A Level · Cambridge (CIE) · 9709

Rebuild connected motion after a string goes slack

Calculate the motion up to a floor impact, carry the unaffected particle’s velocity into a new stage and stop when a new force or static state applies.

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

Solve a table-and-hanging system.
Carry final velocity between constant-acceleration stages.

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

Quick readiness check

Which quantity is carried into the table particle’s slack-string stage under the stated no-jump model?

What you will learn

Identify the event that ends the taut-string common-acceleration stage.
Recalculate the unaffected particle’s forces and motion after slackening.

A boundary event ends the original constraint

A 4 kg particle on a rough horizontal table with μ = 1/4 is linked to a 2 kg hanging mass. Released from rest, it starts moving because the hanging weight 20 N exceeds the 10 N static limit. During taut motion, a = (20 − 10)/6 = 5/3 m s⁻². If the hanging particle falls 0.3 m before hitting the floor, both particles reach speed v with v² = 2(5/3)(0.3) = 1, so v = 1 m s⁻¹ at that event.

Suppose the hanging particle then remains on the floor, the string slackens and there is no instantaneous velocity change of the table particle. The table particle retains speed 1 m s⁻¹ towards the pulley but no longer has string tension. Its only horizontal force is friction 10 N away from the pulley, giving acceleration −10/4 = −2.5 m s⁻² with towards the pulley positive. Equal accelerations and the previous tension no longer apply.

Use a new force diagram and a local clock

From v = 1 under a = −2.5, the table particle takes 0.4 s to stop and moves a further 1²/(2 × 2.5) = 0.2 m. Its total distance is 0.3 + 0.2 = 0.5 m. These numbers use the stated no-impact velocity change for this particle, not a conservation assumption across the hanging particle’s floor impact. There must be enough table length and enough slack for this next stage to occur as specified.

Once the table particle stops with no applied horizontal drive, the needed static friction is zero; it does not reverse merely because the old sliding equation had negative acceleration. On a smooth table, loss of tension instead gives zero acceleration and the unaffected particle continues at constant velocity until another stated event. Describe the new interval and force assumptions so a multi-stage answer does not imply that a slack string still constrains the two particles.

Pause and explain

After the rough-table particle stops with no horizontal drive and slack string, what happens to friction?

Put the idea to work

Worked example

A 4 kg table particle with μ = 1/4 is linked to a 2 kg hanging mass and released from rest. The hanging mass falls 0.3 m to the floor, stays there and the string goes slack; the table particle has no instantaneous velocity change and sufficient table space. Find its further and total travel before stopping.

Show the worked solution

Before impact, a = 5/3 and reached speed satisfies v² = 2(5/3)(0.3) = 1.
After slackening, table friction is 10 N and acceleration is −2.5. Further stopping distance is 1/5 = 0.2 m.
Total table travel is 0.3 + 0.2 = 0.5 m; time after slackening to rest is 1/2.5 = 0.4 s.

Answer Further travel 0.2 m; total table travel 0.5 m; stops 0.4 s after slackening.

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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 rough-table particle with μ = 1/4 is linked to a 2 kg hanging mass, released from rest. The hanging mass falls 0.3 m, stays on the floor and the string slackens without a table-particle velocity jump. Find the reached speed and further stopping distance, with enough table space.

Give me a hint

Separate the taut acceleration from the friction-only deceleration after slackening.

Compare my reasoning

Taut acceleration is (20 − 10)/6 = 5/3 m s⁻², giving reached speed √(2 × 5/3 × 0.3) = 1 m s⁻¹.
After slackening, table acceleration is −10/4 = −2.5 m s⁻².
Further distance is 1²/(2 × 2.5) = 0.2 m; total travel is 0.5 m.

Reached speed 1 m s⁻¹; further distance 0.2 m; total 0.5 m.

Look for these in your work

End the shared-acceleration model at the floor event.
Use the unchanged reached speed in the new stopping calculation.

2 · Independent

A 3 kg rough-table particle with μ = 1/5 is linked to a 2 kg hanging mass and released from rest. The hanging mass falls 5/7 m and then stays on the floor, making the string slack without a table-particle velocity jump. Find reached speed, further stopping time and distance before any other event.

Give me a hint

The table friction is 6 N in both moving stages, but tension disappears in the second.

Compare my reasoning

During taut motion, a = (20 − 6)/5 = 14/5, so v² = 2(14/5)(5/7) = 4 and v = 2 m s⁻¹.
After slackening, a = −6/3 = −2 m s⁻², so stopping time is 2/2 = 1 s.
Further distance is 2²/(2 × 2) = 1 m; total table travel is 5/7 + 1 = 12/7 m.

Reached speed 2 m s⁻¹; further time 1 s; further distance 1 m; total 12/7 m.

Look for these in your work

Calculate both stages from their own resultants.
Stop the friction-only motion at zero velocity and then check rest.

3 · Transfer

A 3 kg particle on a smooth horizontal table is linked to a 2 kg hanging mass and released from rest. After the hanging mass falls 0.5 m it stays on the floor and the string slackens without a table-particle velocity jump. With sufficient table space, find the table particle’s speed and additional distance during the next 2 s.

Give me a hint

On the smooth table, slackening removes its only horizontal force.

Compare my reasoning

Before the event, common acceleration is 20/5 = 4 m s⁻² and v² = 2 × 4 × 0.5 = 4, so speed is 2 m s⁻¹.
After slackening, the horizontal resultant is zero and acceleration is zero.
It continues towards the pulley at 2 m s⁻¹, moving an additional 4 m in the next 2 s; total travel is 4.5 m.

Speed remains 2 m s⁻¹; additional distance 4 m; total table travel 4.5 m.

Look for these in your work

Apply the smooth-contact assumption after the tension vanishes.
Do not confuse zero acceleration with stopping.

Common mistakes

The taut-string constraint ends when the string slackens.
Do not reset the unaffected particle’s velocity at the stated release event.
At final rest, replace sliding friction with the required static balance.

Recall without your notes

What can you explain now?

On a smooth table, what happens to a moving particle when the connecting string becomes slack and no other horizontal force acts?

Compare with the explanation

It continues at its reached constant velocity until another event.

Zero resultant after slackening means zero acceleration, not an instant stop.

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Make it stick

Locate the constraint-changing event, carry the permitted velocity forward and write a new Newton equation for the next stage.

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.