpendicular to AB, with opposite directions, the IC lies:
   A. Outside AB
   B. Between A and B, where the line joining the tips of the vectors crosses AB
   C. At infinity
   D. At A
   Answer: B. The same-direction case gives the IC outside AB; opposite directions put it between A and B. (Key point: Parallel velocities)

3. For a wheel rolling without slipping, the IC is:
   A. The wheel centre
   B. The top point
   C. The contact point, so the top moves at 2 v_O
   D. At infinity
   Answer: C. The contact point has zero velocity, and the top is at distance 2r so it moves at 2v_O. (Key point: Rolling)

4. Is the acceleration of the instantaneous centre zero?
   A. Yes, since its velocity is zero
   B. No; for a rolling wheel the contact point has acceleration omega^2 r toward the centre
   C. Yes for rolling only
   D. Only if omega is constant
   Answer: B. Zero velocity does not mean zero acceleration, so the IC cannot be used to find accelerations. (Key point: Acceleration of the IC)

## Related in Engineering Mechanics

- [Sliding ladder by instantaneous centre](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Asliding-ladder-instantaneous-centre) — Engineering Mechanics, chapter 5
- [Accelerations of points on rolling wheel](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Arolling-wheel-acceleration-points) — Engineering Mechanics, chapter 5
- [Cylinder rolling down an incline](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Acylinder-rolling-down-incline) — Engineering Mechanics, chapter 6
- [Composite body moment of inertia](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Arod-sphere-composite-inertia) — Engineering Mechanics, chapter 6
- [Rolling cylinder attached to a spring](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Arolling-cylinder-spring-energy) — Engineering Mechanics, chapter 7
- [Acceleration analysis of slider-crank](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Acrank-slider-acceleration-numerical) — Engineering Mechanics, chapter 5

## Related in other subjects

- [Velocities in plane motion by instantaneous centre](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Aladder-sliding-wheel-rolling-velocities) — Theory of Mechanism and Machine I, chapter 7
- [Rigid body motion types; double gear velocities](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Adouble-gear-rack-velocity-types-motion) — Applied Mechanics (Dynamics), chapter 4
- [Zero pressure difference under upward acceleration](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Aupward-acceleration-pressure-difference) — Fluid Mechanics, chapter 2
- [Slider crank velocity by instantaneous centre method](https://www.bctengineer.com/question.md?key=semester_6%2Ftheory_of_mechanism_and_machine_i%3Aslider-crank-velocity-instantaneous-centre) — Theory of Mechanism and Machine I, chapter 7