---
title: "Explain general plane motion with example"
subject: "Applied Mechanics (Dynamics)"
chapter: 4
key: "semester_3/applied_mechanics_dynamics:general-plane-motion-explain-example"
source: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/general-plane-motion-explain-example
site: BCT Engineering
---

# Explain general plane motion with example

Applied Mechanics (Dynamics) · Chapter 4: Kinematics of Rigid Bodies

- Asked in: 2075 Chaitra, 4 marks
- Video: [Intro to General Planar Motion - Engineering Dynamics](https://www.youtube.com/watch?v=KDhHTHfPESg) — Devin Berg
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/general-plane-motion-explain-example
- Whole chapter: https://www.bctengineer.com/md/semester_3/applied_mechanics_dynamics/chapter-4

## Question

Explain general plane motion of rigid bodies with suitable example.

## Short answer

General plane motion is motion where all particles stay in parallel planes, combining translation of a reference point and rotation about it. For two points, v_B = v_A + ω×r_B/A, a_B = a_A + α×r_B/A - ω² r_B/A. Example: a rolling wheel, where the centre translates and the wheel rotates.

## Answer

**General plane motion (GPM)** is the motion of a rigid body in which every particle stays in a plane parallel to a fixed plane, and the motion is neither pure translation nor pure rotation about a fixed axis. It is the sum of (i) a translation of a chosen reference point and (ii) a rotation of the body about that point.

For two points $A$ and $B$ of the same rigid body:

$$
\vec v_B = \vec v_A + \vec\omega \times \vec r_{B/A}, \qquad \vec a_B = \vec a_A + \vec\alpha \times \vec r_{B/A} - \omega^2 \vec r_{B/A}
$$

Here $v_{B/A} = \omega\, r_{B/A}$ is perpendicular to $AB$, and $a_{B/A}$ has a tangential part $\alpha r$ and a normal part $\omega^2 r$ towards $A$.

**Examples**
- A wheel rolling on the road: the centre translates while the wheel rotates about the centre.
- The connecting rod of an engine: end $A$ (crank pin) moves on a circle, end $B$ (piston) moves in a straight line.
- A ladder sliding down a wall.

**Worked example.** For a wheel of radius $r$ rolling without slipping with centre speed $v$: the centre $O$ translates with $v$ and the wheel rotates with $\omega = v/r$. The top point has $v_{top} = v + \omega r = 2v$ and the contact point has $v_{contact} = v - \omega r = 0$, so the contact point is the instantaneous centre.

```text
        top: 2v ->
         .---.
   v ->  | O |
         '---'
     contact point: v = 0
```

## Check yourself

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

1. General plane motion of a rigid body can be regarded as:
   A. Pure rotation about a fixed axis
   B. A translation of a reference point plus a rotation about that point
   C. Pure translation only
   D. Two independent rotations
   Answer: B. GPM is the sum of translation of a chosen point and rotation of the body about it. (Key point: What GPM is)

2. For two points A and B of the same rigid body, v_B/A is:
   A. Along AB with magnitude omega r
   B. Zero
   C. Perpendicular to AB with magnitude omega r_B/A
   D. Equal to v_A
   Answer: C. v_B/A = omega x r_B/A is perpendicular to AB with magnitude omega r_B/A. (Key point: Relative velocity)

3. For a wheel rolling without slipping with centre speed v, what are the speeds of the top and contact points?
   A. Top v, contact v
   B. Top 2v, contact 0
   C. Top 2v, contact v
   D. Top 0, contact 2v
   Answer: B. Top: v + omega r = 2v; contact: v - omega r = 0, so the contact point is the instantaneous centre. (Key point: Rolling wheel example)

4. In a_B = a_A + alpha x r - omega^2 r, the normal part of a_B/A is:
   A. alpha r, away from A
   B. omega^2 r, directed towards A
   C. omega r, towards A
   D. omega^2 r, away from A
   Answer: B. The normal component is omega^2 r directed towards the reference point A. (Key point: Acceleration components)

## Related in Applied Mechanics (Dynamics)

- [Angular momentum of rigid body in plane motion](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Aangular-momentum-rigid-body-plane) — Applied Mechanics (Dynamics), chapter 5
- [General plane motion; angular acceleration of arm DE](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Ageneral-plane-motion-arm-de-acceleration) — Applied Mechanics (Dynamics), chapter 4
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## Related in other subjects

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