Work, Energy & Power— Notes, Formulas & Revision

Work done by a constant force F over displacement d is W = F·d·cos θ, where θ is the angle between F and d.

Units: joule (J) = N·m. 1 J = work done by 1 N over 1 m parallel.

Scalar quantity. Can be positive (force along motion), negative (force opposite motion), or zero (force ⊥ motion).

Force perpendicular to displacement does NO work — even though force is exerted.

Gravity does NEGATIVE work when lifting up (F downward, motion upward).

Friction (kinetic) generally does negative work on the moving object — opposes motion.

Vector form: W = F · d (dot product). For non-constant F: integrate.

Work is a TRANSFER of energy — equal to change in KE when only F acts.

Work-energy theorem: net work done on a body equals its change in kinetic energy.

W_net = K_f − K_i = ½m(v_f² − v_i²).

Applies to any net force — constant, variable, gravity, friction, spring, etc.

Derived directly from Newton's 2nd law integrated: F = m·a ⇒ ∫F·dx = ∫ma·dx = ∫mv·dv = ½m·Δ(v²).

Useful for solving problems WITHOUT explicit time analysis — only initial and final states matter.

If multiple forces act, W_net = sum of works done by each (or work by the net force).

Power form: P = dW/dt = F·v.

Used in projectile, pendulum, spring-block, collision, friction-on-incline problems.

Kinetic energy K is the energy a body has due to its motion: K = ½mv².

Scalar quantity, always non-negative. Units: joule (J).

Depends on the reference frame — moving frame sees different v ⇒ different K.

K = p²/(2m), where p = mv. Useful when momentum is known.

Work-energy theorem: W_net = ΔK. Net work changes KE.

Translational KE (linear motion) + Rotational KE = total mechanical KE.

For relativistic speeds: K = (γ − 1)mc². Reduces to ½mv² for v ≪ c.

Energy of motion can be transferred to other forms: heat (friction), elastic PE (spring), gravitational PE (rising), light, sound.

Gravitational potential energy U of mass m at distance r from M: U = −GMm/r. Always negative (taking U → 0 at infinity).

Near Earth's surface (h ≪ R): U ≈ mgh + constant. Linear approximation; offset reference is arbitrary.

Work done by gravity = −ΔU. Falling object loses PE, gains KE.

Energy required to ESCAPE from surface = |U_surface| = GMm/R. This gives escape velocity (½mv² = GMm/R).

Orbiting body: total energy E = ½mv² + U = −GMm/(2r) (for circular orbit). Negative total energy = bound.

PE is a SCALAR. PE differences are physically meaningful; absolute PE has an arbitrary zero.

PE of a two-body system depends only on separation, not on individual positions.

Tidal forces arise from gradients in gravitational PE — different parts of an extended body sit at different U.

Spring obeys Hooke's law: F = −kx (restoring force, opposite to displacement).

Elastic potential energy stored in a stretched/compressed spring: U = ½kx², where x is the displacement from natural length.

Energy stored is ALWAYS positive (x² is positive).

Conservation of energy: ½mv² + ½kx² = constant (for ideal spring-mass system, no friction).

At maximum extension/compression: KE = 0, PE = max. At natural length (passing through): KE = max, PE = 0.

Hooke's law fails for very large extensions — spring may deform plastically or break.

Springs in series: 1/k_eq = 1/k₁ + 1/k₂ (softer combination). Parallel: k_eq = k₁ + k₂ (stiffer).

Used in: oscillators, shock absorbers, vehicle suspensions, accelerometers, energy storage.

Conservation of energy: total energy of an isolated system is constant — only converts between forms.

Mechanical energy = KE + PE. Conserved when only CONSERVATIVE forces act (gravity, spring).

Non-conservative forces (friction, air drag): mechanical energy is NOT conserved; some converts to heat or sound.

Total energy (mechanical + thermal + chemical + EM + nuclear + …): ALWAYS conserved (1st law of thermodynamics).

Examples: pendulum (KE ↔ PE), free fall (PE → KE), spring-mass (KE ↔ U_spring), hydroelectric (PE → KE → electrical), photosynthesis (light → chemical), nuclear (mass → energy).

Friction converts mechanical energy to HEAT — random molecular motion in the contact surfaces.

Energy can change FORM but never be created or destroyed.

Closed-system energy conservation is THE most powerful physical principle.

Power P is the rate of doing work or transferring energy: P = dW/dt.

Units: watt (W) = J/s. 1 kW = 1000 J/s.

Average power: P_avg = W/Δt. Instantaneous: P = F·v (for constant or varying force).

Power required to move at constant velocity against friction: P = F_friction × v.

Acceleration phase: P keeps increasing if you maintain F = F_friction + ma.

Power output of a car engine: V_max occurs when all engine power goes against drag — P = F_drag × V_max.

Vehicles, machines, appliances all rated by power. Human body produces ~75 W steady; up to ~1500 W in short bursts.

Electrical: P = VI = I²R = V²/R. Mechanical: P = F·v.

For a force varying with position, work is the integral W = ∫F·dx along the path.

Equivalently: area under F-vs-x curve from x_i to x_f.

Spring force F = −kx: W (compressing from 0 to x) = ½kx² (taken by the spring); −½kx² done BY the spring on the block.

Gravity near surface (W = mgh): constant force, simple. Gravity far from surface (W = −GMm·(1/r₁ − 1/r₂)): use integral.

Force ⊥ to motion (e.g., circular motion at constant speed): W = 0 instantaneously.

Work depends on path for non-conservative forces (friction, air drag).

For conservative forces, W = −ΔU. Total energy is conserved.

Used in: SHM analysis, escape velocity, pendulum, free-fall in real gravity.

Elastic collision: BOTH momentum AND kinetic energy are conserved.

1D two-body: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ (momentum) AND ½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂² (KE).

Solving these gives final velocities in terms of initial — exact formulas exist (see Formulas section).

Equal masses (m₁ = m₂): velocities are EXCHANGED — incoming particle stops, target moves with original speed.

Massive target (m₂ ≫ m₁): incoming particle bounces back with nearly same speed; target barely moves.

Light target (m₂ ≪ m₁): incoming particle continues nearly unchanged; target shoots off at 2u₁.

Real-world approximations: hard steel balls, atomic collisions (in some regimes), Newton's cradle.

Macroscopically, perfectly elastic collisions don't exist — some energy always lost to heat, sound, deformation.

Inelastic collision: momentum is conserved, but KE is NOT — some converts to heat, sound, deformation.

PERFECTLY inelastic: objects STICK TOGETHER. Maximum KE loss.

Common cases: cars colliding (crumple), bullet embedding in a block, clay balls sticking, two trains coupling.

Perfectly inelastic 1D: m₁u₁ + m₂u₂ = (m₁ + m₂)·v_common.

KE loss in perfectly inelastic: ΔK = ½·(m₁m₂/(m₁+m₂))·(u₁−u₂)² — reduced-mass times relative speed squared.

Real-world collisions are MOSTLY inelastic — perfectly elastic is rare in macroscopic objects.

Coefficient of restitution e: e = 1 for elastic, e = 0 for perfectly inelastic, 0 < e < 1 for partially inelastic.

Used in: crash-test physics, billiards (partial inelasticity), bullets, ballistics pendulum.

Coefficient of restitution e is the ratio of relative speed AFTER collision to relative speed BEFORE: e = |v_2 − v_1|/|u_1 − u_2|.

Range: 0 ≤ e ≤ 1. e = 1: perfectly elastic. e = 0: perfectly inelastic (objects stick together). 0 < e < 1: partially elastic.

Property of MATERIALS — not just objects. Different surface pairs have different e.

Typical values: glass-glass ~0.95, steel-steel ~0.7-0.9, basketball-floor ~0.75, clay-clay ~0.1-0.2.

Determines bounce height: h_n = e^(2n) × h₀ for n bounces.

Ratio of bounce heights: h_after/h_before = e². Direct experimental method.

In oblique collisions: e applies only to NORMAL component of velocity; tangential component (if smooth) is unchanged.

e depends slightly on relative speed and surface conditions — not a constant in all situations.

The Work-Energy Theorem states: Net work done on a body equals the change in its kinetic energy (W_net = ΔKE).

Work done by a constant force: W = F·d·cos θ, where θ is the angle between force and displacement.

Conservative forces (gravity, spring) have associated potential energy. Non-conservative forces (friction) dissipate energy as heat.

For a spring: PE = ½kx², where k is the spring constant and x is the displacement from equilibrium.

In SHM (spring-mass system), energy continuously converts between KE and PE. Total mechanical energy is conserved.

Power is the rate of doing work: P = dW/dt = F·v (instantaneous power).

The area under a Force-displacement graph gives the work done.

At equilibrium position in SHM: KE is maximum and PE is minimum. At extreme positions: KE = 0, PE = maximum.

W = F·d cosθ. Sign depends entirely on the angle between force and displacement.

θ = 0° → +W (force aids motion). θ = 180° → −W (force opposes motion, e.g. friction).

θ = 90° → W = 0 (centripetal force, normal force on a horizontal floor — perpendicular).

Block sliding on rough floor decelerates; KE is dissipated as heat by friction.

Stopping distance d = v₀²/(2μg) — independent of mass.

Total heat generated = initial KE = ½mv₀².