Work-Energy Theorem

🟢 Lite — Quick Review (1h–1d)

Work (W = Fd cos θ): Energy transfer when a force acts through displacement. Measured in joules. Kinetic Energy (KE = ½mv²): Energy of motion; depends on mass and velocity squared. Potential Energy: Stored energy from position — gravitational PE = mgh, elastic PE = ½kx². Power (P = W/t = Fv): Rate of doing work, measured in watts. Work-Energy Theorem: Net work equals change in KE. Conservation: Mechanical energy conserved when only conservative forces act (no friction/air resistance). Efficiency = (Power output ÷ Power input) × 100% — always below 100%.

High-yield exam pointers: Always check the angle θ before calculating work — if force is perpendicular to motion, W = 0. For falling/sliding objects, convert KE ↔ PE directly; do not add extra energy. Power problems often ask which device does more work in less time — compare P = W/t values.

🟡 Standard — Regular Study (2d–2mo)

Definitions and Distinctions

Work occurs only when a force produces displacement in the same direction as the force component. The formula W = Fd cos θ captures this: when θ = 0° (force parallel to motion), cos 0° = 1 and W = Fd; when θ = 90° (force perpendicular to motion), cos 90° = 0 and W = 0. Negative work happens when θ > 90°, meaning the force opposes motion (e.g., friction).

Energy is the capacity to perform work. Kinetic energy (KE = ½mv²) scales with the square of velocity — doubling speed quadruples KE. Potential energy depends on configuration: gravitational PE = mgh requires a reference point for height h, while elastic PE = ½kx² depends on spring constant k and compression/extension distance x.

Work-Energy Theorem

Net work done on an object equals its change in kinetic energy: W_net = ΔKE = ½mv_f² − ½mv_i². This provides an alternative to kinematic equations when forces and displacements are known.

Power and Efficiency

Power measures how fast energy is transferred: P = W/t. For motion at constant velocity, P = Fv is often more convenient. Efficiency compares useful output to total input; no real machine achieves 100% due to dissipative forces.

Common UPCAT Question Patterns

Expect numerical problems requiring: (1) calculating work given force, displacement, and angle; (2) applying energy conservation to find velocity at different heights; (3) computing power from work and time; (4) efficiency calculations for machines.

🔴 Extended — Deep Study (3mo+)

Edge Cases and Mechanism Details

When multiple forces act simultaneously, the work-energy theorem uses the net force: W_net = ΔKE. Individual forces can do positive, negative, or zero work independently. For a block sliding down an inclined plane with friction, gravitational work is positive, friction work is negative, and normal force does zero work — only the net determines KE change.

Gravitational PE (mgh) is path-independent in uniform gravity; only vertical displacement matters. Elastic PE (½kx²) applies only within the spring’s elastic limit — beyond that, permanent deformation occurs and the formula fails. When both gravitational and elastic PE are present (e.g., a mass hanging from a spring), total mechanical energy is the sum of all conservative energy terms.

Connections to Adjacent Topics

The work-energy theorem bridges dynamics and kinematics: W = Fd connects force and displacement, while F = ma links to acceleration. Combined with kinematic equations, this solves problems where direct force analysis is complex. Power concepts extend to electrical energy (P = IV) and thermal systems (rate of heat transfer).

Common Mistakes to Avoid

Mistake	Why It Fails
Using W = Fd without cos θ	Only the force component parallel to displacement does work
Treating PE decrease as energy loss	PE → KE; total mechanical energy conserved (isolated system)
Confusing power and energy	Power is rate; a 10 W bulb uses same energy as a 100 W bulb if run 10× longer
Writing efficiency > 100%	Violates energy conservation; output cannot exceed input

Practice Prompts

A 2 kg ball falls from 5 m height. Using energy conservation (ignoring air resistance), find its speed just before impact: mgh = ½mv² → v = √(2gh) = √(2 × 9.8 × 5) ≈ 9.9 m/s.
A motor lifts a 50 kg load 10 m in 20 s with 80% efficiency. Find input power: Output power = (mgh)/t = (50 × 9.8 × 10)/20 = 245 W. Input power = 245/0.80 = 306 W.

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Work-Energy Theorem