A Level · Cambridge (CIE) · 9709

Distinguish mass weight and an accelerating support force

Use W = mg and resolve vertically to calculate a reaction or cable tension, including upward motion that is slowing down.

Section 4.4 · Newton’s laws of motion

Mechanics · Newton’s laws and connected particles · 4.4

Exam scope and assessment notes

Mechanics (Paper 4, AS & A Level) · Mathematics 9709, 2026–2027, section 4.4. Use g = 10 m s⁻² throughout these problems. Each particle has positive constant mass. Forces and acceleration are constant within each stated stage; linear motion only. Use the resultant force in F = ma. A light taut inextensible string over a smooth pulley has one tension and equal acceleration magnitudes; those constraints cease if the string becomes slack. A rope can pull but cannot push; a light rigid tow-bar may exert tension or thrust. Resistance other than the stated friction is neglected unless explicitly supplied. In our rough-contact model, sliding friction has magnitude μR and opposes motion; at rest, friction adjusts with magnitude at most μR. Check contact and motion assumptions before continuing a solution. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Apply F = ma with a chosen positive direction.
Identify weight, normal reaction and cable tension separately.

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

Quick readiness check

At g = 10 m s⁻², a particle’s weight is 80 N. What is its mass?

What you will learn

Convert between positive mass and gravitational weight.
Find a support force from the signed vertical acceleration and check contact.

The 6 kg particle has weight 60 N downward and a lift-floor reaction 72 N upward. Their resultant 12 N upward gives acceleration 2 m s⁻² upward. The acceleration is labelled separately from the actual force arrows; a floor reaction is not always equal to weight. Arrows are schematic.

Weight is a force; mass is not

The weight of a particle of mass m is mg, directed downwards. With g = 10 m s⁻², a mass of 6 kg has weight 60 N. The numerical values of kilograms and newtons cannot be substituted for each other. If weight is 80 N in this model, mass is 80/10 = 8 kg. A change of support force does not change the particle’s mass or gravitational weight.

An upward normal reaction N from a lift floor acts on a supported particle while its weight acts downwards. With upward positive, N − mg = ma, so N = m(g + a). A suspended particle instead uses cable tension T in that equation. N and T describe different interactions, even when their numerical balances have the same form; draw whichever interaction is actually present.

Acceleration changes the support reading

A 6 kg particle accelerating upward at 2 m s⁻² has N = 6(10 + 2) = 72 N. If it is moving upward but slowing at 3 m s⁻², its acceleration is downward: a = −3, so N = 42 N. Using upward velocity to choose upward acceleration would incorrectly predict 78 N. The support responds to acceleration, not directly to the direction of travel.

A floor can push a particle but cannot pull it downwards. In this simple unrestrained contact model, N must be non-negative. At a = −g the support reading is zero, consistent with free fall. A calculation predicting negative N means the assumed contact cannot continue without another stated constraint; do not silently accept a negative reaction. Similarly, a flexible cable cannot sustain negative tension.

Pause and explain

For upward-positive acceleration a, which support equation is correct?

Put the idea to work

Worked example

A 6 kg particle remains on a lift floor accelerating upward at 2 m s⁻². Find its weight and normal reaction using g = 10.

Show the worked solution

Weight is W = 6 × 10 = 60 N downwards.
Upward force balance is N − 60 = 6 × 2.
Therefore N = 72 N upwards; the positive reaction is compatible with contact.

Answer Weight 60 N; normal reaction 72 N.

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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 4 kg particle rests relative to a lift floor accelerating up at 1.5 m s⁻². Find weight and normal reaction.

Give me a hint

Find mg first, then use an upward-positive force equation.

Compare my reasoning

Weight is 4 × 10 = 40 N.
The normal balance is N − 40 = 4 × 1.5 = 6.
Hence N = 46 N, which is positive and supports the contact assumption.

Weight 40 N; N = 46 N.

Look for these in your work

Distinguish the two forces.
Add ma to weight for upward acceleration.

2 · Independent

A 6 kg particle moves upward with a lift but slows at 3 m s⁻². Find its normal reaction and compare it with weight.

Give me a hint

Slowing during upward motion means acceleration is downward.

Compare my reasoning

Use upward positive and a = −3 m s⁻².
Then N − 60 = 6(−3), giving N = 42 N.
The reading is 18 N less than the unchanged weight of 60 N.

N = 42 N; weight remains 60 N.

Look for these in your work

Choose acceleration from the speed change, not just travel direction.
Retain positive mass and constant gravitational weight.

3 · Transfer

A 5 kg unrestrained particle is assumed to stay on a lift floor accelerating downward at 12 m s⁻². Test that contact model and give the particle’s acceleration if contact is lost and no other force acts.

Give me a hint

A floor cannot exert a downward normal reaction on an unrestrained particle.

Compare my reasoning

Upward positive gives a = −12 and N − 50 = 5(−12), so the assumed N is −10 N.
Negative N is incompatible with this floor contact, so the assumed shared acceleration fails.
After separation, only weight acts: actual acceleration is −10 m s⁻² and N = 0.

Assumed N = −10 N is invalid; contact is lost and a = −10 m s⁻² with N = 0.

Look for these in your work

Flag the physical constraint rather than accepting a negative reaction.
Use a new force model after contact is lost.

Common mistakes

Weight is measured in newtons, mass in kilograms.
Moving up and accelerating up are different statements.
A negative reaction contradicts this unrestrained support model.

Recall without your notes

What can you explain now?

A lift moves up while slowing at 3 m s⁻². Which acceleration enters an upward-positive equation?

Compare with the explanation

a = −3 m s⁻².

The acceleration opposes the upward velocity, so it is directed downwards.

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Make it stick

Compute mg, then use the signed vertical acceleration to find the actual support force and check its admissibility.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

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.