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

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 Momentum ( $H$ )

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 Point

Calculates angular momentum about a fixed point O.

H_O = I_O \omega

Variables

Symbol	Description	Unit
$H_O$	Angular momentum about fixed point O	kg·m²/s
$I_O$	Mass moment of inertia about fixed point 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.

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.