Moment of Inertia (Central Axis)

• Ring: $I = MR^2$
• Disc: $I = MR^2/2$
• Solid sphere: $I = 2MR^2/5$
• Hollow sphere: $I = 2MR^2/3$
• Rod (centre): $I = ML^2/12$
• Rod (end): $I = ML^2/3$
• Disc (diameter): $I = MR^2/4$

Axis Theorems

• Parallel: $I = I_{cm} + Md^2$ → any body
• Perpendicular: $I_z = I_x + I_y$ → flat bodies only

Rolling Race (fastest → slowest)

Solid sphere → Disc → Hollow sphere → Ring

Rolling Acceleration

$a = \frac{g\sin\theta}{1 + K^2/R^2}$

Rolling Speed from Height h

$v = \sqrt{\frac{2gh}{1 + K^2/R^2}}$

$K^{2}$/$R^{2}$ Values

$2/5 (solid sphere) < \1/2 (disc) < \2/3 (hollow sphere) < \1$ (ring)

Angular Momentum Conservation

$I_1\omega_1 = I_2\omega_2 \quad (\tau_{net} = 0)$ $\frac{KE_f}{KE_i} = \frac{I_i}{I_f}$

Torque

\tau = rF\sin\theta \quad [M^1L^2$T^{-2}$]

Angular Momentum

L = I\omega \quad [M^1L^2$T^{-1}$]

Contact Point Velocity

Zero in rolling without slipping. Topmost point velocity = $2v_{cm}$.$