understand the concept of work, and recall and use work done = force × displacement in the direction of the force

Work, Energy and Power – Cambridge IGCSE / A‑Level Physics (9702)

Learning Objectives

• Define work and explain its role as an energy‑transfer process.
• Apply the work‑formula W = F s cosθ for constant forces and for forces that vary with displacement.
• State and use the work‑energy theorem in its most general form.
• Derive and use the expressions for kinetic energy, gravitational potential energy (near‑Earth and in a field), elastic (spring) potential energy, electric potential energy and magnetic energy.
• Calculate power and efficiency for a range of mechanical, electrical and thermal processes.
• Relate work to the first law of thermodynamics and to energy changes in simple harmonic motion.
• Identify and evaluate work in deformation of solids, electrical circuits, magnetic systems and nuclear reactions.
• Plan, carry out and evaluate simple experiments that measure work, energy or power (AO2/AO3).

1. Work – Definition and General Formula

Work is the transfer of energy that occurs when a force (or a set of forces) acts through a displacement.

• Scalar quantity – only its sign (positive, negative or zero) indicates the direction of energy transfer.
• SI unit: 1 J = 1 N·m = kg·m²·s⁻².

1.1 Sign of Work

2. Work‑Energy Theorem (most general form)

The net work done on a system equals the total change in its mechanical energy, plus any change in internal energy caused by non‑conservative forces.

For a particle acted on only by conservative forces, \(W_{\text{net}} = \Delta K\) and the mechanical energy \(K+U\) is conserved.

3. Forms of Energy and Associated Work

3.1 Kinetic Energy (KE)

• Derived directly from the work‑energy theorem for a constant net force.
• Always positive; zero only when the object is at rest.

3.2 Gravitational Potential Energy (GPE)

• Near‑Earth approximation (uniform field):
  \(U_{g}=m g h\)
• General field (point mass M):
  \(U = -\dfrac{G M m}{r}\)
• Zero of potential may be chosen arbitrarily; the change in U is what matters.

3.3 Elastic (Spring) Potential Energy

• Valid only while the spring obeys \(F = kx\).
• Energy is released when the spring returns to its natural length.

3.4 Work in Deformation of Solids (Stress ↔ Strain)

For a material that obeys Hooke’s law in tension or compression, the work done in producing a strain ε is stored as elastic energy.

• \(\sigma\) = stress (N m⁻²), \(\varepsilon\) = strain (dimensionless), \(V\) = volume of the element.
• Useful for questions on stretching a wire, compressing a column or energy stored in a rubber band.

3.5 Electrical Work and Potential Energy

• Work done by an electrostatic force moving a charge q through a potential difference V:
  \(W = q V\)

• Electric potential energy of a system of point charges:
  \(U = \dfrac{1}{4\pi\varepsilon{0}} \dfrac{q{1}q_{2}}{r}\)

• In circuits, the energy supplied by a source of emf E over a time t at current I:
  \(W = E I t\)

3.6 Magnetic Work and Energy

• Force on a moving charge in a magnetic field: \(\vec F = q\vec v \times \vec B\).

  Work is zero because \(\vec F\) is always perpendicular to \(\vec v\) (\(\theta = 90^{\circ}\)).

• Energy stored in an inductor (magnetic field):
  \(U_{L}= \tfrac12 L I^{2}\)

• Work required to move a magnetic dipole \(\vec \mu\) through a change in magnetic field:
  \(W = -\int \vec \mu \cdot d\vec B\)

3.7 Nuclear Energy (mass–energy equivalence)

Energy released in a nuclear reaction is given by Einstein’s relation:

• \(\Delta m\) = loss of mass, \(c = 3.00\times10^{8}\,\text{m s}^{-1}\).
• Often presented as a real‑world application of the work‑energy idea (energy released = work done by the strong nuclear force).

4. Power and Efficiency

• Power – rate of doing work or transferring energy:
  \(P = \dfrac{W}{t}=Fv = \dfrac{\Delta E}{\Delta t}\)

  Units: 1 W = 1 J s⁻¹.

• Efficiency – useful output as a percentage of input:
  \(\eta = \dfrac{W{\text{useful}}}{W{\text{input}}}\times100\%\)

  Accounts for non‑conservative losses (friction, heat, electrical resistance, radiation).

5. First Law of Thermodynamics

• \(Q\) = net heat added to the system (positive if added).
• \(W\) = net work done *on* the system (positive if done on the system, negative if done by the system).
• When only work is involved (e.g. adiabatic compression of an insulated gas), \(\Delta U = W\).
• In isothermal processes for an ideal gas, \(\Delta U = 0\) so \(Q = -W\).

6. Connections to Other Syllabus Areas

6.1 Ideal Gases (Topic 15)

• Isothermal expansion: \(W = nRT\ln\left(\dfrac{Vf}{Vi}\right)\).
• Adiabatic expansion: \(W = \dfrac{PiVi-PfVf}{\gamma-1}\) where \(\gamma = Cp/Cv\).

6.2 Thermodynamics (Topic 16)

• Specific heat: \(Q = mc\Delta T\).
• Latent heat: \(Q = mL\).
• These expressions are frequently combined with the first law when analysing engines, refrigerators or calorimetry experiments.

6.3 Oscillations and Waves (Topics 17‑18)

• For a simple harmonic oscillator:
  \(E{\text{total}} = \tfrac12 kA^{2}=K{\max}=U_{\max}\)

• Energy in a standing wave is distributed between kinetic energy of the medium and potential energy of the deformation; the total energy is constant (ignoring damping).

6.4 Electricity and Magnetism (Topics 19‑21)

• Electrical work in moving charges, energy stored in capacitors (\(U{C}= \tfrac12 C V^{2}\)) and inductors (\(U{L}= \tfrac12 L I^{2}\)).
• Magnetic work is zero for a charge moving in a uniform \(\vec B\) field, but energy is stored in magnetic fields of coils and solenoids.

6.5 Nuclear and Medical Physics (Topics 23‑24)

• Energy released in fission or fusion reactions calculated via \(E=\Delta m c^{2}\).
• Applications: PET scanners, radiotherapy – illustrate the link between mass loss and usable work/energy.

7. Practical / Experimental Skills (AO2/AO3)

• Measuring mechanical work: Use a dynamometer to record force and a metre rule or motion sensor for displacement; calculate \(W = Fs\) (θ = 0°).
• Elastic energy experiment: Stretch a spring, plot force versus extension, determine \(k\) from the slope, then compute \(U_{s}=½kx^{2}\).
• Electrical work: Connect a resistor to a known voltage source, measure current with an ammeter and time with a stopwatch; compute \(W = V I t\).
• Power of a lifting machine: Use a pulley system, measure effort force and distance, compare with load weight × height; calculate efficiency \(\eta\).
• Thermodynamic work: Perform an isothermal expansion of a gas in a piston with a pressure sensor; integrate \(P\,dV\) numerically to obtain work.

8. Worked Examples

Example 1 – Horizontal Push (Mechanical Work)

A 5 kg crate is pushed across a frictionless floor by a constant horizontal force of 20 N over 3 m.

1. Identify: \(F = 20\text{ N},\; s = 3\text{ m},\; \theta = 0^{\circ}\).
2. Calculate: \(W = Fs\cos\theta = (20)(3)(1) = 60\text{ J}\).
3. Result: 60 J of positive work is done on the crate.

Example 2 – Inclined Plane (Work against Gravity)

A 2 kg block is pulled up a smooth 30° incline by a 15 N force parallel to the plane. The block moves 4 m along the plane.

1. Work by the pulling force: \(W_{\text{pull}} = Fs = (15)(4) = 60\text{ J}\).
2. Vertical rise: \(h = 4\sin30^{\circ}=2\text{ m}\).
3. Work done by gravity (negative): \(W_{g}= -mgh = -(2)(9.81)(2) = -39.2\text{ J}\).
4. Net work: \(W_{\text{net}} = 60 - 39.2 = 20.8\text{ J}\) → increase in kinetic energy.

Example 3 – Spring Compression (Elastic PE)

A spring with \(k = 250\;\text{N m}^{-1}\) is compressed 0.12 m from its natural length.

Example 4 – Electrical Work (Charging a Capacitor)

A 10 µF capacitor is charged from 0 V to 12 V by a constant 12 V source.

Example 5 – Magnetic Energy in an Inductor

An inductor of inductance \(L = 0.25\;\text{H}\) carries a current that rises uniformly to 3 A.

Example 6 – Nuclear Energy Release

In a fission reaction 1.0 g of \(\,^{235}\)U undergoes complete fission. Mass defect \(\Delta m = 0.001\text{ g}\).

9. Practice Questions

1. Calculate the work done by a 10 N force acting at 60° to the horizontal when an object moves 5 m horizontally.
2. A 0.5 kg ball is thrown vertically upward with an initial speed of 8 m s⁻¹. Determine the work done by gravity when the ball reaches its highest point.
3. Explain why the work done by the normal reaction force on a block sliding down a frictionless incline is zero.
4. For an ideal gas expanding isothermally from 1.0 L to 3.0 L at 300 K, calculate the work done if the external pressure remains constant at \(1.0\times10^{5}\) Pa.
5. A mass‑spring system oscillates with amplitude 0.10 m and spring constant 80 N m⁻¹. Find the maximum kinetic energy and the total mechanical energy.
6. Determine the electrical work required to move a charge of 5 µC through a potential difference of 250 V.
7. A steel wire of length 2 m, cross‑section 1 mm² and Young’s modulus \(2.0\times10^{11}\) N m⁻² is stretched by 1 mm. Calculate the work stored as elastic energy.
8. Calculate the efficiency of a motor that lifts a 50 kg load through 5 m in 4 s while drawing 0.8 A from a 240 V supply.
9. Using \(E=\Delta mc^{2}\), compute the energy released when 0.2 g of mass is converted entirely into energy.

10. Summary Table – Key Energy Concepts

11. Key Take‑aways

• Work quantifies energy transfer: W = F s cosθ. Only the component of force parallel to the displacement does work.
• The most general work‑energy theorem is Wₙₑₜ = ΔK + ΔUₙ𝚌; for conservative forces it reduces to Δ(K+U)=0.
• Kinetic, gravitational (both near‑Earth and field), elastic, electrical and magnetic energies are the main forms needed for the syllabus.
• When only conservative forces act, total mechanical energy is conserved; otherwise include the work of non‑conservative forces or use the first law.
• Power is the rate of doing work; efficiency measures how much of the input energy appears as useful output.
• Work, heat and internal energy are unified by the first law of thermodynamics.
• In simple harmonic motion, kinetic and elastic energies continuously interchange while the total remains constant.
• Practical skills—measuring forces, distances, voltages, currents and times—are essential for experimental verification of the concepts.