Chapter 6, Problem 1CP

Express Newton’s second law of motion for rotating bodies. That can you say about the angular velocity and angular momentum of a rotating nonrigid body of constant mass if the net torque acting on it is zero?

To determine

The Newton's second law of motion for rotating bodies

Explanation of Solution

The Newton's second law of motion for the bodies in rotation establishes a relationship between the net torque applied externally and the angular acceleration produced in the body. This relationship between the torque and acceleration is relatively complex as compared to the Newton's second law of motion for linear motion of bodies, it is because of the vector nature of moment of inertia.

My opinion about the angular velocity and angular momentum:

The distribution of mass plays an important role in determining the amount of torque to produce an angular acceleration in the body.

The equation of Newton's second law of motion for rotating bodies is restricted to rotation about the principal axis or the axis of symmetry for symmetrical bodies.

Mathematically, the Newton's second law of motion for rotating bodies can be defined as

  $\tau =I\alpha$  .......(1)

Where

  $\begin{array}{l}\tau =\text{ Torque applied on the body}\\ I=\text{ Moment of inertia of the body}\\ \alpha =\text{ Angular acceleration in the body}\end{array}$

Now, when the torque on the body is zero, from the equation (1), the product of $I\text{ and }\alpha$ should be zero.

If the body has some mass, the moment of inertia I can not be zero and hence, angular acceleration $\alpha$ should be zero. This means the body is rotating with constant angular velocity.