11.9:
Angular Momentum: Rigid Body
The angular momentum of a rigid body, made up of tiny particles and rotating about the z-axis with an angular velocity of ω, is the sum of the angular momenta of all such tiny particles.
A particle's angular momentum is equal to the cross product of its position vector and its linear momentum. By expressing the linear momentum in terms of linear velocity, which is tangential, the magnitude of the particle's angular momentum becomes miviri.
Now, expressing linear velocity in terms of angular velocity, an expression for the magnitude of total angular momentum is derived.
Here, the sum of miri2 is equal to the moment of inertia I. Thus, the total angular momentum of the rigid body becomes Iω.
If the rigid body rotates about a symmetrical axis, its moment of inertia is constant, and hence, its angular momentum and angular velocity are parallel to the rotational axis.
11.9:
Angular Momentum: Rigid Body
The total angular momentum of a rigid body can be calculated using the summation of the angular momentum of all the tiny particles rotating in the same plane. Considering all the tiny particles rotating in the x–y plane, the direction of angular momentum of all such particles and that of the rigid body would be perpendicular to the plane of the rotation along the z-axis.
This calculation can get complicated when tiny particles within the rigid body are not rotating in the same plane but have components for rotation along the z-axis. Such particles will have components of their angular momentum perpendicular to the z-axis. The situation becomes easier if the rigid body is symmetrical about the axis of rotation, the z-axis. In such a case, all the angular momentum components perpendicular to the z-axis, from either side of the rigid body, will cancel out. Therefore, if a rigid body has a symmetric axis of rotation, total angular momentum will be the summation of angular momentum of individual tiny particles.
For cases where the axis of rotation is not symmetric, the direction of angular momentum is not along the axis of rotation, but traces a cone around the axis of rotation. That means that there is a net torque acting on the body, even though the angular velocity of the body is constant.
This text is adapted from Openstax, University Physics Volume 1, Section 11.2: Angular Momentum.