Impulse-momentum and eccentric impact

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.