DYNAMIC OF ROTATIONAL MOTION - Coggle Diagram
DYNAMIC OF ROTATIONAL MOTION
Angular Momentum—Objects Rotating About a Fixed Axis
The rotational analog of linear momentum is angular momentum, L:
Then the rotational analog of Newton’s second law is:
In the absence of an external torque, angular momentum is conserved:
Therefore, if an object’s moment of inertia changes, its angular speed changes as well.
Angular momentum is a vector; for a symmetrical object rotating about a symmetry axis it is in the same direction as the angular velocity vector.
Vector Cross Product; Torque as a Vector
The vector cross product is defined as:
The direction of the cross product is defined by a right-hand rule:
The cross product can also be written in determinant form:
Torque can be defined as the vector product of the force and the vector from the point of action of the force to the axis of rotation:
For a particle, the torque can be defined around a point O:
Here, r is the position vector from the particle relative to O.
Angular Momentum of a Particle
The angular momentum of a particle about a specified axis is given by:
Angular Momentum and Torque for a System of Particles; General Motion
The angular momentum of a system of particles can change only if there is an external torque—torques due to internal forces cancel.
This equation is valid in any inertial reference frame. It is also valid for the center of mass, even if it is accelerating:
Angular Momentum and Torque for a Rigid Object
For a rigid object, we can show that its angular momentum when rotating around a particular axis is given by:
A system that is rotationally imbalanced will not have its angular momentum and angular velocity vectors in the same direction. A torque is required to keep an unbalanced system rotating.
Conservation of Angular Momentum
If the net torque on a system is constant,
The total angular momentum of a system remains constant if the net external torque acting on the system is zero.