---
title: "Angular momentum; rate of change equals moment"
subject: "Applied Mechanics"
chapter: 9
key: "semester_1/applied_mechanics:angular-momentum-rate-equals-moment-proof"
source: https://www.bctengineer.com/q/semester_1/applied_mechanics/angular-momentum-rate-equals-moment-proof
site: BCT Engineering
---

# Angular momentum; rate of change equals moment

Applied Mechanics · Chapter 9: Kinetics of Particles and Rigid Body: Force and Acceleration

- Asked in: 2067 Asar, 6 marks
- Video: [Dynamics | Angular momentum and rate of change of angular momentum| Applied mechanics 2](https://www.youtube.com/watch?v=ANXDjqzHiBM) — Sudip Dhakal
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/applied_mechanics/angular-momentum-rate-equals-moment-proof
- Whole chapter: https://www.bctengineer.com/md/semester_1/applied_mechanics/chapter-9

## Question

Define angular momentum and also prove that rate of change of angular momentum is equal to the moment of the force acting on that particle about the same point.

## Answer

#### Angular momentum
The **angular momentum** (moment of momentum) of a particle about a fixed point O is the moment of its linear momentum about O:

H_O = r × m v

where r is the position vector of the particle from O and m v its linear momentum. Its magnitude is H_O = m v r sin φ (φ = angle between r and v), with units kg·m²/s. For motion in a plane in polar coordinates, H_O = m r² θ̇.

#### Proof
Differentiate H_O = r × m v with respect to time:

dH_O/dt = (dr/dt) × m v + r × m (dv/dt)

- The first term is v × m v = m (v × v) = 0.
- In the second term m (dv/dt) = m a = F, by Newton's second law, F being the resultant force.

Therefore **dH_O/dt = r × F = M_O**

The rate of change of the angular momentum about a point equals the moment of the resultant force about the same point. If M_O = 0 (for example for a central force), H_O is constant, which is the principle of conservation of angular momentum.

## Check yourself

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

1. What is the angular momentum of a particle about O?
   A. H_O = m v
   B. H_O = r x m v, the moment of linear momentum, magnitude m v r sin(phi)
   C. H_O = m a r
   D. H_O = F x r
   Answer: B. It is the moment of linear momentum about O, in kg.m^2/s (m r^2 theta_dot in polar plane motion). (Key point: Definition of angular momentum)

2. In dH_O/dt = v x m v + r x m a, why does the first term vanish?
   A. v is constant
   B. r = 0
   C. A vector crossed with itself (parallel) gives zero
   D. Mass is constant
   Answer: C. v x m v = m(v x v) = 0, because the cross product of parallel vectors is zero. (Key point: Why the first term vanishes)

3. What follows if the moment of the force M_O = 0, as for a central force?
   A. H_O is zero
   B. H_O is constant, giving conservation of angular momentum
   C. Velocity must be zero
   D. Momentum doubles
   Answer: B. Since dH_O/dt = M_O, M_O = 0 means H_O does not change. (Key point: Conservation consequence)

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