JEE Physics System Of Particles and Rotational Motion Notes Set 04

Angular Momentum and Its Conservation

For a particle \( \vec L = \vec r\times\vec p \); for a rigid body about a fixed axis \( L = I\omega \). Torque equals the rate of change of angular momentum, \( \vec\tau = d\vec L/dt \). When the net external torque is zero, L stays constant, so \( I_1\omega_1 = I_2\omega_2 \). A skater who pulls in her arms spins faster. In a collision between coaxial rotating bodies, angular momentum is conserved but kinetic energy is not.

Rolling Motion

Pure rolling needs \( v_{cm} = \omega R \). The contact point is at rest, the centre moves at v and the top at 2v. Total kinetic energy \( K = \tfrac12Mv^2\left(1 + \dfrac{k^2}{R^2}\right) \). On an incline of angle \( \theta \): \( a = \dfrac{g\sin\theta}{1 + k^2/R^2} \), and the friction needed is \( f = \dfrac{Mg\sin\theta}{1 + R^2/k^2} \). The k²/R² values are 1/2 (solid cylinder or disc), 1 (ring), 2/5 (solid sphere) and 2/3 (hollow sphere); the smaller the value, the faster the body rolls down.

Work, Power and Energy in Rotation

Rotational kinetic energy is \( \tfrac12I\omega^2 \). Work done by a torque is \( W = \tau\theta \) and power is \( P = \tau\omega \). The work-energy theorem gives \( \tau\theta = \Delta(\tfrac12I\omega^2) \). The analogy with linear motion: mass with I, force with torque, momentum with angular momentum.