Angular momentum
Angular momentum of a particle
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The angular momentum of a particle is the vector product of its position and its linear momentum: \[\mathbf{L} := \mathbf{r} \times \mathbf{p}.\] Angular momentum is a vector quantity. The SI unit of angular momentum is the kilogramme metre squared per second: \[\left[\mathbf{L}\right] = 1\,\mathrm{kg} \cdot \frac{\mathrm{m}}{\mathrm{s}}.\] |
Angular momentum of a rigid object rotating about a fixed axis
The angular momentum of a rigid object rotating about a fixed axis is
\[\mathbf{L} = I \mathbf{ω}.\]
Torque and angular momentum
It can be proved that the rate of change of the angular momentum is the net external torque:
\[\frac{\mathrm{d}\mathbf{L}}{\mathrm{d} t} = \sum\mathbf{τ}.\]
For an object rotating about a fixed axis, we can use \(\mathbf{L} = I \mathbf{ω}\):
\[\frac{\mathrm{d}\mathbf{L}}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t}\left(I \mathbf{ω}\right) = I \frac{\mathrm{d}\mathbf{ω}}{\mathrm{d}t} = I \mathbf{α},\]
which yields the rotational form of Newton‘s second law:
\[\sum\mathbf{τ} = I \mathbf{α}.\]
We have seen that the rate of change of the angular momentum vector equals the net external torque. This also means that if the net external torque is zero, the angular momentum of the system stays constant.
Law of conservation of angular momentum
If the net external torque acting on the system is zero, that is, if the system is isolated, the total angular momentum of a system is constant in both magnitude and direction.
Example: rotating stool
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Neglecting forces of friction and drag, there is no external torque acting on the rotating system, so its angular momentum will stay constant. When the man’s arms are extended, the moment of inertia \(I\) is greater as the weights are farther away from the axis, resulting in a greater \(m R^2\) contribution. Since the angular momentum \(\mathbf{L} = I \mathbf{ω}\) is constant, greater angular momentum means less angular velocity. When he pulls his arms in, the moment of inertia is reduced, and the constancy of the angular momentum means that the stool will spin faster. |