System of Particles & Rotational Motion - Quick Revision
Centre of Mass
- R_cm = Σm_i r_i / M (mass-weighted mean). For 2 equal masses → midpoint; 3 equal masses → centroid.
- Homogeneous symmetric bodies: CM at the geometric centre (may be outside the material, e.g. a ring).
- Motion of CM: P = M V_cm; M A_cm = F_ext — only external forces matter (internal forces cancel). An exploding projectile's CM keeps its parabola.
- F_ext = 0 ⇒ total momentum conserved.
Torque & Angular Momentum
- Torque τ = r × F, |τ| = rF sinθ (N m); zero if line of action passes through the axis.
- Angular momentum l = r × p; τ = dL/dt.
- Conservation of angular momentum: τ_ext = 0 ⇒ L = Iω constant (skater pulls arms in → I↓, ω↑).
- Couple: two equal, opposite forces with different lines of action → rotation without translation; its moment is independent of the reference point.
Equilibrium
- Rigid-body equilibrium needs net force = 0 AND net torque = 0 (independent conditions → partial equilibrium possible).
- Principle of moments (lever): load arm × load = effort arm × effort; M.A. = d2/d1.
- Centre of gravity: point of zero net gravitational torque; = CM in uniform gravity.
Moment of Inertia
- I = Σm_i r_i² (rotational analogue of mass); depends on axis & mass distribution.
- Radius of gyration: I = Mk².
- Parallel axes: I = I_cm + Md². Perpendicular axes (lamina): I_z = I_x + I_y.
- Standard: ring MR²; disc/solid cylinder ½MR²; solid sphere ⅖MR²; rod (centre) ML²/12; ring (diameter) ½MR²; disc (diameter) ¼MR².
Rotation about a Fixed Axis
- Kinematics: ω = ω₀+αt, θ = ω₀t+½αt², ω² = ω₀²+2αθ.
- τ = Iα; L = Iω; K = ½Iω²; P = τω; W = τθ.
- Rolling = translation + rotation; contact point momentarily at rest; KE = ½Mv² + ½Iω². For a rolling solid sphere, rotational KE is 2/7 of the total.