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.
