Motion of System of Particles and Rigid Bodies
What this domain tests
Motion of System of Particles and Rigid Bodies tests centre of mass, torque, angular momentum, moment of inertia, the parallel and perpendicular-axis theorems, and rolling motion.
Core rules
- Centre of mass: R = (Σm_ir_i)/Σm_i.
- Torque τ = r × F; angular momentum L = Iω; τ = dL/dt.
- Moment of inertia depends on the axis: rod about its centre ML²/12, about an end ML²/3; solid cylinder about its axis ½MR²; solid sphere 2/5MR².
- Parallel-axis theorem: I = I_cm + Md². Perpendicular-axis theorem for a lamina: I_z = I_x + I_y.
- Rolling without slipping: v = ωR; total KE = ½mv² + ½Iω².
Worked example
A solid cylinder rolls without slipping at speed v. With I = ½MR² and ω = v/R, total KE = ½Mv² + ½(½MR²)(v²/R²) = ¾Mv². For a ring, I = MR², giving KE = Mv². This is why a ring reaches the bottom of an incline later than a solid cylinder of the same mass and radius.
Traps
Using a moment-of-inertia formula for the wrong axis; applying the parallel-axis theorem with d measured from the surface rather than the centre of mass; treating static friction in rolling without slipping as doing negative work; forgetting that angular momentum is conserved only when external torque is zero.