A Level · Cambridge (CIE) · 9709

Calculate signed work by a constant force

Use the component of a force along its point’s displacement, including positive driving work, negative resistance work and zero perpendicular work.

Section 4.5 · Energy, work and power

Mechanics · Energy, work and power · 4.5

Exam scope and assessment notes

Mechanics (Paper 4, AS & A Level) · Mathematics 9709, 2026–2027, section 4.5. Use g = 10 m s⁻² throughout these problems. Masses are positive and constant. Work by a constant force is Fd cos θ, with θ between force and displacement; scalar-product notation is not required. Kinetic energy is ½mv² and gravitational potential-energy change is mgΔh. State the system and count gravitational work or potential-energy change once, not both. Mechanical energy is conserved only under the stated loss-free model; friction and supplied resistance transfer mechanical energy to other forms. Smooth curved-slide examples use overall energy changes and assume contact throughout, rather than requiring a normal-reaction calculation. Average power is work divided by elapsed time; P = Fv applies to a force along the motion. Constant driving power generally gives variable force and acceleration; instantaneous calculations do not justify constant-acceleration formulae over a whole journey. Check string, contact and motion-state changes before continuing a model. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Resolve a force using sine or cosine.
Distinguish a force from the displacement of its point of application.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

Which part of a constant force contributes to work along a straight displacement?

What you will learn

Calculate work with the angle between force and displacement.
Interpret the sign and units of driving, resisting and perpendicular work.

A constant 25 N pull makes an angle with cosine 0.8 to a horizontal displacement of 8 m. Its along-displacement component is 20 N and its work is +160 J. The upward component can change reaction but does no work on this horizontal displacement. Force and displacement arrows are schematic and represent different physical quantities.

Only the along-displacement component contributes

For a constant force of magnitude F and displacement of magnitude d, the work is W = Fd cos θ, where θ is the angle between their directions. A 25 N pull with cos θ = 4/5 applied while a particle moves 8 m horizontally does work 25 × 8 × 4/5 = 160 J. Its upward component does no work during that horizontal displacement, even though it can change the normal reaction.

Work is measured in joules, with 1 J = 1 N m. It is a scalar and can be positive, zero or negative. Use the displacement of the force’s point of application; force magnitude alone does not specify work. The formula here assumes a constant force and the stated fixed displacement direction. Questions involving a changing force need more information than substituting one instantaneous value into Fd.

The sign follows the force-displacement angle

An opposing resistance Q during forward travel d has θ = 180°, so its work is −Qd. A 12 N resistance during 7 m travel does −84 J. A normal reaction perpendicular to a fixed smooth path does no work in that ideal contact model. A vertical support force also does zero work during a purely horizontal displacement; zero work does not mean that the force is absent.

If θ = 120°, cos θ = −1/2, so a 30 N force during 10 m displacement does −150 J. Use the actual angle between vectors rather than an acute angle copied from an unrelated diagram. When finding net work, calculate each force’s signed work and add them; the result can differ from the work of the drive alone.

Pause and explain

A 12 N resistance opposes forward travel of 7 m. What work does it do?

Put the idea to work

Worked example

A constant 25 N pull acts at angle θ above a horizontal displacement of 8 m, with cos θ = 4/5. Find its work.

Show the worked solution

The component along displacement is 25 × 4/5 = 20 N.
Multiply this component by the point’s displacement: W = 20 × 8.
Work is +160 J; the force has a component in the displacement direction.

Answer Work +160 J.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

A constant 20 N pull acts at angle θ to a 5 m straight displacement, with cos θ = 3/5. Find its signed work.

Give me a hint

Calculate the along-displacement component before multiplying by distance.

Compare my reasoning

The force component is 20 × 3/5 = 12 N in the displacement direction.
Work is 12 × 5 = 60 J.
Its positive sign shows energy transfer by the driving component.

Work +60 J.

Look for these in your work

Use cosine of the angle between force and displacement.
Include the sign and joule units.

2 · Independent

A particle moves 7 m along a horizontal surface against constant resistance 12 N. A perpendicular normal reaction also acts. Find work by the resistance and by the reaction.

Give me a hint

Use angles 180° and 90° to the horizontal displacement.

Compare my reasoning

Resistance work is 12 × 7 × cos 180° = −84 J.
Reaction work is R × 7 × cos 90° = 0, regardless of its magnitude.
Zero reaction work does not remove the reaction from the force model.

Resistance work −84 J; normal-reaction work 0 J.

Look for these in your work

Distinguish negative parallel work from zero perpendicular work.
Do not infer an absent reaction from zero work.

3 · Transfer

A constant force 30 N makes angle 120° with a 10 m displacement. A separate 80 N force is perpendicular to that displacement. Find each work and their total.

Give me a hint

Use the obtuse angle directly and do not turn its cosine positive.

Compare my reasoning

The first work is 30 × 10 × (−1/2) = −150 J.
The perpendicular force contributes 0 J.
Their signed total is −150 J; both forces can still act physically.

Works −150 J and 0 J; total −150 J.

Look for these in your work

Retain the negative cosine for the obtuse angle.
Add signed work values rather than force magnitudes.

Common mistakes

Use the angle between force and displacement.
An opposing force does negative work.
A force can act while doing zero work on the stated displacement.

Recall without your notes

What can you explain now?

Does a perpendicular normal reaction do work on a particle along a fixed smooth path?

Compare with the explanation

No; its component along the displacement is zero.

The force can constrain the path without transferring energy by work in this ideal model.

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Make it stick

Take the force component along its point’s displacement, multiply by displacement and keep the work sign.

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Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.