11.3: Equation of Rotational Dynamics
Angular variables are introduced in rotational dynamics. Comparing the definitions of angular variables with the definitions of linear kinematic variables, it is seen that there is a mapping of the linear variables to the rotational ones. Linear displacement, velocity, and acceleration have their equivalents in rotational motion, which are angular displacement, angular velocity, and angular acceleration. Similar to the rotational variables, a mapping exists from Newton's second law of motion with Newton's second law in rotational motion.
For a particle performing a circular motion about an axis passing through the center of the circular path, the net torque is given as
This is the rotational analog of Newton's second law of motion. The net torque on the particle is equal to the moment of inertia about the rotation axis times the angular acceleration. The above equation can be written in vector form as
If more than one torque acts on a rigid body about a fixed axis, then the sum of the torques equals the moment of inertia times the angular acceleration.
The term I is a scalar quantity and can be positive or negative (counterclockwise or clockwise) depending on the sign of the net torque. As per the convention, counterclockwise angular acceleration is positive. If a rigid body rotates clockwise and experiences a positive torque (counterclockwise), the angular acceleration is positive. Newton's second law for rotation relates torque, the moment of inertia, and the rotational kinematics. This is called the equation for rotational dynamics. With this equation, one can solve a whole class of problems involving force and rotation.