Essential NEET Facts — Work, Energy & Power — Summary

Work formula: W = Fd cos θ; [ $M^{1}$$L^{2}$$T^{-2}$ ]; 1 J = 1 N·m
Work is positive when θ < 90° (force aids motion); zero when θ = 90° (normal force, centripetal force); negative when θ > 90° (friction, braking force)
Variable force work = area under F–x graph = ∫F dx
KE = ½ $mv^{2}$ = $p^{2}$ /(2m); same unit as work (joule); same dimension [ $M^{1}$$L^{2}$$T^{-2}$ ]
Work-Energy Theorem: W_net (ALL forces) = $\Delta K$ E = ½ $mv^{2}$ − ½ $mu^{2}$
Gravitational PE = mgh; reference level is arbitrary; only differences matter
Spring PE = ½ $kx^{2}$ ; spring constant k has units N/m, dimension [ $M^{1}$$L^{0}$$T^{-2}$ ]
Conservative forces (gravity, spring): work is path-independent; PE function exists
Non-conservative forces (friction, viscosity): work is path-dependent; convert ME to heat
Conservation of ME: KE + PE = constant (only when no non-conservative forces act)
Power P = W/t = Fv cos θ; [ $M^{1}$$L^{2}$$T^{-3}$ ]; 1 hp = 746 W
Vertical circle — string: v_top(min) = √(gR); v_bottom(min) = √(5gR); v_side(min) = √(3gR)
Vertical circle — rod: v_top(min) = 0; v_bottom(min) = 2√(gR) = √(4gR)
Elastic collision: e = 1; momentum AND KE conserved; equal-mass case → velocity exchange
Perfectly inelastic: e = 0; momentum conserved; KE NOT conserved; bodies stick; max KE loss
Coefficient of restitution: e = (relative speed of separation)/(relative speed of approach)
KE = $p^{2}$ /(2m): equal momentum → lighter body has greater KE (frequently tested)
Centripetal force does ZERO work (always perpendicular to velocity/displacement)
At top of vertical circle (string): T + mg = $mv^{2}$ /R → T = 0 gives v_min = √(gR)
At bottom of vertical circle (string): T − mg = $mv^{2}$ /R → T_bottom = mg + $mv^{2}$ /R