Work, energy, power & Hooke's law

Learning goals

What students should be able to explain, calculate and investigate after completing Work, Energy, Power and Hooke's Law.

• Define work done and calculate it for a force acting in the direction of motion.
• Define kinetic and potential energy, and apply the principle of conservation of energy.
• Define power and calculate it from work done or force and velocity.
• Calculate the efficiency of a device or process.
• State Hooke's Law and identify the linear region and limit of proportionality on a force-extension graph.
• Higher Level: Calculate the elastic potential energy stored, or work done, when a spring or elastic material is stretched or compressed within the relevant model.
• Describe experiments to verify Hooke's Law and to investigate conservation of energy.

Key definitions

Precise terms, symbols and SI units used in Work, Energy, Power and Hooke's Law.

Work Done

Work is done when a force causes displacement in the direction of that force: W = Fd.

Work is done when a force causes displacement in its direction. No work is done if there is a force but no displacement, or when the displacement is perpendicular to that force.

Kinetic and Potential Energy

Kinetic energy, Ek = ½mv², depends on an object's mass and the square of its speed — doubling the speed quadruples the kinetic energy.

Gravitational potential energy, Ep = mgh, depends on an object's height above a chosen reference level. In a system where only gravity does work (no friction or air resistance), the sum ΔEₚ + ΔEₖ = 0, meaning any loss in potential energy appears as an equal gain in kinetic energy, and vice versa.

Power and Efficiency

Power measures how quickly energy is transferred or work is done, not how much total energy is transferred.

A high-power device transfers energy quickly; it is not necessarily efficient. Efficiency describes the fraction of input transferred usefully, while power describes how rapidly transfer occurs.

• Power is the rate of energy transfer or work done.
• Efficiency compares useful output with total input.
• Use consistent energy units or consistent power units in an efficiency calculation.

Hooke's Law

Hooke's Law states that the extension of a spring is proportional to the applied force, up to the limit of proportionality.

Beyond this limit, the spring may still return to its original length when the force is removed (elastic behaviour) up to the elastic limit, but the relationship between force and extension is no longer linear. Beyond the elastic limit, the spring is permanently deformed. F = ks and F = −ks describe the same physical situation from two different perspectives, not two different laws. F = ks gives the magnitude of the applied (external) force needed to stretch or compress the spring by an extension s. F = −ks gives the restoring force exerted BY the spring on whatever is stretching it — this force always acts in the opposite direction to the extension (back towards the spring's natural length), which is why the minus sign appears. For most Leaving Certificate calculations, using the magnitude form F = ks is sufficient; the minus sign becomes important when Hooke's Law is used to describe simple harmonic motion (the restoring force always pulls back towards equilibrium).

Diagrams and formulae

Key diagrams and equations for reviewing Work, Energy, Power and Hooke's Law.

• Work is done when a force moves its point of application in the direction of the force: W = Fd.
• Hooke's Law graph: the linear (straight-line) region shows F ∝ extension; beyond the limit of proportionality, the graph curves and Hooke's Law no longer applies.

Work is done when a force moves its point of application in the direction of the force: W = Fd.

Hooke's Law graph: the linear (straight-line) region shows F ∝ extension; beyond the limit of proportionality, the graph curves and Hooke's Law no longer applies.

Note: E = mc² (mass–energy equivalence) belongs to the atomic and nuclear physics units and is not used for the mechanical-energy calculations in this unit.

Worked Example

A step-by-step calculation applying the central relationships in Work, Energy, Power and Hooke's Law.

A ball of mass 0.2 kg is dropped from a height of 5 m. Taking g = 9.8 m s⁻² and ignoring air resistance, find its speed just before hitting the ground.

• Loss in Ep = gain in Ek (conservation of energy, no air resistance)
• mgh = ½mv²
• gh = ½v² (mass cancels)
• v² = 2gh = 2 × 9.8 × 5 = 98
• v = √98 ≈ 9.9 m s⁻¹

Practical / Investigation

Experimental planning, measurement and evaluation for Work, Energy, Power and Hooke's Law.

Investigation: Force needed to stretch or compress an object

• Choose a safe object such as elastic cord, foam or a spring. Measure its original length or thickness, apply known forces over a suitable range, and record the extension or compression after each force is applied.
• Plot force against extension or compression and describe the relationship shown. Do not assume it is proportional: identify any curved region, limit of proportionality or permanent deformation supported by the data.
• Repeat readings while loading and unloading where appropriate. Keep the same object and loading direction, read the scale without parallax, and do not exceed a safe deformation.

Experiment: Verifying Hooke's Law

• Apparatus: spring, clamp stand, metre stick, slotted masses, pointer.
• Independent variable: applied force (weight added). Dependent variable: extension of the spring. Controlled variable: the spring used, and the temperature.
• Method: record the unstretched length of the spring; add masses one at a time, recording the new length after each addition; calculate the extension each time (new length − original length).
• Plot force (N) on the y-axis against extension (m) on the x-axis; a straight line through the origin (within the linear region) confirms Hooke's Law. The slope gives the spring constant, k.
• Sources of error: overstretching the spring past its limit of proportionality gives non-linear results; parallax error reading the pointer against the metre stick.
• Improvement: use a set square to read the pointer position perpendicular to the scale, reducing parallax error.

Experiment: Investigating conservation of energy using a falling mass

• Independent variable: height fallen, h. Dependent variable: speed just before landing, v (found using a light gate or motion sensor near the bottom of the fall). Controlled variable: the mass used.
• Method: release the mass from rest at a measured height above the light gate; record the speed as it passes through the gate; repeat for several different heights.
• For each height, calculate the loss in potential energy (mgh) and the gain in kinetic energy (½mv²); if mechanical energy is conserved, these two values should be equal (within experimental error).
• Plot v² (y-axis) against height h (x-axis). Since v² = 2gh, the graph should be a straight line through the origin with slope 2g.
• Sources of error: air resistance removes some mechanical energy, so measured v is usually slightly less than predicted; timing/height measurement errors.
• Improvement: use a dense, compact object to minimise air resistance; use a light gate rather than a stopwatch for precise timing.

• Hooke: measure original length, add known forces, calculate extension and plot F against x only within the elastic/linear region; gradient = k.
• Energy: release from rest at measured height h, measure speed v at the same reference level, compare mgh with ½mv² and plot v² against h.
• Use a safety stop for falling masses, do not overload the spring, repeat readings and quantify departures caused by friction/air resistance.

Quick-Check Questions

Short retrieval questions covering the essential ideas in Work, Energy, Power and Hooke's Law.

1. Define work done and give its unit.

2. Write the formula for kinetic energy, stating the meaning of each symbol.

3. Why is a device's efficiency always less than 100% in practice?

4. What does the slope of a force-extension graph in the linear region represent?

5. State Hooke's Law and its condition.

Common mistakes and exam tips

Common misconceptions and exam-technique checks for Work, Energy, Power and Hooke's Law.

• Avoid: Using the full force when it is not acting in the direction of displacement. Correct idea: For course calculations, identify the force component acting along the displacement and use W = Fd.
• Avoid: Saying doubling speed doubles kinetic energy. Correct idea: Because Ek = ½mv², doubling speed makes kinetic energy four times larger.
• Avoid: Finding a spring constant from the curved part of a force–extension graph. Correct idea: Use the gradient only in the straight-line region where force is proportional to extension.

• State whether friction and air resistance can be neglected before using conservation of mechanical energy.
• For efficiency, compare useful output and total input using matching energy or power units.
• Use only the proportional region of a force–extension graph to determine k.

Exam-style questions

Five exam-style questions with worked answers on Work, Energy, Power and Hooke's Law.

Higher Level — Required Content

Required Higher Level content for this topic.

Within the linear region, the elastic potential energy stored is the area under the force–extension graph.

• k is spring constant in N m⁻¹ and s is extension in metres.
• Use the relationship in Higher Level calculations and check that the spring remains within the linear region.

End-of-unit revision

Formulae, definitions, experiments and mixed revision questions for Work, Energy, Power and Hooke's Law.

• W = Fd (force acting in the direction of displacement)
• Ek = ½mv²
• Ep = mgh
• ΔEₚ + ΔEₖ = 0 (conservation of mechanical energy)
• P = W/t = Fv
• Efficiency (%) = (useful output ÷ total input) × 100
• F = ks (Hooke's Law)

• Work done
• Kinetic and potential energy
• Power
• Efficiency
• Hooke's Law
• Limit of proportionality

• Force and displacement diagram showing W = Fd
• Force-extension graph with the linear region and limit of proportionality clearly labelled

• Verifying Hooke's Law using a spring and slotted masses
• Investigating conservation of energy (e.g. a falling mass, a pendulum, or a trolley on a slope)

• Forgetting Ek depends on v², not v
• Confusing power with total energy/work
• Reading k from beyond the limit of proportionality
• Assuming 100% efficiency

1. A spring has a spring constant of 40 N m⁻¹. Calculate the force needed to extend it by 0.15 m.

2. A pump raises water of mass 500 kg through a height of 12 m in 20 s. Calculate the useful power output of the pump. (g = 9.8 m s⁻²)

3. Explain, in terms of energy transformations, why a real pendulum eventually stops swinging.

4. A motor with an input power of 800 W produces a useful output power of 600 W. Calculate its efficiency.

1. F = ks = 40 × 0.15 = 6 N.

2. W = mgh = 500 × 9.8 × 12 = 58 800 J. P = W/t = 58 800 ÷ 20 = 2940 W.

3. Air resistance and friction at the pivot convert some of the pendulum's mechanical energy (kinetic + potential) into heat and sound energy at each swing, so its total mechanical energy gradually decreases until it stops.

4. Efficiency = (600 ÷ 800) × 100 = 75%.

The table below shows which equations from this unit appear in the official State Examinations Commission Formulae and Tables book, and where.

For this course, use W = Fd where the force acts in the direction of displacement.