A Level · Cambridge (CIE) · 9709

Identify forces and draw a particle free body diagram

Choose the body being modelled, include each force acting on it and distinguish weight, normal contact, friction and string tension.

Section 4.1 · Forces and equilibrium

Mechanics · Forces and equilibrium · 4.1

Exam scope and assessment notes

Mechanics (Paper 4, AS & A Level) · Mathematics 9709, 2026–2027, section 4.1. Paper 1 algebra and trigonometry are assumed. Bodies are modelled as particles: forces act at one point and rotational effects are not considered. Resolve numerically; a scale drawing is not a substitute for calculation. Force diagrams are qualitative unless stated otherwise. Where mass is supplied in this sequence, use the stated g = 10 m s⁻². Check non-negative tension and contact reaction and use F ≤ μR except at limiting friction. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Use signed directions and basic trigonometry.
Distinguish mass in kilograms from force in newtons.

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

Quick readiness check

With g = 10 m s⁻², what is the weight of a 3 kg particle?

What you will learn

Identify external forces acting on a chosen particle.
Draw labelled directions using the supplied contact and string assumptions.

This free-body diagram chooses the 5 kg particle alone, using the stated g = 10 m s⁻². For rest on a rough horizontal surface under a 12 N rightward pull, the forces are weight 50 N down, normal reaction 50 N up and required friction 12 N left. The contact must have enough friction capacity for this assumed rest state. All arrows meet at the particle, and forces exerted by the particle on other bodies are omitted. Arrow lengths are schematic, not proportional to force magnitudes.

Choose one body before drawing its forces

A free-body diagram shows forces acting on the chosen body, not forces it exerts on its surroundings. Model the body as a particle with all force arrows meeting at one point. Weight mg acts vertically downwards; a taut string pulls along its length away from the body; normal contact acts perpendicular to a surface and pushes rather than pulls. Friction, when present, acts along the surface and opposes actual or impending relative sliding. It can be zero when no tangential force is needed.

For a 5 kg particle at rest on a horizontal rough surface, pulled horizontally right by 12 N, use the stated g = 10 m s⁻². Its weight is 50 N down, normal reaction is 50 N up and equilibrium requires friction 12 N left. These magnitudes follow from balance; the model also needs enough available friction to support that rest state. Do not add an extra force called ‘motion’ or put the force exerted on the ground onto the particle’s own diagram.

Model assumptions determine which forces are present

A smooth contact has no friction, while a rough contact permits a tangential friction component. A light flexible string carries tension when taut and cannot push a particle; a slack string supplies no tension. Weight remains vertical on an inclined plane, whereas the normal contact tilts perpendicular to it. On a smooth plane inclined at 30°, a 3 kg particle held by a string parallel to the plane has weight 30 N, tension 15 N up the plane and reaction 15√3 N normal to it.

An extended object is treated as a particle in this Mechanics component. This removes rotational effects from the model; it does not prove a real object cannot tip. State the assumptions before using equilibrium equations. A horizontal surface does not automatically make R = mg: an additional upward pull reduces the reaction, and a downward push increases it. A missing contact or slack string changes the force diagram and invalidates equations that retain that force.

Pause and explain

Which force is absent at a smooth contact?

Put the idea to work

Worked example

A 5 kg particle rests on a rough horizontal surface under a horizontal pull of 12 N to the right. With g = 10 m s⁻², list the forces required for equilibrium.

Show the worked solution

Weight is mg = 50 N vertically downwards.
Vertical balance requires a 50 N upward normal reaction.
Horizontal balance requires friction 12 N left, opposing the tendency to slide right; the available friction must be sufficient.

Answer Weight 50 N down; reaction 50 N up; pull 12 N right; friction 12 N left.

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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 3 kg lamp is at rest, suspended by one vertical taut light string. Use g = 10 m s⁻². Draw its force directions and find the tension.

Give me a hint

Choose the lamp alone and balance its two vertical forces.

Compare my reasoning

Its weight is 3 × 10 = 30 N downwards.
The string pulls the lamp vertically upwards with tension T.
Equilibrium gives T − 30 = 0, so T = 30 N.

Weight 30 N down; tension 30 N up.

Look for these in your work

Include the two forces on the lamp, not forces on the support.
Use weight in newtons and check vertical balance.

2 · Independent

A 4 kg particle rests on a smooth horizontal surface. A string pulls at 20 N, 30° above the rightward horizontal; a separate horizontal force keeps it at rest. Use g = 10 m s⁻². Find the reaction and balancing horizontal force.

Give me a hint

The pull has both horizontal and upward components; smooth contact supplies no friction.

Compare my reasoning

Weight is 40 N, and the string’s upward component is 20sin30° = 10 N.
Vertical balance gives R = 40 − 10 = 30 N.
The rightward string component is 20cos30° = 10√3 N, so the balancing force is 10√3 N left.

Reaction 30 N up; balancing horizontal force 10√3 N left; no friction.

Look for these in your work

Include the string’s vertical component in the reaction.
Use the smooth assumption and balance the horizontal forces separately.

3 · Transfer

A 3 kg particle is held at rest on a smooth 30° incline by a taut string parallel to the plane. Use g = 10 m s⁻². List the three forces and find their magnitudes.

Give me a hint

Resolve the vertical weight parallel and perpendicular to the plane.

Compare my reasoning

Weight is 30 N vertically downwards.
The downslope component is 30sin30° = 15 N, so tension is 15 N upslope.
The perpendicular component is 30cos30° = 15√3 N, so the normal reaction has that magnitude; friction is absent.

Weight 30 N vertical; tension 15 N upslope; reaction 15√3 N perpendicular to the plane.

Look for these in your work

Draw distinct vertical and plane-normal directions.
Use the smooth model without adding friction.

Common mistakes

Include only forces acting on the chosen body.
Keep weight vertical and normal reaction perpendicular to the surface.
A string pulls and a smooth surface supplies no friction.

Recall without your notes

What can you explain now?

Does a particle resting on an inclined plane have weight directed down the plane?

Compare with the explanation

No: weight is vertically downwards.

Its parallel and normal components are obtained by resolving the vertical weight.

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Make it stick

Choose a body, state its assumptions and draw every external force with the correct direction before resolving.

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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.