Newton's laws of motion

A particle $P$ with mass $0.8\,\text{kg}$ rests on a rough horizontal table. The coefficient of friction between $P$ and the table is $\mu$. A force of magnitude $5\,\text{N}$, directed upwards at an angle $\alpha$ above the horizontal, where $\tan\alpha = \frac{3}{4}$, acts on $P$. The particle is just about to slide across the table.

Feb/March 2016

A car with mass $1200\,\text{kg}$ is towing a trailer with mass $800\,\text{kg}$ up a slope making an angle $\alpha$ with the horizontal, where $\sin\alpha = 0.1$. The car and trailer system is treated as two particles joined by a light inextensible cable. The engine provides a driving force of $2500\,\text{N}$, while the resistive forces on the car and trailer are $100\,\text{N}$ and $150\,\text{N}$ respectively.

Feb/March 2016

Particles $A$ and $B$, with masses $0.8\,\text{kg}$ and $0.2\,\text{kg}$ respectively, are joined by a light inextensible string. $A$ rests on a horizontal plane. The string runs over a small smooth pulley $P$ at the edge of the plane, and $B$ is suspended freely. The horizontal part of the string, $AP$, has length $2.5\,\text{m}$. The particles are let go from rest, with both parts of the string taut.

Feb/March 2016

Two particles with masses $1.2\,\text{kg}$ and $0.8\,\text{kg}$ are joined by a light inextensible string that runs over a fixed smooth pulley. They hang vertically. The system starts from rest, and both particles are $0.64\,\text{m}$ above the floor (see diagram). During the motion that follows, the $0.8\,\text{kg}$ particle does not reach the pulley.

Feb/March 2017

Particles $A$ and $B$, with masses $0.8\,\text{kg}$ and $0.2\,\text{kg}$ respectively, are linked by a light inextensible string which runs over a fixed smooth pulley. The particles are suspended vertically. The system is released from rest.

Feb/March 2018

An elevator travels vertically while being held by a cable. The diagram presents a velocity-time graph that represents the elevator’s motion. The graph is made up of $7$ straight line segments. The elevator first accelerates upwards from rest to a speed of $2\,\text{m s}^{-1}$ in $1.5\,\text{s}$, then continues at this speed for $4.5\,\text{s}$, before slowing to rest in $1\,\text{s}$. It then stays at rest for $6\,\text{s}$, before accelerating downwards to a speed of $V\,\text{m s}^{-1}$ over $2\,\text{s}$. The elevator then moves at this speed for $5\,\text{s}$, before slowing to rest in $1.5\,\text{s}$.

Feb/March 2021

A car with mass $m\,\text{kg}$ is pulling a trailer of mass $300\,\text{kg}$ downhill on a straight slope at $3^{\circ}$ to the horizontal at constant speed. Resistive forces act on both the car and the trailer, and the total work done against these resistive forces over a distance of $50\,\text{m}$ is $40\,000\,\text{J}$. The car’s engine is doing no work, and the tow-bar is light and rigid.

Feb/March 2022

A block $B$, with mass $2\,\text{kg}$, rests on a rough inclined plane that makes an angle of $30^\circ$ with the horizontal. A light rope, which is at $20^\circ$ above the line of greatest slope, is fixed to $B$. The rope tension is $T\,\text{N}$. A friction force of $F\,\text{N}$ acts on $B$ (see diagram). The coefficient of friction between $B$ and the plane is $\mu$.

Feb/March 2023

Particles $A$ and $B$ have masses $0.3\,\text{kg}$ and $0.1\,\text{kg}$ respectively. They are fixed to the two ends of a light inextensible string. The string runs over a fixed smooth pulley, and the particles hang vertically below the pulley. At the start, both particles are at a height of $x\,\text{m}$ above the horizontal ground (see diagram). The system is released from rest.

Feb/March 2025

Particles $A$ and $B$, with masses $0.2 \text{ kg}$ and $0.45 \text{ kg}$ respectively, are linked by a light inextensible string of length $2.8 \text{ m}$. The string runs over a small smooth pulley at the edge of a rough horizontal surface, which is $2 \text{ m}$ above the floor. Particle $A$ is kept in contact with the surface at a distance of $2.1 \text{ m}$ from the pulley, while particle $B$ hangs freely. The coefficient of friction between $A$ and the surface is $0.3$. When particle $A$ is released, the system starts to move.

May/June 2010

Particles $A$ and $B$, with masses $0.2\,\text{kg}$ and $0.45\,\text{kg}$ respectively, are joined by a light inextensible string of length $2.8\,\text{m}$. The string goes over a small smooth pulley at the edge of a rough horizontal surface, which is $2\,\text{m}$ above the floor. Particle $A$ is kept in contact with the surface at a distance of $2.1\,\text{m}$ from the pulley, while particle $B$ hangs freely. The coefficient of friction between $A$ and the surface is $0.3$. Particle $A$ is released and the system starts to move.

May/June 2010

The sketch depicts a vertical cross-section through a triangular prism that is secured so that two of its faces make an angle of $60^\circ$ with the horizontal. One face is smooth, while the other is rough. Particles $A$ and $B$, with masses $0.36\,\text{kg}$ and $0.24\,\text{kg}$ respectively, are joined to the ends of a light inextensible string that passes over a small smooth pulley fixed at the highest point of the cross-section. $B$ is kept at rest at a point on the rough face of the cross-section, and $A$ hangs freely in contact with the smooth face (see diagram). When $B$ is released, it moves up the face with acceleration $0.25\,\text{m s}^{-2}$.

May/June 2010

Particles $P$ and $Q$ travel along the line of greatest slope on a smooth inclined plane. $P$ starts from rest at point $O$ on the line, and after $2\,\text{s}$ it passes point $A$ with speed $3.5\,\text{m s}^{-1}$.

May/June 2010

Two rectangular boxes $A$ and $B$ have the same dimensions. They are initially stationary on a rough horizontal floor, with $A$ placed on top of $B$. Box $A$ has mass $200\,\text{kg}$ and box $B$ has mass $250\,\text{kg}$. A horizontal force of magnitude $P\,\text{N}$ acts on $B$ (see diagram). The boxes stay at rest when $P \le 3150$ and begin to move when $P > 3150$.

May/June 2010

The diagram gives the vertical cross-section $OAB$ of a slide. The straight section $AB$ is tangent to the curve $OA$ at $A$. The line $AB$ makes an angle $\alpha$ with the horizontal, where $\sin \alpha = 0.28$. Point $O$ is $10\,\text{m}$ above $B$, and $AB$ is $10\,\text{m}$ long (see diagram). The section of the slide containing the curve $OA$ is smooth, whereas the section containing $AB$ is rough. A particle $P$ of mass $2\,\text{kg}$ is released from rest at $O$ and slides down the slide.

May/June 2012

Particles $P$ and $Q$, with masses $0.6\,\text{kg}$ and $0.4\,\text{kg}$ respectively, are attached to the two ends of a light inextensible string. The string passes over a small smooth pulley fixed at the top of a vertical cross-section of a triangular prism. The prism has its base on horizontal ground, and both sloping faces are smooth. Each sloping face is inclined at angle $\theta$ to the ground, where $\sin \theta = 0.8$. At the beginning, the particles are held stationary on the sloping faces, with the string taut (see diagram). They are then released and travel along lines of greatest slope.

May/June 2012

Block $A$ has mass $3\,\text{kg}$ and is tied to one end of a light inextensible string $S_1$. At the far end of $S_1$ is block $B$ of mass $2\,\text{kg}$, and $B$ is also joined to one end of another light inextensible string $S_2$. The other end of $S_2$ is fastened to a fixed point $O$, so the blocks hang in equilibrium beneath $O$ (see diagram). When $S_2$ breaks, the particles fall. The air resistance on $A$ is $1.6\,\text{N}$ and the air resistance on $B$ is $4\,\text{N}$.

May/June 2012

A block is resting on a rough horizontal plane, and the coefficient of friction between the block and the plane is $1.25$.

May/June 2013

A train with mass $400\,000\text{ kg}$ is travelling along a straight horizontal track. The engine provides a constant power of $1500\text{ kW}$, and the force resisting the train’s motion is $30\,000\text{ N}$.

May/June 2013

A light inextensible string carries a particle $A$ of mass $0.26\text{ kg}$ at one end and a particle $B$ of mass $0.54\text{ kg}$ at the other. Particle $A$ is kept at rest on a rough plane inclined at angle $\alpha$ to the horizontal, where $\sin\alpha = \frac{5}{13}$. The string is taut and runs parallel to a line of greatest slope on the plane. It passes over a small smooth pulley at the top of the plane. Particle $B$ hangs motionless vertically below the pulley. The coefficient of friction between $A$ and the plane is $0.2$. Particle $A$ is released, and the particles begin to move.

May/June 2013

A particle $P$, with mass $0.5\text{ kg}$, is situated on a smooth horizontal plane. Forces with magnitudes $F\text{ N}$, $2.5\text{ N}$ and $2.6\text{ N}$ act horizontally on $P$. Their directions are as shown in the diagram, where $\tan\alpha = \frac{12}{5}$ and $\tan\beta = \frac{7}{24}$.

May/June 2013

A block of weight $30\,\text{N}$ rests on a rough horizontal plane and is connected to a string. With the string horizontal and the tension equal to $24\,\text{N}$, the block is in limiting equilibrium.

May/June 2013

Particles A, of mass $0.26\,\text{kg}$, and B, of mass $0.52\,\text{kg}$, are fastened to the two ends of a light inextensible string. That string runs over a small smooth pulley $P$, which is fixed at the top of a smooth plane. The plane is tilted at an angle $\alpha$ to the horizontal, where $\sin\alpha = \frac{16}{65}$ and $\cos\alpha = \frac{63}{65}$. A is kept at rest $2.5\,\text{m}$ from $P$, with the section $AP$ of the string parallel to a line of greatest slope of the plane. B is hanging freely below $P$ at a point $0.6\,\text{m}$ above the floor (see diagram). A is released and the particles begin to move.

May/June 2013

Particle $A$ has mass $1.26\,\text{kg}$ and particle $B$ has mass $0.9\,\text{kg}$. They are joined to the two ends of a light inextensible string. This string runs over a small smooth pulley $P$, which is fixed at the edge of a rough horizontal table. $A$ is initially held at rest $0.48\,\text{m}$ from $P$, while $B$ hangs vertically below $P$, with its height $0.45\,\text{m}$ above the floor (see diagram). The coefficient of friction between $A$ and the table is $\frac{2}{7}$. $A$ is released and the particles begin to move.

May/June 2013

Particles $A$ with mass $0.25\,\text{kg}$ and $B$ with mass $0.75\,\text{kg}$ are fixed to the two ends of a light inextensible string that runs over a fixed smooth pulley. The arrangement is kept at rest with the string taut and its straight sections vertical. Each particle is initially at a height of $h\,\text{m}$ above the floor. The system is released from rest, and $0.6\,\text{s}$ later, while both particles are moving, the string snaps. In the later motion, particle $A$ does not reach the pulley.

May/June 2014

A particle $P$ with mass $0.2\text{ kg}$ is released from rest at a point $7.2\text{ m}$ above the liquid surface in a container. $P$ drops through the air and then enters the liquid. Air resistance is absent and there is no sudden change in speed as $P$ enters the liquid. When $P$ is $0.8\text{ m}$ below the surface of the liquid, $P$’s speed is $6\,\text{m s}^{-1}$. The only force on $P$ from the liquid is a constant resistance to motion of magnitude $R\text{ N}$. The liquid has depth $3.6\text{ m}$ in the container. $P$ is removed from the container and attached to one end of a light inextensible string. $P$ is placed at the bottom of the container and then pulled vertically upwards with constant acceleration. The resistance to motion of $R\text{ N}$ continues to act. The particle reaches the surface $4\text{ s}$ after leaving the bottom of the container.

May/June 2014

A light inextensible string, of length $5.28\text{ m}$, has particles $A$ and $B$ attached to its ends, with masses $0.25\text{ kg}$ and $0.75\text{ kg}$ respectively. A further particle $P$, of mass $0.5\text{ kg}$, is fixed at the string’s mid-point. Two smooth pulleys $P_1$ and $P_2$ are mounted at opposite ends of a rough horizontal table of length $4\text{ m}$ and height $1\text{ m}$. The string runs over $P_1$ and $P_2$, with particle $A$ kept at rest vertically under $P_1$, the string taut, and $B$ hanging freely below $P_2$. Particle $P$ is touching the table halfway between $P_1$ and $P_2$ (see diagram). The coefficient of friction between $P$ and the table is $0.4$. Particle $A$ is released and the system begins to move with constant acceleration of magnitude $a\,\text{m s}^{-2}$. The tension in the part $AP$ of the string is $T_A\text{ N}$ and the tension in the part $PB$ of the string is $T_B\text{ N}$.

May/June 2014

A smooth inclined plane with length $160\text{ cm}$ is set so that one end is $40\text{ cm}$ above the other end, and the lower end lies on horizontal ground. Particles $P$ and $Q$, with masses $0.76\text{ kg}$ and $0.49\text{ kg}$ respectively, are joined by the ends of a light inextensible string that passes over a small smooth pulley fixed at the top of the plane. Particle $P$ is initially held at rest on the same line of greatest slope as the pulley, while $Q$ hangs vertically below the pulley at a height of $30\text{ cm}$ above the ground (see diagram). $P$ is released from rest. It moves up the plane and does not reach the pulley.

May/June 2014

Particles $A$ and $B$, with masses $0.3\,\text{kg}$ and $0.7\,\text{kg}$ respectively, are joined to the two ends of a light inextensible string. Particle $A$ is initially at rest on a rough horizontal table, and the string passes over a smooth pulley fixed at the table edge. The coefficient of friction between $A$ and the table is $0.2$. Particle $B$ hangs vertically below the pulley, with its height $0.5\,\text{m}$ above the floor (see diagram). The system is released from rest, and $0.25\,\text{s}$ later the string breaks. In the later motion, $A$ does not reach the pulley.

May/June 2015

Particles $P$ and $Q$ have masses $m\,\text{kg}$ and $(1-m)\,\text{kg}$ respectively. They are connected to the ends of a light inextensible string that passes over a smooth fixed pulley. At the start, $P$ is kept at rest with the string taut and the two straight sections of string vertical. Each particle is at a height of $h\,\text{m}$ above horizontal ground. When $P$ is released, $Q$ moves downwards. After that, $Q$ reaches the ground and stops. Fig. 2 gives the velocity-time graph for $P$ during the time that $Q$ is moving downwards.

May/June 2015

A lorry with mass $12\,000\,\text{kg}$ travels up a straight hill that is $500\,\text{m}$ long, beginning at the foot of the hill with speed $24\,\text{m s}^{-1}$ and arriving at the top with speed $16\,\text{m s}^{-1}$. The summit of the hill is $25\,\text{m}$ higher than the level at the bottom. The resistance to motion of the lorry is $7500\,\text{N}$.

May/June 2015

Two particles, with masses $5\,\text{kg}$ and $10\,\text{kg}$, are joined by a light inextensible string passing over a fixed smooth pulley. The $5\,\text{kg}$ particle lies on a rough fixed slope inclined at an angle of $\alpha$ to the horizontal, where $\tan\alpha = \frac{3}{4}$. The $10\,\text{kg}$ particle hangs beneath the pulley (see diagram). The coefficient of friction between the slope and the $5\,\text{kg}$ particle is $\frac{1}{2}$. The particles are released from rest.

May/June 2016

A particle of mass $30\,\text{kg}$ rests on a plane that is inclined at $20^\circ$ to the horizontal. It is initially at rest and is then pulled up the plane by a force of magnitude $200\,\text{N}$ acting parallel to the line of greatest slope.

May/June 2016

Particle $A$, with mass $1.6\,\text{kg}$, is at rest on a horizontal table and is linked to one end of a light inextensible string. This string runs over a small smooth pulley $P$ at the table’s edge. The free end is connected to particle $B$, of mass $2.4\,\text{kg}$, which hangs vertically beneath the pulley. The system is let go from rest, with the string taut and $B$ positioned $0.5\,\text{m}$ above the ground, as the diagram shows. During the resulting motion, $A$ does not reach $P$ before $B$ hits the ground.

May/June 2016

Two particles with masses $1.3\,\text{kg}$ and $0.7\,\text{kg}$ are joined by a light inextensible string passing over a fixed smooth pulley. The particles are initially at the same vertical level, with the string taut. Each particle is $2\,\text{m}$ above a horizontal plane, and each is $4\,\text{m}$ beneath the pulley. The particles are then released from rest.

May/June 2016

The diagram depicts a fixed block with a horizontal upper face and another face that is inclined at an angle of $\theta^\circ$ to the horizontal, where $\sin\theta = \frac{3}{5}$. Particle $A$ of mass $0.3\,\text{kg}$ is at rest on the horizontal face and is connected to one end of a light inextensible string. The string runs over a small smooth pulley $P$ fixed at the block’s edge. The other end is joined to particle $B$ of mass $1.5\,\text{kg}$, which is resting on the sloping face of the block. The system is released from rest with the string taut.

May/June 2017

The diagram shows a particle $A$ of mass $1.6\,\text{kg}$ on a horizontal plane, and a particle $B$ of mass $2.4\,\text{kg}$ on a plane inclined at an angle of $30^\circ$ to the horizontal. The particles are joined by a light inextensible string that goes over a small smooth pulley $P$ fixed at the top of the inclined plane. The distance $AP$ is $2.5\,\text{m}$ and the distance of $B$ from the bottom of the inclined plane is $1\,\text{m}$. A barrier at the bottom of the inclined plane stops any further movement of $B$. The section $BP$ of the string is parallel to a line of greatest slope of the inclined plane. The particles are released from rest, with both sections of the string taut.

May/June 2018

Particles $A$ and $B$, with masses $0.4\,\text{kg}$ and $0.2\,\text{kg}$ respectively, are joined by a light inextensible string. Particle $A$ is supported on a smooth plane inclined at an angle of $\theta^\circ$ to the horizontal. The string goes over a small smooth pulley $P$ fixed at the top of the plane, and $B$ hangs freely $0.5\,\text{m}$ above horizontal ground (see diagram). The particles are released from rest with both parts of the string taut.

May/June 2019

Particles $A$ and $B$, with masses $0.4\,\text{kg}$ and $0.2\,\text{kg}$ respectively, are linked by a light inextensible string that goes over a fixed smooth pulley. Each particle is $0.5\,\text{m}$ above the ground. They hang vertically, as shown in the diagram. The particles are released from rest. In the motion that follows, $B$ does not reach the pulley and $A$ comes to rest once it reaches the ground.

May/June 2019

Particles $A$ and $B$, having masses $1.3\,\text{kg}$ and $0.7\,\text{kg}$ respectively, are linked by a light inextensible string passing over a smooth fixed pulley. Particle $A$ is positioned $1.75\,\text{m}$ above the floor, while particle $B$ is $1\,\text{m}$ above the floor (see diagram). The system is let go from rest with the string taut, and the particles travel vertically. When the particles are level with each other the string breaks.

May/June 2019

Masses $3m$ kg and $2m$ kg, named $A$ and $B$, are fixed to the two ends of a light inextensible string. The string runs over a fixed smooth pulley that is attached to the edge of a plane. The plane makes an angle $\theta$ with the horizontal. $A$ is on the plane while $B$ hangs vertically, $0.8$ m above the floor. The section of string from $A$ to the pulley is parallel to a line of greatest slope on the plane (see diagram). At the beginning, both $A$ and $B$ are at rest.

May/June 2020

Particles $P$ and $Q$, with masses $0.5\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are joined by a light inextensible string. The string is taut, with $P$ positioned vertically above $Q$. A force of magnitude $10\,\text{N}$ acts on $P$ vertically upwards.

May/June 2022

At point $O$ on a straight horizontal test track, racing cars $A$ and $B$ are stationary side by side. The mass of $A$ is $1200\text{ kg}$. A’s engine exerts a constant driving force of $4500\text{ N}$. By the time $A$ reaches point $P$, its speed is $25\text{ m s}^{-1}$. The distance $OP$ is $d\text{ m}$. From $O$ to $P$, the work done against the resistance force acting on $A$ is $75\,000\text{ J}$.

May/June 2022

Particles $A$ and $B$, whose masses are $2.4\,\text{kg}$ and $1.2\,\text{kg}$ respectively, are joined by a light inextensible string that passes over a fixed smooth pulley. $A$ is initially at a height of $2.1\,\text{m}$ above a horizontal plane and $B$ is $1.5\,\text{m}$ above the plane. The particles hang vertically and are released from rest. In the motion that follows, $A$ arrives at the plane and does not rebound, and $B$ does not reach the pulley.

May/June 2022

Particles $P$ and $Q$, with masses $0.3\text{ kg}$ and $0.2\text{ kg}$ respectively, are joined by a light inextensible string. The string goes over a fixed smooth pulley at $B$, which is fitted between two inclined planes. $P$ is on the smooth plane $AB$, inclined at $60^\circ$ to the horizontal. $Q$ is on the plane $BC$, inclined at $30^\circ$ to the horizontal. The string is taut, and the particles are able to move along the lines of greatest slope of the two planes (see diagram).

May/June 2022

Particles $P$ and $Q$, with masses $0.2\,\text{kg}$ and $0.1\,\text{kg}$ respectively, are connected to the two ends of a light inextensible string. This string goes over a fixed smooth pulley $B$ that is attached to two inclined planes. Particle $P$ is situated on a smooth plane $AB$ inclined at $60^{\circ}$ to the horizontal. Particle $Q$ is situated on a plane $BC$ inclined at an angle of $\theta^{\circ}$ to the horizontal. The string is taut, and the particles can travel along the lines of greatest slope of the planes (see diagram).

May/June 2023

An elevator is drawn upward by a cable. It speeds up vertically at $0.4\,\text{m s}^{-2}$ for $5\,\text{s}$, then continues at a steady speed for $25\,\text{s}$. After that, it slows down at $0.2\,\text{m s}^{-2}$ until it is stationary.

May/June 2023

A car with mass $1700\,\text{kg}$ is towing a trailer with mass $300\,\text{kg}$ on a level straight road. A light inextensible cable, parallel to the road, joins the car to the trailer. The constant resistive forces opposing motion are $400\,\text{N}$ on the car and $150\,\text{N}$ on the trailer. The car’s engine has power $14\,000\,\text{W}$.

May/June 2024

A van with mass $4500\,\text{kg}$ is pulling a trailer with mass $750\,\text{kg}$ along a straight hill that slopes at angle $\theta$ to the horizontal, where $\sin \theta = 0.05$. The van and trailer are joined by a light rigid tow-bar parallel to the road. Constant resistive forces act on the van and the trailer, with $2500\,\text{N}$ on the van and $300\,\text{N}$ on the trailer.

May/June 2024

A light inextensible string has one end attached to particle $A$ of mass $3\,\text{kg}$, while the other end is connected to particle $B$ of mass $4\,\text{kg}$. Particle $A$ is on a rough plane inclined at $30^\circ$ to the horizontal, and particle $B$ is on a smooth horizontal plane. A second light inextensible string is joined to $B$; its other end is attached to particle $C$ of mass $5\,\text{kg}$, which hangs vertically. Both strings are taut and pass over small smooth pulleys fixed at the ends of the horizontal plane. The segment of string from $A$ to the pulley is parallel to a line of greatest slope of the inclined plane, and $A$, $B$ and $C$ all lie in the same vertical plane (see diagram). The system is released from rest. In the motion that follows, $C$ moves vertically downwards with acceleration $2\,\text{m s}^{-2}$, and neither $A$ nor $B$ reaches a pulley.

May/June 2025

A van with mass $2500\,\text{kg}$, moving at speed $v\,\text{m s}^{-1}$, is acted on by a resistance force of $kv^2\,\text{N}$. The engine of the van delivers a constant power of $62.5\,\text{kW}$.

May/June 2025

A racing cyclist, with a combined mass together with his cycle of $75\,\text{kg}$, produces power at a rate of $720\,\text{W}$ while travelling along a straight horizontal road. The resistive force opposing the cyclist’s motion is constant and has magnitude $R\,\text{N}$.

Oct/Nov 2011

Particles $A$ and $B$, with masses $0.9\,\text{kg}$ and $0.6\,\text{kg}$ respectively, are joined by the ends of a light inextensible string. The string passes over a fixed smooth pulley. The system is let go from rest while the string is taut, the two straight sections are vertical, and the particles are at the same level above the horizontal floor. In the motion that follows, $B$ does not reach the pulley.

Oct/Nov 2011

Two particles, $A$ and $B$, with masses $0.3\text{ kg}$ and $0.2\text{ kg}$ respectively, are fastened to the two ends of a light inextensible string. $A$ is kept at rest on a rough horizontal table, and the string passes over a small smooth pulley at the table edge. $B$ hangs vertically beneath the pulley. The system is released and $B$ begins to move downwards with acceleration $1.6\text{ m s}^{-2}$.

Oct/Nov 2012

Masses $A$ and $B$, with masses $m\,\text{kg}$ and $(1 - m)\,\text{kg}$ respectively, are joined to the two ends of a light inextensible string that runs over a fixed smooth pulley. The system is let go from rest, with the straight sections of the string initially vertical. $A$ moves vertically downwards, and $0.3\,\text{s}$ afterwards its speed is $0.6\,\text{m s}^{-1}$.

Oct/Nov 2012

$A$, $B$ and $C$ are three points along the line of greatest slope on a plane inclined at $\theta^\circ$ to the horizontal, with $A$ above $B$ and $B$ above $C$. The section between $A$ and $B$ is smooth, while the section between $B$ and $C$ is rough. A particle $P$ is released from rest at $A$ and slides down the line $ABC$. $0.8\,\text{s}$ after leaving $A$, the particle passes through $B$ with speed $4\,\text{m s}^{-1}$.

Oct/Nov 2012

Particles A and B have masses $0.32\,\text{kg}$ and $0.48\,\text{kg}$ respectively. They are joined by a light inextensible string that passes over a small smooth pulley fixed at the edge of a smooth horizontal table. Particle B is initially at rest on the table, $1.4\,\text{m}$ from the pulley. A is hanging vertically beneath the pulley, with its height $0.98\,\text{m}$ above the floor (see diagram). A, B, the string and the pulley all lie in the same vertical plane. B is released and A moves downwards.

Oct/Nov 2012

Particles $P$ and $Q$ move along a straight line on a rough horizontal plane, and friction is the only horizontal force acting on the particles.

Oct/Nov 2013

Particles $A$ and $B$, with masses of $0.3\,\mathrm{kg}$ and $0.7\,\mathrm{kg}$ respectively, are fixed to the two ends of a light inextensible string. This string runs over a fixed smooth pulley. $A$ is initially held at rest and $B$ is hanging freely, while the two vertical sections of the string are straight and both particles are $0.52\,\mathrm{m}$ above the floor (see diagram). $A$ is then released, so both particles begin to move.

Oct/Nov 2013

A cable pulls an elevator straight upward. The velocity-time graph for this motion is shown above.

Oct/Nov 2013

A car with mass 800 kg is travelling along a straight horizontal road, and its engine is doing work at a rate of 22.5 kW.

Oct/Nov 2014

A light inextensible string has a small block $B$ of mass $0.25\,\text{kg}$ fixed at its midpoint. Particles $P$ and $Q$, with masses $0.2\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are attached to the two ends. The string runs over two smooth pulleys mounted at opposite sides of a rough table, and $B$ is in limiting equilibrium on the table between the pulleys, with particles $P$ and $Q$ and block $B$ all lying in the same vertical plane (see diagram).

Oct/Nov 2014

Particles $P$ and $Q$ together have a total mass of $1\text{ kg}$. They are connected by a light inextensible string that runs over a smooth fixed pulley. $P$ is kept at rest while $Q$ hangs freely, and both vertical sections of the string are straight. At the start, both particles are at a height of $h\text{ m}$ above the floor (see Fig. 1). $P$ is released from rest and the particles begin to move with the string taut. Fig. 2 gives the velocity-time graphs for the motion of $P$ and for the motion of $Q$, with vertically upwards taken as the positive direction for velocity.

Oct/Nov 2014

A smooth inclined plane of length $2.5$ m has one end on the horizontal floor, while the other end is raised to a height of $0.7$ m above the floor. Particles $P$ and $Q$, with masses $0.5$ kg and $0.1$ kg respectively, are connected to the ends of a light inextensible string that passes over a small smooth pulley fixed at the top of the plane. Particle $Q$ is kept at rest on the floor directly below the pulley. The string is taut and $P$ is at rest on the plane (see diagram). $Q$ is released and starts to move vertically upwards towards the pulley, while $P$ moves down the plane.

Oct/Nov 2015

A small ring of mass $0.024$ kg is threaded onto a rough horizontal rod that is fixed in place. A light inextensible string is attached to the ring, and it is pulled by a force of magnitude $0.195$ N making an angle $\theta$ with the horizontal, where $\sin\theta = \frac{5}{13}$. When $\theta$ is below the horizontal (see Fig. 1), the ring is in limiting equilibrium.

Oct/Nov 2015

Particles $A$ and $B$, with masses $0.35\,\text{kg}$ and $0.15\,\text{kg}$ respectively, are fixed to the ends of a light inextensible string that runs over a fixed smooth pulley. The system is initially at rest, with $B$ resting on the horizontal floor, the string taut and its straight sections vertical. $A$ is $1.6\,\text{m}$ above the floor (see diagram). $B$ is then released and the system starts to move; $B$ does not reach the pulley.

Oct/Nov 2015

Particles $P$ and $Q$, with masses $0.6\,\text{kg}$ and $0.4\,\text{kg}$ respectively, are joined by a light inextensible string. This string runs over a small smooth light pulley that is fixed at the end of a smooth horizontal table. At first, $P$ is held stationary on the table while $Q$ hangs vertically (see diagram). $P$ is then released.

Oct/Nov 2016

A particle with mass $0.1\,\text{kg}$ is released from rest on a rough plane that is inclined at $20^{\circ}$ to the horizontal. It is stated that, $5$ seconds after release, the particle is moving at $2\,\text{m s}^{-1}$.

Oct/Nov 2016

Particles $P$ and $Q$, whose masses are $7\,\text{kg}$ and $3\,\text{kg}$ respectively, are secured to the two ends of a light inextensible string. The string goes over two small smooth pulleys fixed at the two ends of a horizontal table. Both particles hang vertically beneath the pulleys. Initially, both particles are at rest, $0.5\,\text{m}$ below table level and $0.4\,\text{m}$ above the horizontal floor (see diagram).

Oct/Nov 2016

A particle with mass $0.2\,\text{kg}$ is in equilibrium at rest on a rough plane that is tilted at $20^\circ$ to the horizontal.

Oct/Nov 2017

Particles $P$ and $Q$, each of mass $m\,\text{kg}$, are fastened to the two ends of a light inextensible string. That string passes over a fixed smooth pulley attached to the edge of a rough plane. The plane makes an angle $\alpha$ with the horizontal, where $\tan \alpha = \frac{7}{24}$. Particle $P$ is on the plane and particle $Q$ hangs vertically, as shown in the diagram. The part of the string between $P$ and the pulley is parallel to a line of greatest slope on the plane. The system is in limiting equilibrium.

Oct/Nov 2017

Particles $A$ and $B$, with masses $m\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are fastened to the two ends of a light inextensible string. The string goes over a fixed smooth pulley, and the particles hang freely beneath it. The system is let go from rest, with both particles $0.8\,\text{m}$ above horizontal ground. Particle $A$ reaches the ground at a speed of $0.6\,\text{m s}^{-1}$.

Oct/Nov 2018

A particle is launched from point P with initial speed $u\,\text{m s}^{-1}$ up the line of greatest slope $PQR$ on a rough inclined plane. The distances $PQ$ and $QR$ are each $0.8\,\text{m}$. The particle takes $0.6\,\text{s}$ to move from $P$ to $Q$ and $1\,\text{s}$ to move from $Q$ to $R$.

Oct/Nov 2018

Two particles $P$ and $Q$, with masses $0.3\text{ kg}$ and $0.5\text{ kg}$ respectively, are fastened to the two ends of a light inextensible string. The string goes over a fixed smooth pulley, and the particles hang freely beneath it (see diagram). $Q$ is kept stationary with the string taut at a height of $h\text{ m}$ above a horizontal floor (see diagram). $Q$ is then let go, and both particles begin to move. The pulley is high enough for $P$ never to reach it at any stage. The time for $Q$ to reach the floor is $0.6\text{ s}$.

Oct/Nov 2018

Blocks $A$ and $B$, with masses $4\,\text{kg}$ and $5\,\text{kg}$ respectively, are connected by a light inextensible string. They are at rest on a smooth plane inclined at angle $\alpha$ to the horizontal, where $\tan \alpha = \frac{7}{24}$. The string lies parallel to a line of greatest slope of the plane, with $B$ positioned above $A$. A force of magnitude $36\,\text{N}$ acts on $B$, parallel to a line of greatest slope of the plane (see diagram).

Oct/Nov 2019

Particles $P$ and $Q$, with masses $0.3\,\text{kg}$ and $0.2\,\text{kg}$ respectively, are fastened to the two ends of a light inextensible string. The string runs over a fixed smooth pulley attached to the edge of a smooth plane. The plane is set at an angle $\theta$ to the horizontal, with $\sin\theta = \frac{3}{5}$. $P$ is on the plane, while $Q$ hangs vertically beneath the pulley at a height of $0.8\,\text{m}$ above the floor. The section of string between $P$ and the pulley lies parallel to a line of greatest slope of the plane. $P$ is let go from rest and $Q$ moves vertically downwards.

Oct/Nov 2019

Two particles $A$ and $B$ have masses $m$ kg and $km$ kg respectively, with $k > 1$. They are fastened to the two ends of a light inextensible string. The string passes over a fixed smooth pulley, and the particles hang vertically beneath it. At the start, both particles are $0.81$ m above horizontal ground (see diagram). The system is released from rest, and particle $B$ hits the ground $0.9$ s later. In the later motion, particle $A$ does not reach the pulley.

Oct/Nov 2019

A light inextensible string passes over a fixed smooth pulley and joins particles of masses $0.8\text{ kg}$ and $0.2\text{ kg}$. The system starts from rest, with each particle $0.5\text{ m}$ above a horizontal floor (see diagram). During the motion that follows, the $0.2\text{ kg}$ particle does not reach the pulley.

Oct/Nov 2020

The diagram depicts a particle of mass $5\text{ kg}$ resting on a rough horizontal table, with two light inextensible strings attached to it and passing over smooth pulleys fixed at the table’s edges. Particles of masses $4\text{ kg}$ and $6\text{ kg}$ hang freely from the two string ends. The particle of mass $6\text{ kg}$ is $0.5\text{ m}$ above the ground. The system is in limiting equilibrium.

Oct/Nov 2021

Particle $P$, with mass $0.4\text{ kg}$, is in limiting equilibrium on a plane that is inclined at $30^\circ$ to the horizontal.

Oct/Nov 2022

Block $A$ has mass $80\,\text{kg}$ and is joined by a light, inextensible rope to block $B$, which has mass $40\,\text{kg}$. The rope between the blocks is taut and runs parallel to the line of greatest slope of a plane that is inclined at $20^\circ$ to the horizontal. A force of magnitude $500\,\text{N}$ acting at an angle of $15^\circ$ above that same line of greatest slope is applied to $A$ (see diagram). The blocks move up the plane, there is a resistance force of $50\,\text{N}$ on $B$, and there is no resistance force on $A$.

Oct/Nov 2022

A railway engine with mass $120000\text{ kg}$ is hauling a coach with mass $60000\text{ kg}$ along a straight track that rises at an angle $\alpha$ to the horizontal, where $\sin \alpha = 0.02$. The engine and coach are linked by a light rigid coupling, parallel to the track. The engine supplies a driving force of $125000\text{ N}$, and the constant resistances to motion are $22000\text{ N}$ on the engine and $13000\text{ N}$ on the coach.

Oct/Nov 2023

A block of mass $8\,\text{kg}$ moves down a rough plane inclined at $30^\circ$ to the horizontal, and it begins from rest. The coefficient of friction between the block and the plane is $\mu$. The block accelerates uniformly down the plane at $2.4\,\text{m s}^{-2}$.

Oct/Nov 2023

A light inextensible string passes over a fixed smooth pulley and links two particles of masses $1.8\,\text{kg}$ and $1.2\,\text{kg}$. The particles are hanging vertically. The system is released from rest.

Oct/Nov 2024

A car of mass $1200\,\text{kg}$ is moving at speed $v\,\text{m s}^{-1}$, and the resistive force has magnitude $kv\,\text{N}$. The engine’s maximum power is $92.16\,\text{kW}$. The car is moving on a straight, horizontal road.

Oct/Nov 2024

Particles $A$ and $B$, with masses $0.2\,\text{kg}$ and $0.3\,\text{kg}$ respectively, are attached to the two ends of a light inextensible string. That string runs over a small fixed smooth pulley fixed to the lower end of a rough plane inclined at an angle $\theta$ to the horizontal, where $\sin\theta = 0.6$. Particle $A$ is on the plane, while particle $B$ hangs vertically below the pulley and is $0.25\,\text{m}$ above horizontal ground. The part of the string from $A$ to the pulley is parallel to a line of greatest slope of the plane (see diagram). The coefficient of friction between $A$ and the plane is $1.125$. Particle $A$ is released from rest.

Oct/Nov 2024

A particle with mass 12 kg is to be dragged over a rough horizontal surface by a light inextensible string. The string makes an angle of $30^\circ$ to the plane, and its tension is $T$ N (see diagram). The coefficient of friction between the particle and the plane is 0.5.

Oct/Nov 2024

Particles $A$ and $B$, whose masses are $3\,\text{kg}$ and $5\,\text{kg}$ respectively, are joined by a light inextensible string that runs over a fixed smooth pulley. The string is taut, with each straight section vertical, and the particles are initially held so that $A$ is $1\,\text{m}$ above a horizontal plane while $B$ is $2\,\text{m}$ above the plane (see diagram). They are then released from rest. In the motion that follows, $A$ does not reach the pulley, and after $B$ touches the plane it remains in contact with the plane.

Oct/Nov 2024

A car with mass 900 kg is travelling along a straight horizontal road, and it experiences a constant resistive force of 350 N. At the moment when its speed is $15\text{ m s}^{-1}$, its acceleration is $0.25\text{ m s}^{-2}$.

Oct/Nov 2025

A block $A$ with mass $2\text{ kg}$ and a particle $B$ with mass $0.5\text{ kg}$ are joined by a light inextensible string that is tilted at $15^\circ$ to the horizontal. They are dragged over a horizontal surface with acceleration $1.2\text{ m s}^{-2}$ by a force of magnitude $6\text{ N}$ applied to $A$, which acts at $20^\circ$ above the horizontal, as illustrated in the diagram. The string and the force on $A$ lie in the same vertical plane. The contact between $B$ and the surface is smooth, whereas the contact between $A$ and the surface is rough.

Oct/Nov 2025

A railway locomotive with mass $240\,000\text{ kg}$ is pulling a coach of mass $36\,000\text{ kg}$ down a slope whose angle to the horizontal is $\sin^{-1} 0.04$. The locomotive provides a driving force of $450\,000\text{ N}$, while the resistive forces are $120\,000\text{ N}$ on the locomotive and $15\,000\text{ N}$ on the coach. The coupling linking the locomotive and the coach is light, rigid and parallel to the hill.

Oct/Nov 2025

Particles $A$ and $B$, with masses $2\text{ kg}$ and $6\text{ kg}$ respectively, are fastened to the two ends of a light inextensible string. The string goes over a smooth fixed pulley, and the particles hang vertically beneath the pulley. At the start, both particles are at rest at a height of $3.2\text{ m}$ above the horizontal ground level (see diagram). Particle $A$ is then projected vertically downwards with speed $1.2\text{ m s}^{-1}$.

Oct/Nov 2025

The sketch represents particle $P$ with mass $6\text{ kg}$ on a rough plane inclined at $30^\circ$ to the horizontal. Two light inextensible strings are fastened to $P$. Each string passes over a small smooth pulley fixed at an end of the plane. The non-vertical sections of the strings run parallel to a line of greatest slope of the plane. Particles $Q$ and $R$, with masses $5\text{ kg}$ and $2\text{ kg}$ respectively, hang vertically from the free ends of the strings. Both strings are taut, and the system is released from rest. It is stated that the tension in the string attached to $Q$ is twice the tension in the string attached to $R$.

Oct/Nov 2025