Forces Momentum And Energy

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Original, AI-authored English practice with worked guidance. Selected topics only; independent teacher review is pending. This is not a complete syllabus, official past-paper collection or predicted exam score. Check your own exam-board specification: these concepts are not certified against a particular board or exam year.

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Cambridge International AS & A Level Mathematics (9709), examination year 2027: AI-assisted conceptual alignment to official sources, not teacher or exam-board certification. Syllabus mapping and remaining coverage.

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Original Cambridge 2027 concept practice for source-backed gaps. Indicative self-marking only, not an official paper or complete syllabus coverage.

Selected subtopics: Static friction on an incline · Coalescence and kinetic energy · Work and instantaneous power

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Question 1A 4.0 kg particle rests on a rough plane inclined at 30 degrees to the horizontal. Take g=10 m

s^{-2}

and coefficient of friction mu=0.40. A force P acts up the plane. Find the normal reaction, the friction required for equilibrium, and the full interval of P>=0 that can maintain rest. Give limiting values to 3 significant figures and state friction directions at each end.6 indicative marks

Show your reasoning6 indicative marksCalculator allowed

A 4.0 kg particle rests on a rough plane inclined at 30 degrees to the horizontal. Take g=10 m $s^{-2}$ and coefficient of friction mu=0.40. A force P acts up the plane. Find the normal reaction, the friction required for equilibrium, and the full interval of P>=0 that can maintain rest. Give limiting values to 3 significant figures and state friction directions at each end.

Cambridge 2027: Direct conceptual match

Independently resolved the forces and solved the two-sided limiting inequality; verified both friction signs and 6.14/33.9 N bounds.

9709: 4.1 Forces and equilibrium; candidate bullet 1 (bullet ordinal, not an official subcode): Identify forces and draw force diagrams
9709: 4.1 Forces and equilibrium; candidate bullet 2 (bullet ordinal, not an official subcode): Resolve forces and compute resultants
9709: 4.1 Forces and equilibrium; candidate bullet 3 (bullet ordinal, not an official subcode): Use zero resultant for particle equilibrium
9709: 4.1 Forces and equilibrium; candidate bullet 4 (bullet ordinal, not an official subcode): Separate normal contact and friction components
9709: 4.1 Forces and equilibrium; candidate bullet 6 (bullet ordinal, not an official subcode): Use limiting friction and F<=mu*R

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Your response to question 1

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Worked guidance

Draw weight, normal, P and a friction arrow of unspecified sign. Resolve the normal and parallel directions separately; rest requires zero resultant in each.
The contact force consists of the normal plus the friction component. The surface is rough, so treating it as smooth would incorrectly allow only P=20 N.

Self-marking points

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The weight components are 40sin30=20 N down the plane and 40cos30=20sqrt(3) N into it.
The normal reaction is R=20sqrt(3) N; the applied force has no normal component.
Writing friction positive up the plane gives F=20-P in equilibrium.
Static friction satisfies |20-P|<=mu*R=8sqrt(3), so 20-8sqrt(3)<=P<=20+8sqrt(3).
The bounds are approximately 6.14 N and 33.9 N.
At the lower bound friction acts up the plane; at the upper it acts down. Between them its magnitude need not equal mu*R.

Examiner tip

Use the inequality for static friction; equality applies only at a limiting endpoint.

Question 2Two particles of masses 2.0 kg and 3.0 kg move along a smooth horizontal line at +5.0 m s^-1 and -1.0 m s^-1. They collide and coalesce; external impulse is negligible. Find the common velocity and loss of kinetic energy. Afterwards a constant 4.0 N force acts in the positive direction for a further displacement of 3.0 m. Find the final speed and instantaneous power of that force at the end.6 indicative marks

Show your reasoning6 indicative marksCalculator allowed

Two particles of masses 2.0 kg and 3.0 kg move along a smooth horizontal line at +5.0 m s^-1 and -1.0 m s^-1. They collide and coalesce; external impulse is negligible. Find the common velocity and loss of kinetic energy. Afterwards a constant 4.0 N force acts in the positive direction for a further displacement of 3.0 m. Find the final speed and instantaneous power of that force at the end.

Cambridge 2027: Direct conceptual match

Computed signed momentum 7, energies 26.5/4.9/16.9 J, speed exactly 2.6 and end power 10.4 W.

9709: 4.3 Momentum; candidate bullet 1 (bullet ordinal, not an official subcode): Use signed one-dimensional linear momentum
9709: 4.3 Momentum; candidate bullet 2 (bullet ordinal, not an official subcode): Conserve momentum in direct impacts, including coalescence
9709: 4.5 Energy, work and power; candidate bullet 1 (bullet ordinal, not an official subcode): Compute work for a constant force and displacement angle
9709: 4.5 Energy, work and power; candidate bullet 2 (bullet ordinal, not an official subcode): Calculate gravitational potential and kinetic energies
9709: 4.5 Energy, work and power; candidate bullet 3 (bullet ordinal, not an official subcode): Use external work and energy conservation
9709: 4.5 Energy, work and power; candidate bullet 4 (bullet ordinal, not an official subcode): Use average power and instantaneous P=Fv

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Your response to question 2

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Worked guidance

Use signed velocities in the momentum balance. Coalescence conserves momentum but not kinetic energy, so calculate the two energies separately.
The later smooth-track segment is a separate energy balance. The work adds to the post-collision energy, not the pre-collision energy.

Self-marking points

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Initial momentum is 2*5+3*(-1)=7 kg m $s^{-1}$ .
The common velocity is +7/5=+1.4 m $s^{-1}$ .
Initial KE is 26.5 J and final collision KE is 4.9 J, so the loss is 21.6 J.
The subsequent force does work 4*3=12 J.
Final KE is 16.9 J, so final speed sqrt(2*16.9/5)=2.60 m $s^{-1}$ to 3 significant figures.
Power at the end is Fv=4*2.60=10.4 W using the unrounded speed.

Examiner tip

Do not confuse a force's instantaneous power with its total work or with average power.

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