Work and Energy
Work and Energy is the bridge between forces and motion. The PMDC MDCAT 2026 syllabus expects you to handle work as a dot product, distinguish kinetic from potential energy, apply the work-energy theorem, calculate power and efficiency. Expect 2-3 MCQs.
Work
Work is done when a force produces a displacement of its point of application. For a constant force F producing a displacement d:
W = F·d·cosθ (or W = F·d in vector form)
SI unit: joule (J) = N·m. Work is a scalar despite being defined from two vectors.
Sign of work
- θ < 90° ⇒ W > 0 (positive work, e.g. pulling a sled).
- θ = 90° ⇒ W = 0 (e.g. centripetal force does no work in uniform circular motion).
- θ > 90° ⇒ W < 0 (e.g. friction).
Energy
Energy is the capacity to do work. SI unit: joule (J). Energy exists in many forms — kinetic, potential, thermal, chemical, nuclear, electromagnetic — and can be transformed from one to another.
Law of conservation of energy: energy can neither be created nor destroyed; it can only be transformed. The total energy of an isolated system is constant.
Kinetic Energy
Kinetic energy (KE) is the energy possessed by a body by virtue of its motion:
KE = ½mv²
KE is a scalar, always non-negative. Doubling the speed quadruples the KE; doubling the mass only doubles the KE.
Relation to momentum
Since p = mv, we have KE = p²/(2m). For two bodies of equal momentum, the lighter one has more KE.
Potential Energy
Potential energy (PE) is the energy a body possesses by virtue of its position or configuration.
| Energy type | Formula | Depends on | SI unit | Example |
|---|---|---|---|---|
| Kinetic energy | ½mv² (= p²/2m) | Mass, speed | J | Moving car, falling raindrop |
| Gravitational PE (near Earth) | mgh | Mass, height, g | J | Object lifted above ground |
| Gravitational PE (general) | U = −GMm/r | Distance from centre | J | Satellite, planet in orbit |
| Elastic PE (spring) | ½kx² | Spring constant, extension | J | Stretched spring, bow string |
| Rotational KE | ½Iω² | Moment of inertia, angular speed | J | Spinning wheel, rotating disc |
| Heat (thermal) | mcΔT | Mass, specific heat, T change | J | Warming water |
| Mass-energy | E = mc² | Mass | J | Nuclear reactions |
PE is a relative quantity — it depends on the choice of reference level. Only changes Δ(PE) are physically meaningful.
Conservative vs Non-conservative forces
| Property | Conservative | Non-conservative |
|---|---|---|
| Path dependence of work | Independent of path; depends only on initial & final positions | Depends on path |
| Work in a closed loop | Zero | Non-zero (energy dissipated) |
| Associated PE function? | Yes (PE definable) | No |
| Mechanical energy conserved? | Yes — KE + PE = constant | No — mechanical energy decreases |
| Examples | Gravity, spring (elastic), electrostatic, magnetostatic | Friction, air drag, viscous drag, tension in inelastic string |
Absolute Potential Energy
Near Earth's surface PE = mgh works fine, but for objects in space we need a more general expression. The absolute gravitational potential energy of a mass m in the field of Earth (mass M, radius R) at distance r from Earth's centre is:
U(r) = −GMm/r
The reference is taken at infinity, where U = 0. PE is negative everywhere in a gravitational field because work must be done against attractive gravity to bring a mass from r to infinity.
Escape velocity
The minimum launch speed required for a body to leave Earth's gravitational field permanently. Setting KE = |U|:
½mve² = GMm/R ⇒ ve = √(2GM/R) = √(2gR) ≈ 11.2 km/s.
Work-Energy Theorem
The net work done on a body equals the change in its kinetic energy:
Wnet = ΔKE = ½mv² − ½mu²
This theorem is universal — it applies whether the force is constant or variable, and whether the path is straight or curved. It is often quicker than working with F = ma directly.
Conservation of mechanical energy
If only conservative forces (gravity, springs) do work, the total mechanical energy E = KE + PE is conserved. Example: a pendulum at the bottom of its swing has all KE; at the top, all PE; in between, a mix — but the sum is constant.
Power
Power is the rate of doing work (or rate of energy transfer):
P = W/t = dW/dt
SI unit: watt (W) = J/s. 1 horsepower (hp) = 746 W. For a constant force moving with velocity v:
P = F·v = F·v·cosθ
Energy Losses and Efficiency
In real systems, some input energy is always lost to friction, heat, sound, deformation, etc. The efficiency of a machine is the ratio of useful output energy to total input energy:
η = (Useful output / Total input) × 100%
- Efficiency is always < 100% in practice (ideal machines would be 100%).
- For a heat engine the maximum efficiency is the Carnot value ηmax = 1 − Tc/Th.
- The remaining energy is dissipated as heat — not destroyed.
Worked MCQs
Five MCQs that capture the high-yield testing patterns for this chapter.
Q1. A 50 N force pushes a box through 4 m at an angle of 60° to the horizontal direction of motion. Work done is:
W = Fd·cosθ = 50 × 4 × cos60° = 50 × 4 × 0.5 = 100 J.
Q2. If the speed of a body is doubled, its kinetic energy becomes:
KE = ½mv². Since KE ∝ v², doubling v makes KE go up by a factor of 4.
Q3. A 2 kg ball is raised through a height of 5 m. Taking g = 10 m/s², the gain in gravitational PE is:
ΔPE = mgh = 2 × 10 × 5 = 100 J.
Q4. A motor lifts a 50 kg load through 10 m in 5 s. Power output (g = 10 m/s²):
Work done = mgh = 50 × 10 × 10 = 5000 J. Power = W/t = 5000/5 = 1000 W.
Q5. A machine takes in 500 J of energy and produces 350 J of useful output. Its efficiency is:
η = (output/input) × 100% = (350/500) × 100% = 70%. The remaining 150 J is lost as heat / sound.
Quick Recap
- W = F·d·cosθ; SI unit joule (J).
- KE = ½mv² = p²/(2m).
- Gravitational PE = mgh; spring PE = ½kx²; absolute U = −GMm/r.
- Work-energy theorem: Wnet = ΔKE.
- Conservation of mechanical energy when only conservative forces act.
- P = W/t = F·v; SI unit watt.
- Efficiency η = output/input × 100%.
- Escape velocity ve = √(2gR) ≈ 11.2 km/s for Earth.