Work is the transfer of energy by a force acting over a displacement. Only the component of force along the direction of motion does work — a force perpendicular to motion transfers no energy.
Sign of Work
| Case | Angle θ | Result |
|---|---|---|
| Force aids motion | 0° ≤ θ < 90° | W > 0 (positive work) |
| Force opposes motion | 90° < θ ≤ 180° | W < 0 (negative work) |
| Force perpendicular to motion | θ = 90° | W = 0 (no work done) |
Kinetic energy is the energy an object possesses due to its motion. It depends on the square of speed — doubling speed quadruples kinetic energy.
ΔEk = ½(1200)(8² − 20²) = ½(1200)(64 − 400) = −201 600 J
The brakes did −201.6 kJ of work on the car (negative → removed energy).
Gravitational potential energy is stored energy due to an object's height above a reference level. The reference level (Ep = 0) is arbitrary — only changes in height matter.
- m = mass (kg), g = 9.8 m/s², h = vertical height above reference (m)
- Choose the reference level that makes the problem simplest
- Only the vertical component of height matters — not the path taken
A compressed or stretched spring stores energy. Hooke's Law describes the relationship between force and deformation.
- k = spring constant (N/m) — stiffness of the spring
- x = compression or extension from natural length (metres)
- x is the deformation, not the absolute position of the end
In a closed system with no friction or air resistance, the total mechanical energy (kinetic + potential) remains constant. Energy transforms between forms but is never created or destroyed.
A pendulum continuously converts between gravitational PE and kinetic energy. At the extremes of its swing, it momentarily stops — all energy is potential. At the bottom, it moves fastest — all energy is kinetic.
| Position | Kinetic Energy | Potential Energy |
|---|---|---|
| Top of swing (extreme) | Zero (v = 0) | Maximum |
| Bottom of swing | Maximum | Zero (reference level) |
Height and Speed Formulas
Where L = pendulum length and θ = angle from vertical.
h = 0.8(1 − cos 25°) = 0.8(1 − 0.906) = 0.0752 m
v = √(2 × 9.8 × 0.0752) = √1.474 ≈ 1.21 m/s at the bottom
Power is the rate at which work is done (or energy is transferred). Two machines can do the same work — the one that does it faster has greater power.
- Average power: total work done divided by total time
- Instantaneous power: power at a specific instant, P = Fv
- 1 watt = 1 joule per second; 1 kilowatt = 1000 W; 1 horsepower ≈ 746 W
No real machine converts 100% of input energy into useful output — some is always lost to heat, sound, or friction. Efficiency measures how well a machine converts energy.
- Efficiency is always ≤ 100% (second law of thermodynamics)
- Can also be expressed as a decimal (0 to 1) rather than a percentage
- Input power = useful output power ÷ efficiency (as a decimal)
Efficiency = (858 / 1009) × 100% ≈ 85%
| Mistake | What to do instead |
|---|---|
| Using W = Fd when θ ≠ 0° | Always use W = F·d·cosθ. W = Fd only when the force is exactly along the displacement. |
| Forgetting cosθ entirely | Identify the angle between force and displacement before writing the work formula. |
| Using slant height for Ep | h in Ep = mgh must be the vertical height — not the length along a ramp or path. |
| Using velocity component instead of full speed in Ek | Ek = ½mv² uses the full speed magnitude, not a single component. |
| Confusing Ep and Ek at pendulum extremes | At top: Ep = max, Ek = 0. At bottom: Ek = max, Ep = 0. |
| Efficiency > 100% | Efficiency can never exceed 100%. Check signs and formula if you get this result. |
| Using position instead of deformation for spring PE | x in Ee = ½kx² is the compression or extension from natural length, not position. |