11.3:
Equation of Rotational Dynamics
Consider a point mass performing a circular motion about the center of rotation; Newton's second law of motion gives the force acting on this point mass.
The linear acceleration can be written in terms of the angular acceleration, and multiplying both sides with the radius of the circular path gives an expression for the torque.
Recalling the definition of the moment of inertia, the torque applied on a point mass can be rewritten. This is Newton's second law of rotation.
In the vector form of Newton's second law of rotation, the torque and angular acceleration are in the same direction.
This equation of rotational motion can be generalized to any rigid body rotating about a fixed axis. If multiple forces are acting on the rigid body, then the equation of rotational dynamics is expressed using the summation of all the torques.
The net torque includes only external forces, as all the internal forces cancel out due to Newton's third law of motion.
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.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/10-7-newtons-second-law-for-rotation