Angular Momentum The Cross Product

The angular momentum is defined to be \[\vec{L} = \vec{r} \times \vec{p}\]

The angular momentum can be thought of as the spin. Any extended object which is rotating contains angular momentum.

This cross product is a bit complex to get your head around at first, but it is ESSENTIAL to truly understand dynamics of spinning systems. The angular momentum is also a fundamental quantity in more advanced topics like quantum mechanics and the proporties of atoms.

This quantity \(\vec{L}\) should be thought of as the rotational analog of the linear momentum \(\vec{p}\). Whne studying linear momentum, we could write down a conserveation equation using impulse (\(\int \vec{F_{ext}} \, dt = \Delta \vec{p}\)). For rotaitonal systems, we have a slightly modified version: \(\int \vec{\tau_{ext}}\,dt = \Delta \vec{L}\). If there is zero external torque, the angular momentum (spin) will be constant in time \((\Delta \vec{L} = 0\)).

Fixed vs Free

There are 2 important cases to consider when studying rotational colliosions:

  • Fixed rotation point: One of the objects is fixed to rotate about a pivot (ball bearing, a nail, etc). Before the collision, the system will have some value of angular momentum. After the collision takes place, the angular momentum will be conserved, but will also immediately start changing. The rate of change in the angular momentum (\(dp/dt\)) will be equal to the overall external torque. (this is analogous to an external force, such as friction, changing the overall momentum)
  • Free rotation: The objects are free to rotate (and translate) after the collision. In this type of collision, the angular and linear momentum will both be conserved (for all time). In order to conserve angular momentum \(\vec{L}\), the rotation must take place about the Center of Mass.

When the objects are free to rotate, it is advisable to start the problem by calculating the position of the center of mass before you start calculating the angular momentum. If you calculate using this axis, the final angular momentum will be proportional to the overall angular velocity of the system after the collision (\(\vec{\omega} = \tfrac{\vec{L}}{I}\)).