Kinetics of Rigid Bodies: Impulse and Momentum

Kinetics of Rigid Bodies: Impulse and Momentum

$Learning Objectives$

Define linear and angular impulse and momentum for a rigid body.
Apply the principle of impulse and momentum to solve kinetics problems involving force, mass, velocity, and time.
Analyze systems of interacting rigid bodies.
Apply conservation of linear and angular momentum to collision and impact problems.
Understand the mechanics of eccentric impact.

This method relates forces and moments acting over a time interval to the change in both linear and angular momentum of the body. It is particularly powerful for impact problems and situations involving variable forces, providing a direct relation between time and velocity.

$Civil Engineering Applications$

Impulse and momentum methods are frequently applied when examining impulsive impact forces in civil structures:

Vehicular Collision: Analyzing impact forces of cars or ships colliding with bridge piers over short time durations.
Pile Driving: Studying the rapid transfer of momentum from the falling pile hammer to the driven pile.
Dynamic Impact: Evaluating the response of slabs when struck by dropped objects in industrial facilities.

$Impulse$

The integral of a force or couple moment over the time interval during which it acts. It represents the total "push" or "twist" given to a body.

$Momentum$

The product of a body's mass or mass moment of inertia and its corresponding velocity. Linear momentum is $m\mathbf{v}$ , and angular momentum is $I\mathbf{\omega}$ .

Principle of Impulse and Momentum

The principle states that the initial momentum of the system plus the sum of all external impulses acting on the system equals the final momentum of the system.

Linear Impulse-Momentum Equation

The change in linear momentum of the center of mass G is equal to the linear impulse of external forces.

m(v_G)_1 + \sum \int_{t_1}^{t_2} \mathbf{F} \, dt = m(v_G)_2

Variables

Symbol	Description	Unit
$m$	Mass of the rigid body	kg
$(v_G)_1, (v_G)_2$	Initial and final velocities of the center of mass	m/s
$\mathbf{F}$	External forces acting on the body	N
$t_1, t_2$	Initial and final times	s

Linear Impulse Changes Center-of-Mass Momentum

Track the vector balance from the initial to final center-of-mass momentum over the impact interval.

Use the vector equation m v_G1 + ΣJ_ext = m v_G2 component by component. Momentum is conserved in a component only when the net external impulse in that component is zero or negligible.

Initial momentum of the center of mass

Add the net external force impulse over the interval

Obtain final center-of-mass momentum

Angular Impulse-Momentum Equation

The change in angular momentum about the center of mass G is equal to the angular impulse of external moments about G.

I_G \omega_1 + \sum \int_{t_1}^{t_2} M_G \, dt = I_G \omega_2

Variables

Symbol	Description	Unit
$I_G$	Mass moment of inertia about the center of mass	kg·m²
$\omega_1, \omega_2$	Initial and final angular velocities	rad/s
$M_G$	External moments about the center of mass	N·m
$t_1, t_2$	Initial and final times	s

Angular Impulse Changes Spin About G

Relate the angular impulse of external moments about the center of mass to the change in angular momentum.

For planar motion, use H_G = I_Gω. Angular momentum about G is conserved only when the net external moment impulse about G is zero or negligible.

Initial angular momentum about G

Add external moment impulse about G

Obtain final angular momentum about G

$H$

Angular momentum represents the quantity of rotation of a body, which is the product of its moment of inertia and its angular velocity.

Angular Momentum About Center of Mass

Calculates angular momentum about the center of mass G.

H_G = I_G \omega

Variables

Symbol	Description	Unit
$H_G$	Angular momentum about G	kg·m²/s
$I_G$	Mass moment of inertia about G	kg·m²
$\omega$	Angular velocity	rad/s

Angular Momentum About a Fixed Axis Through O

For planar rigid-body rotation about a fixed axis through O, angular momentum about O is the mass moment of inertia about that same axis times the signed angular velocity.

H_O = I_O \omega

Variables

Symbol	Description	Unit
$H_O$	Angular momentum about fixed point O	kg·m²/s
$I_O$	Planar mass moment of inertia about the fixed axis through O	kg·m²
$\omega$	Angular velocity	rad/s

System of Rigid Bodies

When multiple rigid bodies interact, the principles of linear and angular impulse-momentum can be applied to the entire system.

$System of Interacting Bodies$

For a system of interacting rigid bodies, the internal forces between them generate equal and opposite impulses that sum to zero. Therefore, only external forces create a net impulse on the system.

Linear Impulse-Momentum for a System

The total linear momentum of the system is the sum of the linear momenta of each individual body.

(\mathbf{L}_{sys})_1 + \sum \int_{t_1}^{t_2} \mathbf{F}_{ext} dt = (\mathbf{L}_{sys})_2

Variables

Symbol	Description	Unit
$(\mathbf{L}_{sys})_1, (\mathbf{L}_{sys})_2$	Initial and final total linear momenta of the system	kg·m/s
$\mathbf{F}_{ext}$	External forces acting on the system	N
$t_1, t_2$	Initial and final times	s

Angular Impulse-Momentum for a System

The total angular momentum of the system about a fixed point O is the sum of the angular momenta of each body about O.

(\mathbf{H}_{sys})_1 + \sum \int_{t_1}^{t_2} (M_O)_{ext} dt = (\mathbf{H}_{sys})_2

Variables

Symbol	Description	Unit
$(\mathbf{H}_{sys})_1, (\mathbf{H}_{sys})_2$	Initial and final total angular momenta about O	kg·m²/s
$(M_O)_{ext}$	External moments about O acting on the system	N·m
$t_1, t_2$	Initial and final times	s

Conservation of Momentum

The conservation laws are powerful when analyzing the interaction of bodies where external impulses are negligible, such as during collisions or explosions in free space.

$Conservation of Linear Momentum$

If there is no net external linear impulse acting on the system ( $\sum \int \mathbf{F} \, dt = 0$ ), then the total linear momentum is conserved.

Conservation of Linear Momentum Equation

Expresses that the initial total linear momentum equals the final total linear momentum.

\mathbf{L}_1 = \mathbf{L}_2

Variables

Symbol	Description	Unit
$\mathbf{L}_1$	Total initial linear momentum	kg·m/s
$\mathbf{L}_2$	Total final linear momentum	kg·m/s

$Conservation of Angular Momentum$

If there is no net external angular impulse acting about a point P ( $\sum \int M_P \, dt = 0$ ), then the total angular momentum about that point is conserved. This often occurs when all external forces pass through a common point P or are zero.

Conservation of Angular Momentum Equation

Expresses that the initial total angular momentum about P equals the final total angular momentum about P.

(\mathbf{H}_P)_1 = (\mathbf{H}_P)_2

Variables

Symbol	Description	Unit
$(\mathbf{H}_P)_1$	Total initial angular momentum about point P	kg·m²/s
$(\mathbf{H}_P)_2$	Total final angular momentum about point P	kg·m²/s

Eccentric Impact

$Eccentric Impact$

An eccentric impact occurs when the line of action of the impulsive forces acting on a rigid body does not pass through its center of mass, resulting in both translation and rotation.

When a rigid body is subjected to an eccentric impact, it undergoes a change in both translational and rotational velocity.

$Eccentric Impact Principles$

The point of impact experiences a large impulsive force $\int \mathbf{F} dt$ .
This impulsive force changes the linear momentum: $m(\mathbf{v}_G)_1 + \int \mathbf{F} dt = m(\mathbf{v}_G)_2$ .
Because the line of impact does not pass through the center of mass $G$ , the force creates an angular impulse, changing the angular momentum: $I_G \omega_1 + \int \mathbf{r} \times \mathbf{F} dt = I_G \omega_2$ .
If the object is constrained by a pin, analyzing angular momentum about the fixed pivot eliminates the need to solve for the unknown pin reaction impulses.

Why Moments About the Fixed Pivot Remove Its Reaction Impulse

Compare the moment arm of the pivot reaction impulse with the eccentric impact impulse.

For planar rotation about a fixed point P, H_P = I_Pω. This simplification requires P to remain fixed during the impact interval.

Take moments about G

The pin reaction impulse can have a moment arm from G
Its angular impulse may remain unknown

Take moments about fixed P

The pin reaction impulse passes through P
Its moment about P is zero; the eccentric impact impulse contributes r_Q/P × J_Q

Rigid-Body Impact: Impulse–Momentum Equation Selection

Choose the linear and angular impulse–momentum equations that match the impact unknowns and constraints.

Keep external force and moment impulses unless their resultants are zero or negligible over the impact interval in the selected direction or about the selected point. For planar rotation about a fixed pivot P, angular momentum is H_P = I_P ω; an impulsive reaction whose line of action passes through P has zero moment about P. This pivot simplification requires P to remain fixed during the interval.

Start rigid-body impact analysis → Set system, impact interval, axes, constraints, and unknowns; Set system, impact interval, axes, constraints, and unknowns → Is center-of-mass velocity change needed?; Is center-of-mass velocity change needed? — Yes → Write linear impulse–momentum about G; Is center-of-mass velocity change needed? — No → No linear velocity unknown; continue to angular check; Write linear impulse–momentum about G → Is external linear impulse negligible in the component?; Is external linear impulse negligible in the component? — Yes → Conserve total linear momentum in this component; Is external linear impulse negligible in the component? — No → Retain external linear impulse; Conserve total linear momentum in this component → Is angular velocity change needed?; Retain external linear impulse → Is angular velocity change needed?; No linear velocity unknown; continue to angular check → Is angular velocity change needed?; Is angular velocity change needed? — Yes → Is there a fixed pivot with an unknown reaction impulse through it?; Is angular velocity change needed? — No → No angular velocity unknown; solve selected relations; Is there a fixed pivot with an unknown reaction impulse through it? — Yes → Use angular impulse–momentum about fixed P; Is there a fixed pivot with an unknown reaction impulse through it? — No → Use angular impulse–momentum about G; Use angular impulse–momentum about fixed P → Is net external moment impulse about the chosen point negligible?; Use angular impulse–momentum about G → Is net external moment impulse about the chosen point negligible?; Is net external moment impulse about the chosen point negligible? — Yes → Conserve angular momentum about the chosen point; Is net external moment impulse about the chosen point negligible? — No → Retain external moment impulse; Conserve angular momentum about the chosen point → Solve selected equations; check signs and units; Retain external moment impulse → Solve selected equations; check signs and units; No angular velocity unknown; solve selected relations → Solve selected equations; check signs and units; Solve selected equations; check signs and units → Impact response determined

Start rigid-body impact analysis: terminator
Set system, impact interval, axes, constraints, and unknowns: process
Is center-of-mass velocity change needed?: decision
Write linear impulse–momentum about G: process
Is external linear impulse negligible in the component?: decision
Conserve total linear momentum in this component: process
Retain external linear impulse: process
No linear velocity unknown; continue to angular check: process
Is angular velocity change needed?: decision
Is there a fixed pivot with an unknown reaction impulse through it?: decision
Use angular impulse–momentum about fixed P: process
Use angular impulse–momentum about G: process
Is net external moment impulse about the chosen point negligible?: decision
Conserve angular momentum about the chosen point: process
Retain external moment impulse: process
No angular velocity unknown; solve selected relations: process
Solve selected equations; check signs and units: process
Impact response determined: terminator

$Interactive Simulation$

Interact with the simulation below to explore eccentric impact on a rod.

Angular Momentum: Bullet Striking a Rod

System Parameters

Bullet Mass (

m

)0.010 kg

Bullet Speed (

v_0

)400 m/s

Rod Mass (

M

)4 kg

Rod Length (

L

)1.0 m

Impact Point (

d/L

)50% from pivot

Angular Momentum Solver

Conservation of

H_O

m v_0 d = I_{\text{total}}\,\omega

I_{\text{total}} = \tfrac{1}{3}ML^2 + md^2

= 1.333 + 0.0025 = 1.3358\,\text{kg·m}^2

H_0 = 2.0000\,\text{kg·m}^2/\text{s}

\omega = 1.50\,\text{rad/s}

KE before:800.0 J

KE after:1.5 J

Energy lost:798.5 J (99.8%)

Bullet (m = 0.010 kg)

Rod (M = 4 kg)

Impact point d = 0.50 m

Key Takeaways

Impulse-Momentum Principle ( $L_1 + \Sigma I = L_2$ ) applies to both linear ( $mv$ ) and angular ( $I\omega$ ) momentum.
System of Rigid Bodies: Internal impulses cancel; only external forces and moments change the total linear and angular momentum.
Angular Momentum ( $H = I\omega$ ) is conserved when the net external moment is zero.
Impact Problems often use conservation of angular momentum about a fixed pivot to eliminate unknown reaction impulses.
Eccentric Impact imparts both a linear velocity change and a rotational velocity change due to the moment arm of the impulse.

Previous TopicKinetics of Rigid Bodies: Work and Energy - Examples & Applications

Quiz Me

Next TopicKinetics of Rigid Bodies: Impulse and Momentum - Examples & Applications

Prev Next

Quiz Me

Learning Objectives

Civil Engineering Applications

Impulse

Momentum

Principle of Impulse and Momentum

Linear Impulse-Momentum Equation

Angular Impulse-Momentum Equation

Angular Momentum (HHH)

Angular Momentum About Center of Mass

Angular Momentum About a Fixed Axis Through O

System of Rigid Bodies

System of Interacting Bodies

Linear Impulse-Momentum for a System

Angular Impulse-Momentum for a System

Conservation of Momentum

Conservation of Linear Momentum

Conservation of Linear Momentum Equation

Conservation of Angular Momentum

Conservation of Angular Momentum Equation

Eccentric Impact

Eccentric Impact

Eccentric Impact Principles

Interactive Simulation

System Parameters

Angular Momentum Solver

$Learning Objectives$

$Civil Engineering Applications$

$Impulse$

$Momentum$

$H$

$System of Interacting Bodies$

$Conservation of Linear Momentum$

$Conservation of Angular Momentum$

$Eccentric Impact$

$Eccentric Impact Principles$

$Interactive Simulation$