Work, Energy and Power - Quick Revision
Work
- W = F·d = Fd cosθ (scalar; via the dot product). Positive (θ<90°), negative (θ>90°, e.g. friction at 180°), zero (θ=90°, or zero force/displacement).
- Variable force: W = ∫F dx = area under the F-x graph.
Energy
- Kinetic energy K = ½mv² (scalar; K = p²/2m).
- Work-energy theorem: W_net = K_f − K_i. (Integral/scalar form of Newton's 2nd law; holds in all inertial frames.)
- Potential energy: gravitational V = mgh; spring V = ½kx². Force from PE: F = −dV/dx.
Conservative vs non-conservative
- Conservative (gravity, spring): work path-independent, zero over a closed loop, derivable from a PE.
- Non-conservative (friction): closed-loop work ≠ 0, no PE.
- Conservation of mechanical energy: K + V = constant when only conservative forces do work.
Power
- P = W/t (average) = F·v (instantaneous). Scalar; SI unit watt; 1 hp = 746 W; kWh is energy (= 3.6×10⁶ J).
Collisions
- Momentum is conserved in ALL collisions.
- Elastic: KE also conserved. Equal masses (1-D) exchange velocities; light hits heavy → light reverses; equal-mass glancing collision → 90° apart.
- Inelastic: KE not conserved (→ heat/sound). Completely inelastic: bodies move together, v = (m₁u₁+m₂u₂)/(m₁+m₂).
- Coefficient of restitution e = (separation speed)/(approach speed); e=1 elastic, e=0 perfectly inelastic.
Common traps
- Work by friction/braking is negative (θ=180°), NOT perpendicular.
- Only TOTAL momentum/energy of the system is conserved in collisions, not each body's.
- kWh is a unit of energy, not power.