Dynamics: Impulse-Momentum for Rigid Bodies

The principles of impulse and momentum are indispensable tools for analyzing the motion of rigid bodies subjected to sudden, large forces over short time intervals, such as those experienced in impacts, collisions, or explosions. While Newton’s second law in its standard form is ideal for continuous forces, the impulse-momentum approach is far more powerful when dealing with forces acting over very brief durations, where the exact time-history of the force is unknown or highly complex. This framework elegantly connects the cumulative effect of an impulse to a measurable change in the body's momentum, bypassing the need to integrate complex acceleration profiles.

Linear Impulse-Momentum for Rigid Bodies

The linear impulse-momentum equation for a rigid body is a direct extension of the particle principle. It states that the linear impulse—the integral of a force over time—applied to a body equals the change in its linear momentum, which is the product of its total mass and the velocity of its center of mass. The vector equation is:

$$\sum {\int }_{t_1}^{t_2}\vec{F}dt=m{\vec{v}}_{G_2}-m{\vec{v}}_{G_1}$$

Here, $m$ is the body's mass, and ${\vec{v}}_{G_1}$ and ${\vec{v}}_{G_2}$ are the velocities of the mass center $G$ at times $t_1$ and $t_2$, respectively. The sum of the integrals is the total linear impulse from all external forces. A crucial insight is that internal forces, which occur in equal and opposite pairs, generate no net impulse on the system. In many impact scenarios, non-impulsive forces (like weight or spring forces) can be neglected during the infinitesimal impact duration because their impulse is negligible compared to the massive impulsive contact forces.

Example: A crate of mass 50 kg slides on a smooth floor at 4 m/s when it is struck horizontally by a hammer blow, imparting an impulse of 150 N·s. Find the crate's new velocity. Applying the principle in the direction of impact: $150=50(v_2-4)$. Solving gives $v_2=7$ m/s. The simplicity of this calculation highlights the method's utility.

Angular Impulse-Momentum: Two Key Points

A rigid body in motion possesses angular momentum about any point, which changes due to applied angular impulses. The choice of reference point is critical and leads to two primary, equally valid equations.

About the Mass Center (G): This is often the most convenient formulation. The angular impulse-momentum equation about the mass center states that the sum of the angular impulses (the time integral of moments) of all external forces about $G$ equals the change in the body's angular momentum about $G$.

$$\sum {\int }_{t_1}^{t_2}{\vec{M}}_{G}dt={\vec{H}}_{G_2}-{\vec{H}}_{G_1}$$

For planar motion, the angular momentum about $G$ is $H_G=I_G\omega$, where $I_G$ is the mass moment of inertia about $G$ and $\omega$ is the angular velocity. Thus, the scalar form is $\int M_{G}dt=I_G({\omega }_2-{\omega }_1)$.

About a Fixed Point (O): If a rigid body rotates about a fixed point O (or an instantaneous center of zero velocity that is fixed in an inertial frame), you can also sum moments about that point. The angular impulse-momentum equation about a fixed point is:

$$\sum {\int }_{t_1}^{t_2}{\vec{M}}_{O}dt={\vec{H}}_{O_2}-{\vec{H}}_{O_1}$$

For rotation about a fixed point $O$, $H_O=I_O\omega$. It is vital to remember that you cannot generally use a moving point (unless it is the center of mass) in this simple form without accounting for extra terms related to the momentum of $G$.

Conservation of Angular Momentum

The principle of conservation of angular momentum is a powerful special case that arises when the net angular impulse about a chosen point is zero. If $\sum \int \vec{M}dt=0$, then ${\vec{H}}_1={\vec{H}}_2$. This conservation can apply about the mass center $G$ or a fixed point $O$.

This is frequently observed in systems with no external moments. For example, a diver tucking in their limbs to increase spin ($\omega$ increases as $I_G$ decreases, keeping $H_G=I_G\omega$ constant) or a satellite using reaction wheels. In rigid body impact problems, if an impact force's line of action passes directly through $G$ or a fixed point $O$, it generates no moment about that point, so angular momentum about that point is conserved during the impact event.

Impact Problems: Rotation and Eccentricity

Real-world collisions often involve rotating rigid bodies and eccentric impacts, where the line of action of the impact force does not pass through the mass center of the bodies involved. This eccentricity creates an impulsive moment, which changes the bodies' rotational motion as well as their translational motion.

Solving these problems requires the simultaneous application of the linear and angular impulse-momentum equations for each body. For planar impact between two bodies A and B, the standard solution procedure involves three key equations applied during the period of deformation/restitution:

1. Linear Impulse-Momentum for the system or a single body.
2. Angular Impulse-Momentum for each body (typically about its own mass center for simplicity).
3. The Coefficient of Restitution (e) equation, which relates the relative speeds of the points of contact normal to the contact surfaces before and after impact.

Consider a slender rod of mass $m$ and length $L$, initially at rest, struck perpendicularly by an impulse $P$ at a point a distance $d$ from its center. We want to find the resulting velocity of $G$ and angular velocity.

• Linear: $P=m(v_{G_2}-0)$ ⇒ $v_{G_2}=P/m$.
• Angular about $G$: $P\cdot d=I_G({\omega }_2-0)$ ⇒ ${\omega }_2=(P\cdot d)/(m{L}^2/12)=(12Pd)/(m{L}^2)$.

The rod thus gains both translation and rotation. If $d=0$ (central impact), ${\omega }_2=0$ and only translation occurs.

Common Pitfalls

1. Misapplying Angular Impulse-Momentum about a Moving Point: A common error is using the simple form $\int M_{A}dt=\Delta H_A$ for an arbitrary moving point $A$ that is not the mass center or a fixed point. This equation is only valid for $G$ or a fixed point $O$ in an inertial frame. For other points, a more complex relation involving the linear momentum of $G$ is required.
2. Neglecting Impulsive Moments in Eccentric Impacts: When an impact force is eccentric, it always creates an impulsive moment that changes angular velocity. Failing to account for this by only using the linear impulse-momentum equation will yield an incomplete and incorrect solution. Always check the line of action relative to $G$.
3. Incorrect Restitution Equation Setup: The coefficient of restitution $e$ is defined using the relative velocities of the points of contact along the line of impact, not the velocities of the mass centers. For rotating bodies, the velocity of the contact point is ${\vec{v}}_G+\vec{\omega }\times {\vec{r}}_{\text{contact/G}}$. Using ${\vec{v}}_G$ directly is a critical mistake.
4. Including Non-Impulsive Forces in the Impulse Calculation: During the very short duration of an impact, forces like weight, spring forces, or normal reaction forces on smooth surfaces typically produce negligible impulse. Including them complicates the equations unnecessarily. Identify and exclude non-impulsive forces to focus the analysis on the dominant impulsive loads.

Summary

• The linear impulse-momentum principle relates the net linear impulse on a rigid body to the change in the linear momentum vector of its mass center: $\sum \int \vec{F}dt=m\Delta {\vec{v}}_G$.
• The angular impulse-momentum principle is most straightforwardly applied about the body's mass center $G$ or a fixed point $O$: $\sum \int {\vec{M}}_{G/O}dt=\Delta {\vec{H}}_{G/O}$. For planar motion, $H_G=I_G\omega$ and $H_O=I_O\omega$.
• Conservation of angular momentum (${\vec{H}}_1={\vec{H}}_2$) applies about a point if the net angular impulse about that point is zero, a common condition in central impacts or systems with no external moments.
• Solving impact problems for rotating rigid bodies, especially eccentric impacts, requires the simultaneous application of linear and angular impulse-momentum equations for each body, coupled with the coefficient of restitution equation applied at the correct contact points.
• Successful analysis hinges on correctly selecting the point for angular momentum calculation, using the velocity of contact points (not mass centers) in the restitution equation, and neglecting non-impulsive forces during the impact interval.