Rotational Motion — Quick Review (10 Sentences) — Summary

The centre of mass of a semicircular ring is $2R/\pi$ from the centre, and that of a semicircular disc is $4R/3\pi$ from the centre.
Moment of inertia $I = \sum m_i r_i^2$ [kg $m^{2}$ ] measures resistance to rotational acceleration, with standard values: ring $MR^2$ , disc $\frac{1}{2}MR^2$ , solid sphere $\frac{2}{5}MR^2$ , hollow sphere $\frac{2}{3}MR^2$ .
The parallel axis theorem $I = I_{cm} + Md^2$ is valid for every body (2D and 3D), while the perpendicular axis theorem $I_z = I_x + I_y$ applies only to flat planar bodies.
Torque $\tau = rF\sin\theta$ [N m] and angular momentum $L = I\omega$ [kg $m^{2}$ /s] are the rotational analogues of force and linear momentum, related by $\tau = dL/dt$ .
When the net external torque on a system is zero, angular momentum is conserved: $I_1\omega_1 = I_2\omega_2$ .
Conservation of angular momentum does not imply conservation of kinetic energy — kinetic energy changes whenever the moment of inertia changes.
Rolling without slipping requires $v_{cm} = \omega R$ , making the contact-point velocity exactly zero.
The total kinetic energy of a rolling body is $\frac{1}{2}mv^2(1 + K^2/R^2)$ , combining translational and rotational contributions.
On an inclined plane, rolling acceleration is $a = g\sin\theta/(1 + K^2/R^2)$ , so the body with the smallest $K^2/R^2$ (solid sphere, $2/5$ ) always reaches the bottom first, regardless of mass or radius.
In NEET, the most targeted results are the moment of inertia of a disc about a tangent in its plane ( $5MR^2/4$ ), the rolling race order, and angular momentum conservation problems.