Law of Rotation

If a body rotates purely about an axis S with angular acceleration $\alpha$ and net torque acting on the body about s is $\tau$]_s, then – $\tau$]_s = I$\alpha$

Illustration 1: A triangular plate of uniform thickness and density is made to rotate about an axis perpendicular to the plane of the paper and (a) passing through A, (b) passing through B, by the application of the same force, F, at C (midpoint of AB) as shown in the figure. Is the angular acceleration in both the cases be the same ?

Solution: $\tau$= I$\alpha$ $\therefore$ $\alpha =\frac{\tau }{I}$

$\tau$= Force perpendicular distance Torque is same in both the cases. But since I will be different due to different mass distribution about the axis.

$\therefore$ $\alpha$ will be different for different cases

ROTATIONAL WORK AND ENERGY

The rotational work done by a force about the fixed axis of rotation is defined as , W_rot = $\int \tau d\theta$

Where $\tau$ the torque is produced by the force, and $d\theta$ is the infinitesimally small angular displacement about the axis.

The rotational kinetic energy of a body about a fixed rotational axis is defined as, $K_{rot}=\frac{1}{2}I{\omega }^2$

Where I is the moment of inertia about the axis.

Work–Energy Theorem: In complete analog to the work energy theorem for the translator motion, it can be stated for rotational motion as: W_rot = $\Delta$K_rot

The net rotational work done by the forces is equal to the change in rotational kinetic energy of the body.

Conservation of Mechanical Energy: In the absence of dissipative work done by non–conservative forces, the total mechanical energy of a system is conserved.

$\Delta K+\Delta U=0$ Or K_f + U_f = K_i + U_i