Rotational Motion — Essential NEET Facts — Summary

CM of semicircular ring: $2R/\pi$ from centre (not R, not R/2).
CM of semicircular disc: $4R/3\pi$ from centre.
CM of hemisphere: $3R/8$ from flat face.
CM of triangular lamina: $h/3$ from the base.
MI of ring (about axis): $MR^2$ ; disc (about axis): $\frac{1}{2}MR^2$ ; solid sphere (diameter): $\frac{2}{5}MR^2$ ; hollow sphere (diameter): $\frac{2}{3}MR^2$ .
Rod about centre: $\frac{ML^2}{12}$ ; rod about end: $\frac{ML^2}{3}$ (4× larger — ratio is always 4).
Disc about diameter: $\frac{MR^2}{4}$ (derived via perpendicular axis theorem: $2I_d = MR^2/2$ ).
Disc about tangent in plane: $\frac{5MR^2}{4}$ (most common NEET target: parallel axis adds $MR^2$ ).
Solid sphere about tangent: $\frac{7MR^2}{5}$ (parallel axis: $2/5 + 1 = 7/5$ ).
Parallel axis theorem: $I = I_{cm} + Md^2$ — valid for any body, 2D or 3D.
Perpendicular axis theorem: $I_z = I_x + I_y$ — valid only for flat/planar (2D) bodies.
Torque: $\tau = rF\sin\theta$ [M $L^{2}$ $T^{-2}$ ]. Angular momentum: $L = I\omega$ [M $L^{2}$ $T^{-1}$ ].
Newton's 2nd (rotation): $\tau = dL/dt = I\alpha$ .
Conservation of L: $I_1\omega_1 = I_2\omega_2$ when $\tau_{ext} = 0$ .
Contact point velocity in rolling without slipping: zero (translational and rotational components cancel).
Total rolling KE: $\frac{1}{2}mv^2(1 + K^2/R^2)$ .
Incline acceleration: $a = g\sin\theta/(1 + K^2/R^2)$ .
Rolling race winner: solid sphere (smallest $K^2/R^2 = 2/5$ ) → disc → hollow sphere → ring (slowest).
Result is independent of mass and radius — only shape matters in rolling.
Radius of gyration: $K = \sqrt{I/M}$ (units: metres).