Kinetics of Rigid Bodies: Impulse and Momentum

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.

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.

Equations of Impulse and Momentum

  1. Linear Impulse-Momentum: m(vG)1+∑∫t1t2F dt=m(vG)2m(v_G)_1 + \sum \int_{t_1}^{t_2} \mathbf{F} \, dt = m(v_G)_2 The change in linear momentum of the center of mass GG is equal to the linear impulse of external forces.
  2. Angular Impulse-Momentum (about GG): IGω1+∑∫t1t2MG dt=IGω2I_G \omega_1 + \sum \int_{t_1}^{t_2} M_G \, dt = I_G \omega_2 The change in angular momentum about GG is equal to the angular impulse of external moments about GG.

Angular Momentum (HH)

  • About Center of Mass (GG): HG=IGωH_G = I_G \omega
  • About Fixed Point (OO): HO=IOωH_O = I_O \omega

System of Rigid Bodies

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

Impulse-Momentum for a System

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.
The total linear momentum Lsys\mathbf{L}_{sys} is the sum of the linear momenta of each individual body:
(Lsys)1+∑∫t1t2Fextdt=(Lsys)2(\mathbf{L}_{sys})_1 + \sum \int_{t_1}^{t_2} \mathbf{F}_{ext} dt = (\mathbf{L}_{sys})_2
Where Lsys=∑mi(vG)i\mathbf{L}_{sys} = \sum m_i (\mathbf{v}_{G})_i.
Similarly, the total angular momentum Hsys\mathbf{H}_{sys} about a fixed point OO is the sum of the angular momenta of each individual body about OO:
(Hsys)1+∑∫t1t2(MO)extdt=(Hsys)2(\mathbf{H}_{sys})_1 + \sum \int_{t_1}^{t_2} (M_O)_{ext} dt = (\mathbf{H}_{sys})_2
Where Hsys=∑(ri×mivi+IG,iωi)\mathbf{H}_{sys} = \sum (\mathbf{r}_i \times m_i \mathbf{v}_i + I_{G,i} \mathbf{\omega}_i).

Conservation of Momentum

The conservation laws are powerful when analyzing the interaction of bodies where external impulses are negligible (e.g., collisions).

Conservation Principles

  • Linear Momentum Conserved: If ∑∫F dt=0\sum \int \mathbf{F} \, dt = 0 (no net external linear impulse), then L1=L2\mathbf{L}_1 = \mathbf{L}_2. This often happens in free-space collisions or explosions.
  • Angular Momentum Conserved: If ∑∫MP dt=0\sum \int M_P \, dt = 0 (no net external angular impulse about point PP), then (HP)1=(HP)2(\mathbf{H}_P)_1 = (\mathbf{H}_P)_2. This happens when all external forces pass through a common point PP or are zero.

Eccentric Impact

When a rigid body is subjected to an eccentric impact—that is, an impact where the line of action of the impulsive forces does not pass through the mass center—it will undergo a change in both translational and rotational velocity.

Eccentric Impact Principles

  • The point of impact (e.g., a ball hitting a bat) experiences a large impulsive force ∫Fdt\int F dt.
  • This impulsive force changes the linear momentum of the body: m(vG)1+∫Fdt=m(vG)2m(\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 GG, the impulsive force creates an angular impulse. Thus, it changes the angular momentum of the body: IGω1+∫r×Fdt=IGω2I_G \omega_1 + \int \mathbf{r} \times \mathbf{F} dt = I_G \omega_2.
  • If an object is constrained at a pin (like a pendulum), the reaction at the pin also creates an impulse. Analyzing the angular momentum about the fixed pivot avoids having to solve for the pin reaction.
Interact with the simulation below to explore eccentric impact on a rod.

Angular Momentum Conservation: Bullet & Rod

Initial Ang. Mom. (HOH_O)2.00 kg·m²/s
Total Inertia (I_total)1.336 kg·m²
Final Ang. Vel. (ω\omega)1.50 rad/s
Key Takeaways
  • Impulse-Momentum Principle (L1+ΣI=L2L_1 + \Sigma I = L_2) applies to both linear (mvmv) and angular (IωI\omega) momentum.
  • System of Rigid Bodies: Internal impulses cancel; only external forces and moments change the total linear and angular momentum.
  • Angular Momentum (H=IωH = 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.
  • Radius of Gyration (kk) relates mass and moment of inertia (I=mk2I = mk^2).
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