Comprehensive theory, key formulas, diagrams, and memory aids for Work, Energy and Power.
In physics, work is done when a force causes displacement in the direction of the force. The definition is precise: $$W = Fs\cos\theta$$
Where: - $F$ = magnitude of the applied force (N) - $s$ = displacement (m) - $\theta$ = angle between the force and displacement
Work is a scalar quantity measured in joules (J), where 1 J = 1 N m.
Key special cases: - If $\theta = 0°$ (force parallel to displacement): $W = Fs$ — maximum work. - If $\theta = 90°$ (force perpendicular to displacement, e.g., centripetal force): $W = 0$ — no work done. - If $\theta = 180°$ (force opposing displacement, e.g., friction): $W = -Fs$ — negative work (energy removed from object).
On a force–displacement graph, the area under the graph equals the total work done — even if the force is not constant.
The kinetic energy (KE) of an object is the energy it possesses due to its motion: $$E_k = \frac{1}{2}mv^2$$
Where $m$ is mass (kg) and $v$ is speed (m s⁻¹). KE is always positive (or zero) and is a scalar.
The work-energy theorem states that the net work done on an object equals its change in kinetic energy: $$W_{net} = \Delta E_k = \frac{1}{2}mv^2 - \frac{1}{2}mu^2$$
This is derived directly from Newton's Second Law and the SUVAT equation $v^2 = u^2 + 2as$: multiplying both sides by $\frac{1}{2}m$ gives $\frac{1}{2}mv^2 - \frac{1}{2}mu^2 = mas = Fs$.
Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field: $$E_p = mgh$$
Where $m$ = mass (kg), $g$ = gravitational field strength (9.81 N kg⁻¹ near Earth's surface), and $h$ = height above a chosen reference level (m).
GPE is relative — it depends on the reference level chosen, but changes in GPE are absolute and do not depend on the choice of reference.
When a spring or elastic material is stretched or compressed, it stores elastic potential energy: $$E_{elastic} = \frac{1}{2}kx^2$$
Where $k$ is the spring constant (N m⁻¹) and $x$ is the extension or compression from the natural (unstretched) length. This equation applies only within the elastic limit (when Hooke's Law is obeyed). It is derived as the area of the triangle under a force–extension graph: $E = \frac{1}{2}Fx = \frac{1}{2}(kx)x$.
The principle of conservation of energy states that energy cannot be created or destroyed — it can only be converted from one form to another. The total energy of a closed system remains constant.
In the absence of non-conservative forces (friction, air resistance): $$E_k + E_p = \text{constant}$$
This leads to: $\frac{1}{2}mv^2 + mgh = \text{constant}$
For a ball dropped from rest at height $h$, on reaching the ground: $\frac{1}{2}mv^2 = mgh$, giving $v = \sqrt{2gh}$.
When friction or air resistance is present, some mechanical energy is converted to thermal (heat) energy. The total energy is still conserved, but useful mechanical energy decreases: $$E_{k,initial} + E_{p,initial} = E_{k,final} + E_{p,final} + E_{thermal}$$
graph TD
A[Total Mechanical Energy] --> B[Kinetic Energy: ½mv²]
A --> C[Gravitational PE: mgh]
A --> D[Elastic PE: ½kx²]
B <-->|Interconversion| C
A --> E[Thermal Energy via Friction]
E --> F[Energy dissipated - cannot be recovered as mechanical energy]
Power is the rate of doing work or the rate of energy transfer: $$P = \frac{W}{t} = \frac{\Delta E}{\Delta t}$$
Power is measured in watts (W), where 1 W = 1 J s⁻¹.
When a constant force $F$ acts on an object moving at velocity $v$: $$P = Fv$$
This is because $P = W/t = Fs/t = Fv$.
Average power uses the total work and total time. Instantaneous power is the power at any specific moment.
A car of mass 1200 kg accelerates from 0 to 30 m s⁻¹ in 10 s on a level road.
Change in KE = $\frac{1}{2}(1200)(30^2) = 540{,}000$ J
Average power = $\frac{540{,}000}{10} = 54{,}000$ W = 54 kW
No energy conversion is 100% efficient — some energy is always dissipated, usually as heat and/or sound.
$$\text{Efficiency} = \frac{\text{Useful output energy}}{\text{Total input energy}} \times 100\%$$
Or equivalently: $$\text{Efficiency} = \frac{\text{Useful output power}}{\text{Total input power}} \times 100\%$$
Efficiency is a dimensionless ratio, expressed as a percentage or a decimal between 0 and 1. A motor that converts 800 J from 1000 J input has efficiency = 80%.
Sources of energy loss in machines: - Friction in moving parts → heat - Electrical resistance → heat (I²R losses) - Sound waves from vibrating parts - Incomplete combustion in engines
Improving efficiency involves reducing these losses through better materials, lubrication, insulation, and design.
Energy exists in many forms that can be converted between each other:
| Form | Description |
|---|---|
| Kinetic | Motion of objects |
| Gravitational PE | Position in gravitational field |
| Elastic PE | Deformation of elastic materials |
| Thermal | Random kinetic/potential energy of particles |
| Chemical | Energy stored in molecular bonds |
| Nuclear | Energy in nuclear bonds (from E = mc²) |
| Electromagnetic | Energy of electric and magnetic fields |
| Sound | Mechanical vibrations transmitted through media |