Class 11 · Chapter 5

Work, Energy & Power

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Work, Energy & Power Short Notes

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Condensed revision points for Work, Energy & Power — for quick recall before exams, not a substitute for the full notes.

Constant force: W = F · d = Fd cosθ.
Variable force (1-D): W = ∫ F(x) dx between initial and final x.
Graphically: W = area under the F–x curve (area below the axis counts as negative).
Work is a scalar; carries a sign, not a direction.

Positive (θ < 90°): force favours motion — e.g. gravity on a falling body, a stretched spring's stretching force.
Negative (θ > 90°): force opposes motion — e.g. friction, braking force.
Zero: F = 0, or displacement = 0, or θ = 90° — e.g. centripetal force in circular motion, tension in a swinging pendulum, pushing an immovable wall.

KE = ½mv² — always ≥ 0, frame-dependent.
Work-energy theorem: W_total (by ALL forces) = ΔKE = ½mv_f² − ½mv_i².
Use this even when friction/other non-conservative forces are present — it never fails for a single body in an inertial frame.

Conservative: work is path-independent, depends only on initial/final positions; work over a closed loop = 0. Examples: gravity, spring force, electrostatic force.
Non-conservative: work depends on path taken. Example: friction, viscous/air drag.
Central force: always directed along the line to a fixed centre, magnitude depends only on distance. All inverse-square-law forces (gravity, Coulomb) are central, and all central forces are conservative.

Defined only for conservative forces. PE = work done against the force to bring the body from a zero-reference point to its current position.
F = −dU/dx (one dimension). ΔU = −W_conservative.
U can be +ve, −ve, or 0 depending on the chosen reference; only ΔU has physical meaning.

Equilibrium points: where dU/dx = 0.
Stable: d²U/dx² > 0 (U is a local minimum) — small displacement creates a restoring force.
Unstable: d²U/dx² < 0 (U is a local maximum) — small displacement pushes the body further away.
Neutral: d²U/dx² = 0 and U constant nearby — no restoring or repelling force.

E = K + U = constant, when only conservative internal forces act (no net external work).
ΔK + ΔU = 0 — kinetic and potential energy trade off, but their sum is fixed.
Breaks down the moment friction or any external force does net work — switch to the work-energy theorem instead.

Block dropped from height h onto vertical spring, compresses by x: mg(h + x) = ½kx².
Block (speed v) hits horizontal spring on smooth surface: max compression x_max = v√(m/k).
Mass lowered gently onto vertical spring: equilibrium extension x = mg/k.
Mass released suddenly (not lowered slowly) from natural length: max extension x_max = 2mg/k — double the equilibrium value.

Average power: P_avg = ΔW/Δt = total work / total time.
Instantaneous power: P = dW/dt = F · v.
SI unit: watt (W) = J/s. 1 hp = 746 W. Dimensions: [ML²T⁻³].
Area under P–t graph = work done. Slope of W–t graph = instantaneous power; slope of chord = average power.
Efficiency η = (work output)/(energy input) — always < 1 for real machines.

Work done by centripetal force / normal reaction / tension perpendicular to motion is always zero — don't add it into work-energy equations.
Don't confuse 'work done by gravity' with 'change in gravitational PE' — they're equal in magnitude but opposite in sign (W_gravity = −ΔU).
Check whether a problem hands you mechanical-energy conservation (only conservative forces) or needs the full work-energy theorem (friction/external force present) before picking a method.
Spring PE uses x measured from the spring's natural (relaxed) length, never from some other arbitrary reference.
For motion starting from rest under constant power, v ∝ t^(1/2) and distance x ∝ t^(3/2) — a frequently tested result.

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