---
title: "Impulse-momentum and eccentric impact"
subject: "Engineering Mechanics"
chapter: 8
key: "semester_3/engineering_mechanics:angular-momentum-rigid-body-eccentric-impact"
source: https://www.bctengineer.com/q/semester_3/engineering_mechanics/angular-momentum-rigid-body-eccentric-impact
site: BCT Engineering
---

# Impulse-momentum and eccentric impact

Engineering Mechanics · Chapter 8: Plane Motion of Rigid Bodies: Impulse and Momentum Method

- Asked in: year not given, 6 marks
- Video: [Impact in Dynamics Explained: Central, Oblique & Eccentric Impact (2026)](https://www.youtube.com/watch?v=1Y7z8qfuVCI) — ADANE TEACH📚
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/engineering_mechanics/angular-momentum-rigid-body-eccentric-impact
- Whole chapter: https://www.bctengineer.com/md/semester_3/engineering_mechanics/chapter-8

## Question

Explain the principle of impulse and momentum for a rigid body in plane motion. State the conditions for conservation of linear and angular momentum. Explain eccentric impact and give the equations used to analyse it.

## Short answer

The impulse-momentum principle: linear m v_G1 + Σ∫F dt = m v_G2; angular I_G ω₁ + Σ∫M_G dt = I_G ω₂. Conservation if external impulse is zero. Eccentric impact uses these plus restitution e = (v'_sep)/(v_app) along line of impact; angular momentum about the pin is conserved if no external moment.

## Answer

#### Impulse and momentum for a rigid body in plane motion
The linear and angular momenta of the body are $m\mathbf{v}_G$ and $I_G\omega$ (angular momentum about G). The principle states that the initial momenta plus the impulses of the external forces equal the final momenta:

$$
m\mathbf{v}_{G1} + \sum\int\mathbf{F}\,dt = m\mathbf{v}_{G2}
$$

$$
I_G\omega_1 + \sum\int M_G\,dt = I_G\omega_2
$$

Angular momentum and angular impulse can also be taken about any fixed point O, provided the moment of the linear momentum is included: $H_O = I_G\omega + m v_G d$ ($d$ is the perpendicular distance from O to the line of $\mathbf{v}_G$). For fixed-axis rotation, $H_O = I_O\omega$.

#### Conservation
- **Linear momentum** is conserved in a direction if the external linear impulse in that direction is zero.
- **Angular momentum** about a point (or axis) is conserved if the angular impulse of the external forces about it is zero, for example when the forces pass through the point, or for an isolated rotating system (a diver, an ice skater, a man walking on a platform).

#### Eccentric impact
Eccentric impact occurs when the line of impact does not pass through the mass centre of one or both colliding bodies; the impulse then changes both the linear and the angular velocity. The usual case is a particle (or body) striking a rigid body that is free to rotate about a pin.

The following equations are used (smooth contact, impulse $P$ along the line of impact):
1. Linear impulse-momentum for each body along the line of impact.
2. Angular impulse-momentum for the rotating body about its pin (or G), with impulse arm $d$: $P\,d = I\,(\omega_2 - \omega_1)$. For the two bodies together, angular momentum about the pin is conserved, because the pin impulse has no moment about the pin.
3. Restitution: $e = \dfrac{(v_2' - v_1')_n}{(v_1 - v_2)_n}$, the ratio of the separation speed to the approach speed of the contact points, measured along the line of impact.

The reaction at the pin during impact is an impulsive force, unless the impact is at the centre of percussion, where the pin impulse is zero.

## Check yourself

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

1. What does the angular impulse-momentum principle state about G?
   A. I_G omega_1 + angular impulse of external moments = I_G omega_2
   B. I_G omega_1 = m v_G
   C. Angular impulse equals change in kinetic energy
   D. Sum M_G = m v_G
   Answer: A. Initial angular momentum plus angular impulse equals final angular momentum. (Key point: Impulse-momentum equations)

2. When is angular momentum about a point conserved?
   A. When the body is rigid
   B. When the angular impulse of external forces about that point is zero
   C. When kinetic energy is conserved
   D. When linear momentum is conserved
   Answer: B. Zero external angular impulse about the point leaves H unchanged. (Key point: Conservation of angular momentum)

3. What defines eccentric impact?
   A. The impact is perfectly elastic
   B. The line of impact does not pass through the mass centre of one or both bodies
   C. The bodies stick together
   D. The collision happens at an angle to the ground
   Answer: B. The impulse then changes both linear and angular velocity. (Key point: Eccentric impact)

4. Restitution in eccentric impact is the ratio of:
   A. Final to initial kinetic energy
   B. Separation speed to approach speed of contact points along the line of impact
   C. Final to initial angular velocity
   D. Impulse of deformation to total mass
   Answer: B. e = (v2' - v1')n / (v1 - v2)n measured at the contact points. (Key point: Restitution)

## Related in Engineering Mechanics

- [Equations of motion for plane motion](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Aequations-plane-motion-rigid-body) — Engineering Mechanics, chapter 6
- [Principle of impulse and momentum](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Aimpulse-momentum-principle-particle) — Engineering Mechanics, chapter 4
- [Kinetic energy and work of rigid body](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Akinetic-energy-rigid-body-work-energy) — Engineering Mechanics, chapter 7
- [Plane motion and relative velocity](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Aplane-motion-types-relative-velocity) — Engineering Mechanics, chapter 5
- [Lagrange's equations from D'Alembert's principle](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Alagrange-equations-derivation-dalembert) — Engineering Mechanics, chapter 9

## Related in other subjects

- [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
- [Impulsive motion, eccentric impact; rod with spring](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Arod-spring-4kg-impulsive-eccentric) — Applied Mechanics (Dynamics), chapter 6, asked 2×
- [Uniformly accelerated motion; angular momentum definition](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Auniformly-accelerated-motion-angular-momentum) — Applied Mechanics, chapter 8
- [Linear momentum and impulse-momentum equation](https://www.bctengineer.com/question.md?key=semester_5%2Ffluid_machines%3Alinear-momentum-impulse-equation) — Fluid Machines, chapter 1