---
title: "Angular momentum of rigid body in plane motion"
subject: "Applied Mechanics (Dynamics)"
chapter: 5
key: "semester_3/applied_mechanics_dynamics:angular-momentum-rigid-body-plane"
source: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/angular-momentum-rigid-body-plane
site: BCT Engineering
---

# Angular momentum of rigid body in plane motion

Applied Mechanics (Dynamics) · Chapter 5: Plane Motion of Rigid Bodies: Forces, Moments, and Accelerations

- Asked in: 2076 Asoj, 4 marks
- Video: [Angular Momentum of a Rigid Body in Planar Motion | Physics & Mechanical Engineering](https://www.youtube.com/watch?v=C98fvWtbcE4) — Derivability
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/angular-momentum-rigid-body-plane
- Whole chapter: https://www.bctengineer.com/md/semester_3/applied_mechanics_dynamics/chapter-5

## Question

Define angular momentum for a rigid body in plane motion with examples.

## Short answer

Angular momentum in plane motion: about the mass centre, H_G = ¯ I ω; about any point O, H_O = ¯ I ω + m v d, where d is the perpendicular distance from O to the linear momentum vector. Examples: a flywheel (20 kg, k=0.3 m, 100 rad/s) gives H = 180 kg·m²/s; a rolling uniform disk gives H_G = 5 kg·m²/s, H_C = 15 kg·m²/s.

## Answer

**Angular momentum** of a rigid body in plane motion is the sum of the moments of the momenta of all its particles about a point. For a body of mass $m$ in plane motion with angular velocity $\omega$, the momenta of the particles are equivalent to the linear momentum $m\bar{\vec v}$ at the mass centre $G$ together with a couple $\bar I\omega$.

#### About the mass centre
$$
H_G = \bar I\,\omega
$$

where $\bar I$ is the moment of inertia about the axis through $G$ perpendicular to the plane. This is true whether or not $G$ is moving.

#### About any other point O
$$
H_O = \bar I\omega + (\bar{\vec r}\times m\bar{\vec v}) = \bar I\omega + m\bar v\,d
$$

where $d$ is the perpendicular distance from $O$ to the line of action of $m\bar{\vec v}$.

#### Special cases
| Motion | Angular momentum |
|---|---|
| Rotation about fixed axis $O$ | $H_O = I_O\omega$ |
| Pure translation | $H_G = 0$, $H_O = m\bar v d$ |
| Rolling without slipping | $H_C = I_C\omega$ about the contact point (instantaneous centre) |

The rate of change of angular momentum equals the sum of moments: $\sum M_G = \dot H_G = \bar I\alpha$.

#### Examples
1. A flywheel of mass 20 kg and radius of gyration 0.3 m turning at 100 rad/s about its axis: $H = mk^2\omega = 20(0.3)^2(100) = 180$ kg m²/s.
2. A uniform disk (mass 10 kg, radius 0.5 m) rolling at $\bar v = 2$ m/s: $\omega = 4$ rad/s, $H_G = \tfrac12(10)(0.5)^2(4) = 5$ kg m²/s; about the contact point $H_C = 5 + 10(2)(0.5) = 15$ kg m²/s, which equals $I_C\omega = \tfrac32(10)(0.25)(4)= 15$ kg m²/s.

## Check yourself

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

1. For a rigid body in plane motion, the angular momentum about its mass centre is:
   A. m v d
   B. I_bar omega, whether or not G moves
   C. I_bar alpha
   D. m v^2
   Answer: B. H_G = I_bar omega, regardless of motion of G. (Key point: Angular momentum about G)

2. The angular momentum about any other point O is:
   A. I_bar omega only
   B. m v d only
   C. I_bar omega + m v_bar d
   D. I_bar omega - m v_bar d always
   Answer: C. Add the moment of the mass-centre linear momentum, m v_bar d, about O. (Key point: About another point)

3. Uniform disk 10 kg, r = 0.5 m, v = 2 m/s rolling: H_G and H_C (contact point) are:
   A. 5 and 15 kg m^2/s
   B. 5 and 5 kg m^2/s
   C. 10 and 20 kg m^2/s
   D. 15 and 5 kg m^2/s
   Answer: A. omega = 4 so H_G = 0.5(10)(0.25)(4) = 5; H_C = 5 + 10(2)(0.5) = 15. (Key point: Rolling example)

4. The rate of change of angular momentum about G equals:
   A. Sum of forces
   B. Sum of moments about G, which is I_bar alpha
   C. m a
   D. Zero always
   Answer: B. Sum M_G = dH_G/dt = I_bar alpha. (Key point: Rate of change)

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