Forces and equilibrium

Three coplanar forces, with magnitudes $50\,\text{N}$, $40\,\text{N}$ and $30\,\text{N}$, are applied at point $O$ in the directions shown in the diagram, where $\tan\alpha = \frac{7}{24}$.

Feb/March 2016

Particle $P$, with mass $1.6\,\text{kg}$, hangs in equilibrium from two light inextensible strings fixed at points $A$ and $B$. These strings are inclined at $20^\circ$ and $40^\circ$ respectively to the horizontal (see diagram).

Feb/March 2017

The three coplanar forces illustrated in the diagram are balanced.

Feb/March 2018

A particle with mass $12\,\text{kg}$ rests on a rough plane inclined at an angle of $25^\circ$ to the horizontal. A force of magnitude $P\,\text{N}$ is applied to the particle. This force is horizontal, and the particle is on the point of moving up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is $0.8$.

Feb/March 2018

A ring $P$ of mass $0.03\,\text{kg}$ is fitted on a rough vertical rod. A light inextensible string is fastened to the ring and pulled upwards at an angle of $15^\circ$ to the horizontal. The tension in the string is $2.5\,\text{N}$ (see diagram). The ring is in limiting equilibrium and is just about to slide up the rod.

Feb/March 2019

Four coplanar forces with magnitudes $F\,\text{N}$, $5\,\text{N}$, $25\,\text{N}$ and $15\,\text{N}$ act at point $P$ in the directions shown in the diagram. Since the forces are in equilibrium,

Feb/March 2019

Coplanar forces with magnitudes $F\ \text{N}$, $3\ \text{N}$, $6\ \text{N}$ and $4\ \text{N}$ are applied at point $P$, as indicated in the diagram.

Feb/March 2020

Particle $Q$ has mass $0.2\,\text{kg}$ and is kept in equilibrium by two light inextensible strings $PQ$ and $QR$. $P$ is a fixed point on a vertical wall, while $R$ is a fixed point on a horizontal floor. The angles that strings $PQ$ and $QR$ make with the horizontal are $60^\circ$ and $30^\circ$ respectively (see diagram).

Feb/March 2021

A block with mass $5\text{ kg}$ is dragged across a rough level floor by a force of magnitude $X\text{ N}$ acting at $30^\circ$ above the horizontal (see diagram). It starts from rest and covers $2\text{ m}$ in the first $5\text{ s}$ of its motion.

Feb/March 2021

Four forces acting at one point lie in the same plane. Their magnitudes are $10\,\text{N}$, $F\,\text{N}$, $G\,\text{N}$ and $2F\,\text{N}$. The directions of the forces are shown in the diagram.

Feb/March 2022

The diagram depicts a block $D$ of mass $100\text{ kg}$ held up by two inclined struts $AD$ and $BD$, each fixed at an angle of $45^\circ$ to points $A$ and $B$ respectively on a horizontal floor. The block is additionally secured by a vertical rope $CD$ attached to a fixed point $C$ on a horizontal ceiling. The tension in rope $CD$ is $500\text{ N}$, and the block is in equilibrium.

Feb/March 2023

A $600\,\text{kg}$ crate is pulled up the line of greatest slope on a rough plane, using a rope fixed to a winch, at a steady speed of $2\,\text{m s}^{-1}$. The plane is inclined at $30^\circ$ to the horizontal, and the rope is parallel to the plane. The winch operates at a steady rate of $8\,\text{kW}$.

Feb/March 2024

At one point, four coplanar forces act. Their magnitudes are $F\text{ N}$, $2F\text{ N}$, $3F\text{ N}$ and $30\text{ N}$. Their directions are illustrated in the diagram.

Feb/March 2024

At a point, three coplanar forces with magnitudes $40\,\text{N}$, $30\,\text{N}$ and $X\,\text{N}$ act in the directions indicated in the diagram.

Feb/March 2025

A block of mass $12\,\text{kg}$ rests on a rough plane inclined at an angle $\alpha$ to the horizontal, where $\alpha = \tan^{-1}(0.5)$. A force of $X\,\text{N}$ acts on the block, directed straight up the plane. The coefficient of friction between the block and the plane is $\mu$.

Feb/March 2025

A ring of mass $0.8 \text{ kg}$, small in size, is placed on a rough rod fixed horizontally. The ring is in equilibrium and experiences a force of magnitude $7 \text{ N}$ acting upward at $45^{\circ}$ to the horizontal.

May/June 2010

Three coplanar forces with magnitudes $250 \text{ N}$, $160 \text{ N}$ and $370 \text{ N}$ are applied at a point $O$ in the directions indicated in the diagram, with angle $\alpha$ defined by $\sin \alpha = 0.28$ and $\cos \alpha = 0.96$.

May/June 2010

A rough horizontal rod carries a small ring of mass $0.8\,\text{kg}$, and the ring is held in equilibrium while a force of magnitude $7\,\text{N}$ acts on it upwards at $45^\circ$ to the horizontal.

May/June 2010

Three coplanar forces with magnitudes $250\,\text{N}$, $160\,\text{N}$ and $370\,\text{N}$ act at point $O$ in the directions shown in the diagram, where the angle $\alpha$ satisfies $\sin\alpha = 0.28$ and $\cos\alpha = 0.96$.

May/June 2010

Three forces acting in one plane are applied at a single point. Their magnitudes are $5.5\,\text{N}$, $6.8\,\text{N}$ and $7.3\,\text{N}$, and the directions in which they act are shown in the diagram.

May/June 2010

A small smooth ring $R$ with weight $8.5\,\text{N}$ is carried by a light inextensible string passing through it. The two string ends are fixed at points $A$ and $B$, with $A$ vertically above $B$. A horizontal force of magnitude $15.5\,\text{N}$ acts on $R$, and the ring is in equilibrium with angle $ARB = 90^\circ$. The segment $AR$ of the string is inclined at angle $\theta$ to the horizontal, while $BR$ is inclined at angle $\theta$ to the vertical (see diagram). The tension in the string is $T\,\text{N}$.

May/June 2011

A block with mass $11\,\text{kg}$ is initially stationary on a rough plane that is inclined at $30^\circ$ to the horizontal. A force is applied to the block up the plane, along a line of greatest slope. When the force has magnitude $2X\,\text{N}$, the block is just about to move down the plane, and when the force has magnitude $9X\,\text{N}$, the block is just about to move up the plane.

May/June 2011

The velocity-time graph shown represents a parachutist’s vertical descent. The motion has four phases: free fall with the parachute shut; slowing down at a steady rate with the parachute open; descending at constant speed with the parachute open; and stopping immediately on reaching the ground.

May/June 2011

The three coplanar forces in the diagram act at a point $P$ and are balanced.

May/June 2011

A small smooth ring $R$, with mass $0.6\,\text{kg}$, is placed on a light inextensible string of length $100\,\text{cm}$. One end of the string is fixed at point $A$. A small bead $B$ of mass $0.4\,\text{kg}$ is attached to the other end of the string, and it is threaded on a fixed rough horizontal rod that passes through $A$. The system is in equilibrium, with $B$ located $80\,\text{cm}$ from $A$.

May/June 2011

A car with mass $1250\,\text{kg}$ moves on a straight horizontal road while its engine delivers power at a steady rate of $P\,\text{W}$. The resistive force opposing the car’s motion remains constant and has magnitude $R\,\text{N}$. At a speed of $19\,\text{m s}^{-1}$, its acceleration is $0.6\,\text{m s}^{-2}$, and at a speed of $30\,\text{m s}^{-1}$, its acceleration is $0.16\,\text{m s}^{-2}$.

May/June 2011

A small block of mass $1.25\,\text{kg}$ rests on a horizontal surface. Three horizontal forces, with the magnitudes and directions shown in the diagram, act on the block. The angle $\theta$ is such that $\cos\theta = 0.28$ and $\sin\theta = 0.96$. A horizontal frictional force also acts on the block, and the block is in equilibrium.

May/June 2011

At point $O$, forces of magnitudes $13\,\text{N}$ and $14\,\text{N}$ act in the directions indicated in the diagram. Their resultant has magnitude $15\,\text{N}$.

May/June 2012

A small ring of mass $0.2\,\text{kg}$ is placed on a fixed vertical rod. The end $A$ of a light inextensible string is fastened to the ring. The other end $C$ of the string is fixed to a point on the rod above $A$. A horizontal force of magnitude $8\,\text{N}$ acts at the point $B$ of the string, where $AB = 1.5\,\text{m}$ and $BC = 2\,\text{m}$. The system is in equilibrium, the string is taut and $AB$ is at right angles to $BC$ (see diagram).

May/June 2012

A block is drawn across level ground in a straight path by a force of fixed magnitude that acts at an angle of $60^{\circ}$ above the horizontal. The work done by the force in moving the block through $5\,\text{m}$ is $75\,\text{J}$.

May/June 2012

At point $P$, three coplanar forces with magnitudes $F\,\text{N}$, $12\,\text{N}$ and $15\,\text{N}$ act in equilibrium in the directions indicated in the diagram.

May/June 2012

A ring of mass $4\,\text{kg}$ is fastened to one end of a light string. The ring is threaded onto a fixed horizontal rod, and the string is pulled at an angle of $25^{\circ}$ below the horizontal (see diagram). When the tension in the string is $T\,\text{N}$, the ring is in equilibrium. The coefficient of friction between the ring and the rod is $0.4$.

May/June 2012

A smooth ring $R$ of mass $0.16\,\text{kg}$ is passed onto a light inextensible string. The string ends are fixed at points $A$ and $B$. A horizontal force of magnitude $11.2\,\text{N}$ acts on $R$, in the same vertical plane as $A$ and $B$. The ring is in equilibrium. The string is taut, with angle $ARB = 90^{\circ}$, and the segment $AR$ of the string is inclined at an angle of $\theta^{\circ}$ to the horizontal (see diagram). The tension in the string is $T\,\text{N}$.

May/June 2012

A block weighing $6.1\,\text{N}$ is stationary on a plane inclined at angle $\alpha$ to the horizontal, with $\tan \alpha = \frac{11}{60}$. The coefficient of friction between the block and the plane is $\mu$. A force of magnitude $5.9\,\text{N}$, applied parallel to the line of greatest slope, acts on the block.

May/June 2012

A particle $P$ with mass $2.1\,\text{kg}$ is connected to one end of each of two light inextensible strings. Their other ends are fixed to points A and B, which lie on the same horizontal line. $P$ is suspended in equilibrium $40\,\text{cm}$ beneath the level of A and B, and the strings $PA$ and $PB$ are $50\,\text{cm}$ and $104\,\text{cm}$ long respectively (see diagram).

May/June 2013

A car has mass $800\,\text{kg}$. Its engine supplies constant power $P\,\text{kW}$ as the car travels along a straight horizontal road. The resistive force opposing the motion is constant and equal to $R\,\text{N}$. At a speed of $14\,\text{m s}^{-1}$, the car’s acceleration is $1.4\,\text{m s}^{-2}$, and at a speed of $25\,\text{m s}^{-1}$, the acceleration is $0.33\,\text{m s}^{-2}$.

May/June 2013

A box of small size with mass $40\,\text{kg}$ is pulled across a rough horizontal floor by three men. Two of them exert horizontal forces of magnitudes $100\,\text{N}$ and $120\,\text{N}$, directed at angles of $30^\circ$ and $60^\circ$ respectively to the positive $x$-direction. The third man exerts a horizontal force of magnitude $F\,\text{N}$ at an angle of $\alpha^\circ$ to the negative $x$-direction (see diagram). The combined effect of the three horizontal forces on the box is a resultant in the positive $x$-direction with magnitude $136\,\text{N}$.

May/June 2013

A rough plane is set at an angle of $\alpha^{\circ}$ to the horizontal. A particle with mass $0.25\,\text{kg}$ remains in equilibrium on the plane. The normal reaction on the particle has magnitude $2.4\,\text{N}$.

May/June 2014

At one point, four coplanar forces are applied. Their magnitudes are $5\,\text{N}$, $4\,\text{N}$, $3\,\text{N}$ and $7\,\text{N}$, and the directions in which they act are indicated in the diagram.

May/June 2014

A and $B$ are two fixed points on a vertical wall, with $A$ directly above $B$. A particle $P$ of mass $0.7\text{ kg}$ is connected to $A$ by a light inextensible string of length $3\text{ m}$. $P$ is also connected to $B$ by a light inextensible string of length $2.5\text{ m}$. $P$ is kept in equilibrium at a distance $2.4\text{ m}$ from the wall by a horizontal force of magnitude $10\text{ N}$ acting on $P$ (see diagram). The two strings are taut, and the $10\text{ N}$ force acts in the plane $APB$, which is perpendicular to the wall.

May/June 2014

Block $B$, with mass $7\text{ kg}$, is initially at rest on rough horizontal ground. A force of size $X\text{ N}$ acts on $B$ at an angle of $15^\circ$ to the upward vertical (see diagram).

May/June 2014

A particle $P$ with weight $1.4\text{ N}$ is connected to one end of a light inextensible string $S_1$ of length $1.5\text{ m}$ and also to one end of another light inextensible string $S_2$ of length $1.3\text{ m}$. The opposite end of $S_1$ is fixed to a wall at a point $0.9\text{ m}$ vertically above a point $O$ on the wall. The opposite end of $S_2$ is fixed to the wall at a point $0.5\text{ m}$ vertically below $O$. The particle is maintained in equilibrium at the same horizontal level as $O$ by a horizontal force of magnitude $2.24\text{ N}$ acting away from the wall and perpendicular to it (see diagram).

May/June 2014

At point O, which is the origin of the x-axis and y-axis, three horizontal forces of magnitudes $F\,\text{N}$, $63\,\text{N}$ and $25\,\text{N}$ are acting. These forces are in equilibrium. The force with magnitude $F\,\text{N}$ is inclined at an angle $\theta$ anticlockwise to the positive $x$-axis. The force of magnitude $63\,\text{N}$ is directed along the negative $y$-axis. The force of magnitude $25\,\text{N}$ is directed at $\tan^{-1}0.75$ clockwise from the negative $x$-axis (see diagram).

May/June 2015

A small ring $R$ is fastened to one end of a light inextensible string of length $70\,\text{cm}$. The string passes through a fixed rough vertical wire. Its other end is tied to a point $A$ on the wire, vertically above $R$. A horizontal force of magnitude $5.6\,\text{N}$ acts at the point $J$ of the string, which is $30\,\text{cm}$ from $A$ and $40\,\text{cm}$ from $R$. The system is in equilibrium, with both parts $AJ$ and $JR$ taut and angle $AJR$ equal to $90^\circ$ (see diagram).

May/June 2015

A car with mass $860\,\text{kg}$ moves on a level straight road. The engine supplies power $P\,\text{W}$, while the resistive force opposing the motion is $R\,\text{N}$. At one point, the car has speed $4.5\,\text{m s}^{-1}$ and acceleration $4\,\text{m s}^{-2}$. At another point, its speed is $22.5\,\text{m s}^{-1}$ and its acceleration is $0.3\,\text{m s}^{-2}$.

May/June 2015

At a point, four coplanar forces with magnitudes $4\,\text{N}$, $8\,\text{N}$, $12\,\text{N}$ and $16\,\text{N}$ are applied. Their directions are illustrated in Fig. 1.

May/June 2015

A small box of mass $5\,\text{kg}$ is dragged at a steady speed of $2.5\,\text{m s}^{-1}$ down the line of greatest slope on a rough plane inclined at $10^{\circ}$ to the horizontal. The pulling force is $20\,\text{N}$ and acts downward, parallel to the plane’s line of greatest slope.

May/June 2015

Four coplanar forces with magnitudes $50\,\text{N}$, $48\,\text{N}$, $14\,\text{N}$ and $P\,\text{N}$ act at a point in the directions shown in the diagram. The system is in equilibrium. Given that $\tan\alpha = \frac{7}{24}$, determine the values of $P$ and $\theta$.

May/June 2016

Coplanar forces with magnitudes $7\,\text{N}$, $6\,\text{N}$ and $8\,\text{N}$ act at a point in the directions illustrated in the diagram. If $\sin \alpha = \frac{3}{5}$, find the magnitude and direction of the resultant of the three forces.

May/June 2016

A block of mass $2.5\,\text{kg}$ rests on a plane inclined at $30^\circ$ to the horizontal. A light string, which is $20^\circ$ above the line of greatest slope, keeps the block in equilibrium. The tension in the string is $T\,\text{N}$, as shown in the diagram. The coefficient of friction between the block and plane is $\frac{1}{4}$. The block is in limiting equilibrium and is on the point of moving up the plane.

May/June 2016

The coplanar forces in the diagram are balanced.

May/June 2016

A particle of mass $15\,\text{kg}$ rests stationary on a rough plane inclined at an angle of $20^\circ$ to the horizontal. The coefficient of friction between the particle and the plane is $0.2$. A force of magnitude $X\,\text{N}$, applied parallel to a line of greatest slope of the plane, is then used to keep the particle in equilibrium.

May/June 2016

A particle weighing $25\text{ N}$ has two light inextensible strings fixed to it. Each string goes over a smooth fixed pulley and then supports a hanging particle of weight $A\text{ N}$ or $B\text{ N}$ at the lower end. The parts of the strings that slope away make angles of $30^\circ$ and $40^\circ$ respectively with the vertical (see diagram). The system is in equilibrium.

May/June 2017

As the diagram shows, particle $A$ has mass $0.8\text{ kg}$ and rests on a plane inclined at $30^\circ$ to the horizontal, while particle $B$ has mass $1.2\text{ kg}$ and rests on a plane inclined at $60^\circ$ to the horizontal. The particles are linked by a light inextensible string that passes over a small smooth pulley $P$ fixed at the top of the planes. The segments $AP$ and $BP$ of the string are parallel to the lines of greatest slope of the two planes. Both particles are released from rest, with the string taut in both sections.

May/June 2017

The four coplanar forces shown in the diagram are balanced.

May/June 2017

Particles $A$ and $B$, with masses $m\text{ kg}$ and $4\text{ kg}$ respectively, are joined by a light inextensible string which runs over a fixed smooth pulley. $A$ rests on a rough fixed slope inclined at $30^{\circ}$ to the horizontal ground. $B$ is suspended vertically beneath the pulley and lies $0.5\text{ m}$ above the ground (see diagram). The coefficient of friction between the slope and $A$ is $0.2$.

May/June 2017

The diagram depicts three coplanar forces acting at point $O$. Their magnitudes are $6\,\text{N}$, $8\,\text{N}$ and $10\,\text{N}$. The angle formed by the $6\,\text{N}$ force and the $8\,\text{N}$ force is $90^\circ$. The forces are in equilibrium.

May/June 2018

A particle $P$ of mass $8\,\text{kg}$ lies on a smooth plane inclined at $30^\circ$ to the horizontal. A force of magnitude $100\,\text{N}$, acting in the vertical plane that contains the line of greatest slope and making an angle of $\theta^\circ$ with that line, acts on $P$ (see diagram).

May/June 2018

The diagram depicts a triangular block whose sloping faces make angles of $45^{\circ}$ and $30^{\circ}$ with the horizontal. Particle $A$, with mass $0.8\text{ kg}$, is on the face inclined at $45^{\circ}$, while particle $B$, with mass $1.2\text{ kg}$, is on the face inclined at $30^{\circ}$. The particles are joined by a light inextensible string that passes over a small smooth pulley $P$ fixed at the top of the faces. The segments $AP$ and $BP$ of the string run parallel to the lines of greatest slope on the respective faces. The particles are let go from rest with both parts of the string taut. During the ensuing motion, neither particle reaches the pulley and neither one reaches the bottom of a face.

May/June 2018

A man of mass $80\,\text{kg}$ runs along a horizontal road while a constant resistance force of magnitude $P\,\text{N}$ acts opposite his motion. The total work done by the man as he increases his speed from $4\,\text{m s}^{-1}$ to $5.5\,\text{m s}^{-1}$ over a distance of $60\,\text{m}$ is $1200\,\text{J}$.

May/June 2018

The diagram shows three coplanar forces with magnitudes $3\,\text{N}$, $2\,\text{N}$ and $P\,\text{N}$.

May/June 2018

A particle with mass $20\,\text{kg}$ rests on a rough plane inclined at an angle of $60^\circ$ to the horizontal. Equilibrium is produced by a force of magnitude $P\,\text{N}$ acting on the particle, parallel to a line of greatest slope on the plane. The maximum possible value of $P$ is twice the minimum possible value of $P$.

May/June 2018

At a point, coplanar forces with magnitudes $8\,\text{N}$, $12\,\text{N}$ and $18\,\text{N}$ act in the directions indicated in the diagram.

May/June 2018

A particle with mass $3\,\text{kg}$ lies on a rough plane that is tilted at an angle of $20^\circ$ to the horizontal.

May/June 2018

The diagram shows a system of coplanar forces acting through a single point. Their magnitudes are $78\,\text{N}$, $50\,\text{N}$ and $112\,\text{N}$, and the angles $\alpha$ and $\theta$ are measured to the horizontal as shown.

May/June 2019

A particle with mass $1.3\,\text{kg}$ is at rest on a rough plane inclined at angle $\theta$ to the horizontal, where $\tan\theta = \frac{12}{5}$. The coefficient of friction between the particle and the plane is $\mu$.

May/June 2019

Forces that are coplanar, with magnitudes $40\,\mathrm{N}$, $32\,\mathrm{N}$, $P\,\mathrm{N}$ and $17\,\mathrm{N}$, act at a point in the directions indicated in the diagram. The system is in a state of equilibrium.

May/June 2019

A car’s mass is $1000\,\text{kg}$. If it moves at a constant speed of $v\,\text{m s}^{-1}$, with $v>2$, the resistive force on the car is $(Av + B)\,\text{N}$, where $A$ and $B$ are constants. On a horizontal road, the car can maintain a steady speed of $18\,\text{m s}^{-1}$ when the engine output is $36\,\text{kW}$. It can also go up a hill making an angle of $\theta$ to the horizontal, with $\sin\theta = 0.05$, at a constant speed of $12\,\text{m s}^{-1}$ when the engine output is $21\,\text{kW}$.

May/June 2019

Three coplanar forces with magnitudes $12\,\text{N}$, $24\,\text{N}$ and $30\,\text{N}$ are applied at one point in the directions indicated in the diagram.

May/June 2019

At point $A$, three coplanar forces with magnitudes $100\,\text{N}$, $50\,\text{N}$ and $50\,\text{N}$ act, as shown in the diagram. The value of $\cos \alpha$ is $\frac{4}{5}$.

May/June 2020

The diagram depicts a ring of mass $0.1\,\text{kg}$ passed onto a fixed horizontal rod. The rod is rough, with coefficient of friction between the ring and the rod equal to $0.8$. A force of magnitude $T\,\text{N}$ acts on the ring at an angle of $30^\circ$ to the rod, directed downwards in the vertical plane that contains the rod. At the start, the ring is at rest.

May/June 2020

Four coplanar forces of magnitudes $20\text{ N}$, $P\text{ N}$, $3P\text{ N}$ and $4P\text{ N}$ act at a point in the directions shown in the diagram. The system remains in equilibrium.

May/June 2020

A particle with mass $2.5\text{ kg}$ is kept in equilibrium on a rough plane inclined at $20^\circ$ to the horizontal by a force of magnitude $T\text{ N}$ acting at an angle of $60^\circ$ to a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is $0.3$.

May/June 2020

The diagram shows four coplanar forces of magnitudes $40\,\text{N}$, $20\,\text{N}$, $50\,\text{N}$ and $F\,\text{N}$ acting at a point in the directions indicated. These four forces are in equilibrium.

May/June 2020

At point O, three coplanar forces with magnitudes 10 N, 25 N and 20 N act in the directions shown in the diagram.

May/June 2021

Three coplanar forces with magnitudes $34\,\text{N}$, $30\,\text{N}$ and $26\,\text{N}$ are applied at a point in the directions indicated in the diagram. Since $\sin \alpha = \frac{5}{13}$ and $\sin \theta = \frac{8}{17}$,

May/June 2021

Four forces lie in one plane and act at a single point. Their magnitudes are $20\,\text{N}$, $30\,\text{N}$, $40\,\text{N}$ and $F\,\text{N}$. Their directions are indicated in the diagram, with $\sin \alpha = 0.28$ and $\sin \beta = 0.6$.

May/June 2021

Particle $P$, with mass $0.3\,\text{kg}$, is stationary on a rough plane that makes an angle $\theta$ with the horizontal, where $\sin\theta = \frac{7}{25}$. A force of magnitude $4\,\text{N}$, applied horizontally in the vertical plane containing a line of greatest slope of the plane, acts on $P$ (see diagram). The particle is on the point of moving up the plane.

May/June 2021

A $300\,\text{kg}$ crate is motionless on rough horizontal ground. The coefficient of friction between the crate and the ground is $0.5$. A force of size $X\,\text{N}$, making an angle $\alpha$ above the horizontal, acts on the crate, with $\sin \alpha = 0.28$.

May/June 2022

Three coplanar forces with magnitudes $20\,\text{N}$, $100\,\text{N}$ and $F\,\text{N}$ act at a point. Their directions are indicated in the diagram.

May/June 2022

Forces with magnitudes $60\,\text{N}$, $20\,\text{N}$, $16\,\text{N}$ and $14\,\text{N}$ lie in the same plane and act at a point in the directions indicated in the diagram.

May/June 2022

A block with mass $12\,\text{kg}$ rests on a plane inclined at $24^{\circ}$ to the horizontal. It is connected to a light string that lies at an angle of $36^{\circ}$ above the line of greatest slope. The string tension is $65\,\text{N}$ (see diagram). The coefficient of friction between the block and the plane is $\mu$. The block is in limiting equilibrium and is just about to move up the plane.

May/June 2022

The diagram depicts a block of mass $10\,\text{kg}$ hanging under a horizontal ceiling from two strings, $AC$ and $BC$, whose lengths are $0.8\,\text{m}$ and $0.6\,\text{m}$ respectively; these strings are fixed to points on the ceiling. The angle $ACB = 90^\circ$. A horizontal force of magnitude $F\,\text{N}$ acts on the block, and the block is in equilibrium.

May/June 2022

Four coplanar forces act at a point, with magnitudes of $F\,\text{N}$, $10\,\text{N}$, $50\,\text{N}$ and $40\,\text{N}$. Their directions are indicated in the diagram.

May/June 2023

Forces of magnitudes $30\text{ N}$, $15\text{ N}$, $33\text{ N}$ and $P\text{ N}$ act together at a point in the directions shown in the diagram, where $\tan \alpha = \frac{4}{3}$. The forces are in equilibrium.

May/June 2023

A particle of mass $0.6\,\text{kg}$ rests on a rough plane inclined at $35^\circ$ to the horizontal. It is held in equilibrium by a horizontal force of magnitude $P\,\text{N}$ acting in a vertical plane that contains a line of greatest slope (see diagram). The coefficient of friction between the particle and the plane is $0.4$.

May/June 2023

A car with mass $1500\,\text{kg}$ is pulling a trailer of mass $m\,\text{kg}$ on a level straight road. The car and trailer are joined by a tow-bar that is horizontal, light and rigid. A resistance force of $F\,\text{N}$ acts on the car and a resistance force of $200\,\text{N}$ acts on the trailer. The engine of the car provides a driving force of $3200\,\text{N}$, the car accelerates at $1.25\,\text{m s}^{-2}$ and the tension in the tow-bar is $300\,\text{N}$.

May/June 2023

A smooth ring $R$ of mass $0.2\,\text{kg}$ is placed on a light string $ARB$. The two ends of the string are fixed at points $A$ and $B$, with $A$ vertically above $B$. The string is taut, and $ABR = 90^\circ$. The angle between the segment $AR$ of the string and the vertical is $60^\circ$. The ring is kept in equilibrium by a force of magnitude $X\,\text{N}$, acting on the ring in a direction perpendicular to $AR$ (see diagram).

May/June 2023

The forces, with magnitudes $20\,\text{N}$ and $F\,\text{N}$, act at point $P$ in the directions indicated in the diagram.

May/June 2024

Four coplanar forces with magnitudes $P\,\text{N}$, $10\,\text{N}$, $16\,\text{N}$ and $2\,\text{N}$ act at a point in the directions indicated in the diagram. The forces are in equilibrium.

May/June 2024

A particle with mass $0.8\,\text{kg}$ rests on a rough plane inclined at an angle of $28^\circ$ to the horizontal. It is maintained in equilibrium by a force of magnitude $T\,\text{N}$. This force acts at an angle of $35^\circ$ above the line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is $0.2$.

May/June 2024

A particle of mass $0.2\,\mathrm{kg}$ is fastened to one end of a light inextensible string. The opposite end of the string is fixed at a point on a vertical wall. The particle is held in equilibrium by a force of magnitude $X\,\mathrm{N}$, acting at right angles to the string, with the string taut and inclined at an angle of $30^\circ$ to the wall (see the diagram).

May/June 2024

A block with mass $12\,\text{kg}$ is pulled upward by a rope on a rough plane. The plane makes an angle of $20^\circ$ to the horizontal. The rope acts parallel to the line of greatest slope of the plane. The coefficient of friction between the block and the plane is $0.4$. The block’s acceleration is $2\,\text{m s}^{-2}$.

May/June 2025

Three coplanar forces with magnitudes $P\,\text{N}$, $5\,\text{N}$ and $10\,\text{N}$ act at a point $O$, as indicated in the diagram. Their resultant has magnitude $Q\,\text{N}$ and is directed perpendicular to the force of magnitude $P\,\text{N}$.

May/June 2025

A particle $P$ of mass $m\,\text{kg}$ is connected to one end of a light inextensible string of length $2.6\,\text{m}$. The opposite end of the string is fixed to a point on a horizontal ceiling, and the string is taut. The particle is maintained in equilibrium by a force of magnitude $35\,\text{N}$, acting in a vertical plane which is perpendicular to the ceiling and includes the string. This force acts perpendicular to the string (see diagram). The tension in the string is $T\,\text{N}$ and the vertical distance of $P$ below the ceiling is $2.4\,\text{m}$.

May/June 2025

A van with mass $4500\,\text{kg}$ is pulling a trailer whose mass is $350\,\text{kg}$ along a straight horizontal road. The van and trailer are joined by a light rigid tow-bar that is parallel to the road. Resistance forces of $X\,\text{N}$ act on the van and $120\,\text{N}$ act on the trailer. The driving force provided by the van’s engine is $2500\,\text{N}$. The tension in the tow-bar is $T\,\text{N}$, and the van’s acceleration is $0.4\,\text{m s}^{-2}$.

May/June 2025

Three blocks $P$, $Q$ and $R$, with masses $25\,\text{kg}$, $20\,\text{kg}$ and $m\,\text{kg}$ respectively, are held in equilibrium by three light inextensible strings $OP$, $OQ$ and $OR$. The strings $OP$ and $OR$ each pass over a small fixed smooth pulley, $A$ and $B$ respectively, and $P$ and $R$ hang vertically beneath the pulleys. Block $Q$ hangs vertically below point $O$. The angle between $OA$ and the vertical is $30^\circ$ and the angle $BOQ$ is $\alpha^\circ$ (see diagram).

May/June 2025

Three coplanar forces with magnitudes $17\,\text{N}$, $51\,\text{N}$ and $34\,\text{N}$ act at point $O$ in the directions indicated in the diagram, with $\tan \alpha = \frac{15}{8}$.

May/June 2025

The diagram depicts three particles $A$, $B$ and $C$ suspended in equilibrium, with each one attached to the end of a string. The other ends of the three strings are fastened together at the point $X$. The strings supporting $A$ and $C$ pass over smooth fixed horizontal pegs $P_1$ and $P_2$ respectively. The weights of $A$, $B$ and $C$ are $5.5\,\text{N}$, $7.3\,\text{N}$ and $W\,\text{N}$ respectively, and the angle $P_1XP_2$ is a right angle.

Oct/Nov 2010

Particles $P$ and $Q$, with masses $0.2\,\text{kg}$ and $0.5\,\text{kg}$ respectively, are joined by a light inextensible string. This string goes over a smooth pulley at the edge of a rough horizontal table. $P$ hangs freely, while $Q$ rests on the table. A force of magnitude $3.2\,\text{N}$ is applied to $Q$, directed upwards and away from the pulley, making an angle of $30^\circ$ to the horizontal.

Oct/Nov 2010