Probability

A uniform solid hemisphere, with weight $60\text{ N}$ and radius $0.8\text{ m}$, is in limiting equilibrium on a rough horizontal plane with its curved face in contact with the plane. The hemisphere’s axis of symmetry makes an angle $\theta$ with the horizontal, where $\cos\theta = 0.28$. A horizontal force of magnitude $P\text{ N}$ is applied at the lowest point of the circular rim to keep the hemisphere in equilibrium (see diagram).

Feb/March 2016

A particle $P$ with mass $0.6\text{ kg}$ is connected to one end of a light elastic string whose natural length is $0.8\text{ m}$ and whose modulus of elasticity is $24\text{ N}$. The opposite end of the string is fixed at point $A$, and $P$ hangs in equilibrium.

Feb/March 2016

One end of a light inextensible string is fixed to the top point $A$ of a solid sphere that is held in place, with centre $O$ and radius $0.6\text{ m}$. The string’s other end is joined to a particle $P$ of mass $0.2\text{ kg}$, which is in contact with the sphere’s smooth surface. The angle $AOP$ is $60^\circ$ (see diagram). The sphere applies a contact force of magnitude $R\text{ N}$ on $P$, and the tension in the string is $T\text{ N}$.

Feb/March 2016

A particle $P$ is projected at a speed of $20\,\text{m s}^{-1}$, making an angle of $60^\circ$ below the horizontal, from a point $O$ that is $30\,\text{m}$ above level ground.

Feb/March 2017

Particles P and Q have masses 0.4\,\text{kg} and m\,\text{kg} respectively. P is connected to the fixed point A by a light inextensible string of length 0.5\,\text{m}, which makes an angle of 60^\circ with the vertical. P and Q are linked by a light inextensible vertical string. Q is connected to the fixed point B, vertically below A, by a light inextensible string. BQ is taut and horizontal. The particles move in horizontal circles about an axis through A and B with constant angular speed \omega\,\text{rad s}^{-1} (see diagram). The tension in the string between P and Q is 1.5\,\text{N}.

Feb/March 2017

A light elastic string has natural length $0.6\text{ m}$ and modulus of elasticity $24\text{ N}$. One end is fixed at point $O$, and the other end is attached to a particle $P$ of mass $0.4\text{ kg}$, which is in equilibrium hanging vertically beneath $O$.

Feb/March 2017

A uniform rectangular block has a square base $ABCD$ with $AB = BC = 0.4\,\text{m}$. The block’s height is $h\,\text{m}$. It stands on a rough plane inclined at $30^\circ$ to the horizontal with its base in contact with the plane. The block does not slide. It is given that the block is on the point of toppling when the diagonal $AC$ lies along a line of greatest slope.

Feb/March 2018

A small ball $B$ is joined to one end of a light elastic string whose natural length is $0.4\,\text{m}$ and whose modulus of elasticity is $12\,\text{N}$. The opposite end of the string is fixed at point $A$. The ball is thrown vertically downwards at speed $1\,\text{m s}^{-1}$ from a point $0.4\,\text{m}$ vertically beneath $A$, and it attains its maximum speed at the position $0.7\,\text{m}$ below $A$.

Feb/March 2018

One end of a light inextensible string of length $0.4\,\text{m}$ is fixed to the lowest point of a hemisphere of radius $0.4\,\text{m}$, with the hemisphere’s axis vertical. A particle $P$ of mass $0.3\,\text{kg}$ is fastened to the other end of the string. The string is taut and is inclined at an angle of $30^\circ$ to the horizontal. $P$ travels on the smooth inner surface of the hemisphere in a horizontal circle (see diagram).

Feb/March 2018

A particle is projected at a speed of $24\,\text{m s}^{-1}$ at an angle of $30^\circ$ above the horizontal. Determine the speed and the direction of motion of the particle at the instant $4\,\text{s}$ after projection.

Feb/March 2019

A particle $P$ with mass $0.3\,\text{kg}$ is connected to a fixed point $A$ by a light elastic string whose natural length is $0.8\,\text{m}$ and modulus of elasticity is $16\,\text{N}$. The particle $P$ travels in a horizontal circle with centre $O$. It is given that $AO$ is vertical and that angle $OAP$ is $60^\circ$ (see diagram).

Feb/March 2019

A particle $P$ with mass $0.3\,\text{kg}$ is fastened to one end of a light elastic string whose natural length is $0.6\,\text{m}$ and modulus of elasticity is $24\,\text{N}$. The other end of the string is attached to a fixed point $O$. Particle $P$ is released from rest at the point $0.4\,\text{m}$ vertically below $O$.

Feb/March 2019

A particle $P$ is projected horizontally from point $O$ along a rough horizontal surface. The coefficient of friction between $P$ and the surface is $0.2$. A horizontal force of magnitude $0.06t$ N, acting away from $O$, is exerted on $P$, where $t$ is the time after projection. $P$ is at rest when $t = 4$.

Feb/March 2019

Box A has $7$ red balls and $1$ blue ball. Box B has $9$ red balls and $5$ blue balls. One ball is taken at random from box A and placed into box B. Then a ball is taken at random from box B. The tree diagram below shows the possible colours of the balls selected.

Feb/March 2020

Georgie owns a red scarf, a blue scarf and a yellow scarf. On each day, she chooses to wear exactly one of these scarves. The probabilities of selecting the three colours are $0.2$, $0.45$ and $0.35$ respectively. If she wears a red scarf, she always also wears a hat. If she wears a blue scarf, the probability that she wears a hat is $0.4$. If she wears a yellow scarf, the probability that she wears a hat is $0.3$.

Feb/March 2021

A factory makes chocolates in three flavours, lemon, orange and strawberry, in the ratio $3:5:7$ respectively. Nell checks the chocolates on the production line by selecting them at random one at a time.

Feb/March 2022

Let the chance that it rains on any particular day be $x$. If it is raining, the probability that Aran wears a hat is $0.8$, whereas if it is not raining, the probability that he wears a hat is $0.3$. No matter whether it is raining, if Aran is wearing a hat, the probability that he wears a scarf is $0.4$. If he is not wearing a hat, the probability that he wears a scarf is $0.1$. On a randomly selected day, the probability that it is not raining and Aran is wearing neither a hat nor a scarf is $0.36$.

Feb/March 2023

Marco has four boxes, labelled $K$, $L$, $M$ and $N$. He arranges them in a single row as $K$, $L$, $M$, $N$, with $K$ on the far left. Marco also has four coloured marbles: one red, one green, one white and one yellow. He puts one marble into each box, chosen at random. Events $A$ and $B$ are defined as follows: $A$: The white marble is in box $L$ or box $M$. $B$: The red marble lies to the left of both the green marble and the yellow marble.

Feb/March 2023

A bag holds 9 blue marbles and 3 red marbles. One marble is picked at random from the bag. If the marble picked is blue, it is placed back into the bag. If the marble picked is red, it is not placed back into the bag. A second marble is then picked at random from the bag.

Feb/March 2024

During the previous year, an online store sold a large number of computers. $55\%$ of these computers were made by company $F$, $30\%$ by company $G$ and $15\%$ by company $H$. A random sample of $3$ customers, each of whom bought a computer from this store, is selected.

Feb/March 2025

Eddie owns 16 toy cars: 8 are white, 5 are black and 3 are silver. He puts every car into a bag, then chooses three at random without replacement.

Feb/March 2025

A particle with mass $0.24\,\text{kg}$ is connected to one end of a light inextensible string of length $2\,\text{m}$. The string’s other end is fixed at a point. The particle travels at constant speed around a horizontal circle. The string is inclined at angle $\theta$ to the vertical (see diagram), and the tension in the string is $T\,\text{N}$. The particle has acceleration of magnitude $7.5\,\text{m s}^{-2}$.

May/June 2010

A lamina of uniform density and weight $15\,\text{N}$ is shaped as trapezium $ABCD$, with the measurements shown in the diagram. It is freely hinged at $A$ to a fixed point. One end of a light inextensible string is fastened to the lamina at $B$. The lamina is in equilibrium with $AB$ horizontal; the string is taut, lies in the same vertical plane as the lamina, and is inclined at an angle of $30^\circ$ above the horizontal (see diagram).

May/June 2010

A uniform solid cone of height $30\text{ cm}$ and base radius $r\text{ cm}$ stands with its axis vertical on a rough horizontal plane. The plane is then gradually tilted, and the cone stays in equilibrium until the plane’s angle of inclination reaches $35^\circ$, at which point the cone topples. The diagram shows a cross-section through the cone.

May/June 2010

A particle with mass $0.24\text{ kg}$ is fixed to one end of a light inextensible string of length $2\text{ m}$. The opposite end of the string is secured at a fixed point. The particle travels at constant speed in a horizontal circle. The string is inclined at an angle $\theta$ to the vertical (see diagram), and the tension in the string is $T\text{ N}$. The particle’s acceleration has magnitude $7.5\text{ m s}^{-2}$.

May/June 2010

A uniform lamina weighing $15\text{ N}$ is shaped as a trapezium $ABCD$, with the measurements shown in the diagram. The lamina is hinged freely at $A$ to a fixed point. One end of a light inextensible string is fastened to the lamina at $B$. The lamina is in equilibrium with $AB$ horizontal; the string is taut, lies in the same vertical plane as the lamina, and is inclined at an angle of $30^\circ$ above the horizontal (see diagram).

May/June 2010

A particle $P$ with mass $0.35\text{ kg}$ is fastened to the midpoint of a light elastic string whose natural length is $4\text{ m}$. The two ends of the string are fixed at points $A$ and $B$, which are $4.8\text{ m}$ apart and lie at the same horizontal height. In equilibrium, $P$ hangs at a point $0.7\text{ m}$ vertically beneath the midpoint $M$ of $AB$ (see diagram).

May/June 2010

Particles $P$ and $Q$ are launched together from point $O$ on a horizontal plane with speed $40\ \text{m s}^{-1}$. At different times later on, both particles pass through point $A$, whose horizontal and vertically upward displacements from $O$ are $40\ \text{m}$ and $15\ \text{m}$ respectively.

May/June 2010

$AB$ names the diameter of a uniform semicircular lamina with radius $0.3\ \text{m}$ and mass $0.4\ \text{kg}$. The lamina is hinged to a vertical wall at $A$, and $AB$ is inclined at $30^{\circ}$ to the vertical. One end of a light inextensible string is fixed to the lamina at $B$, and the other end is fixed to the wall vertically above $A$. The lamina is in equilibrium in a vertical plane perpendicular to the wall, with the string horizontal (see diagram).

May/June 2010

A light pair of equal-length inextensible strings fastens a small ball $B$ of mass $0.4\ \text{kg}$ to fixed points $P$ and $Q$ on a vertical axis. Each string is taut, and both make an angle of $30^{\circ}$ with the vertical. The ball moves in a horizontal circle (see diagram).

May/June 2010

A particle P of mass 0.5 kg travels in a straight line across a smooth horizontal plane. At time t s, P has displacement x m from a chosen point on the line, and its velocity is v m s^{-1}. It is given that when t = 0, x = 0 and v = 9. The motion of P is resisted by a force of magnitude 3√v N.

May/June 2010

One end of a light elastic string with natural length $3\ \text{m}$ and modulus of elasticity $24\ \text{N}$ is fixed at point $O$. A particle $P$ of mass $0.4\ \text{kg}$ is attached to the free end of the string. $P$ is projected vertically downwards from $O$ at an initial speed of $2\ \text{m s}^{-1}$. When the string has extension $x\ \text{m}$, the speed of $P$ is $v\ \text{m s}^{-1}$.

May/June 2010

A uniform triangular lamina has weight $19\,\text{N}$, with $AB = 0.22\,\text{m}$ and $AC = BC = 0.61\,\text{m}$. Its plane is vertical. Point $A$ is in contact with a rough horizontal plane, and $AB$ is vertical. The lamina is kept in equilibrium by a light elastic string of natural length $0.7\,\text{m}$, which passes over a small smooth peg $P$ and is fixed to $B$ and $C$. The part of the string joined to $B$ is horizontal, whereas the part joined to $C$ is vertical (see diagram).

May/June 2011

A particle $P$ with mass $0.5\,\text{kg}$ is connected to the vertex $V$ of a fixed solid cone by a light inextensible string. $P$ is on the smooth curved surface of the cone and travels in a horizontal circle of radius $0.1\,\text{m}$, with its centre on the cone’s axis. The cone’s semi-vertical angle is $60^\circ$ (see diagram).

May/June 2011

One end of a light elastic string whose natural length is $0.5\,\text{m}$ and whose modulus of elasticity is $12\,\text{N}$ is fastened to a fixed point $O$. The other end is joined to a particle $P$ of mass $0.24\,\text{kg}$. $P$ is projected vertically upwards with speed $3\,\text{m s}^{-1}$ from a point $0.8\,\text{m}$ vertically beneath $O$.

May/June 2011

A particle $P$ with mass $0.4\,\text{kg}$ is connected to a fixed point $A$ by a light inextensible string. The string makes an angle of $60^\circ$ to the vertical. $P$ travels at constant speed in a horizontal circle of radius $0.2\,\text{m}$. The centre of the circle lies vertically below $A$ (see diagram).

May/June 2011

The ends of a light elastic string of natural length $0.8\,\text{m}$ and modulus of elasticity $\lambda\,\text{N}$ are fastened to fixed points $A$ and $B$, which are $1.2\,\text{m}$ apart at the same horizontal level. A particle of mass $0.3\,\text{kg}$ is attached to the middle of the string, and released from rest at the midpoint of $AB$. The particle falls vertically by $0.32\,\text{m}$ before it comes to instantaneous rest.

May/June 2011

One end of a light elastic string, whose natural length is $0.3\,\text{m}$ and whose modulus of elasticity is $6\,\text{N}$, is fastened to a fixed point $O$ on a smooth horizontal plane. Its other end is attached to a particle $P$ of mass $0.2\,\text{kg}$, and $P$ travels on the plane in a circular path with centre $O$. The angular speed of $P$ is $\omega\,\text{rad s}^{-1}$.

May/June 2011

A sphere $S$ of mass $m\,\mathrm{kg}$ is inside a smooth hollow bowl with a vertical axis and a sloping face that makes an angle of $60^\circ$ to the horizontal. $S$ travels at constant speed round a horizontal circle of radius $0.6\,\mathrm{m}$ (see Fig. 1). $S$ touches both the flat base and the sloping side of the bowl (see Fig. 2).

May/June 2012

A light elastic string has a natural length of $2.4\,\mathrm{m}$ and an elasticity modulus of $21\,\mathrm{N}$. A particle $P$, with mass $m\,\mathrm{kg}$, is fixed to the midpoint of the string. The two ends of the string are fastened to fixed points $A$ and $B$, which are $2.4\,\mathrm{m}$ apart and lie at the same horizontal height. $P$ is projected vertically upward from the midpoint of $AB$ with speed $12\,\mathrm{m\,s^{-1}}$. In the ensuing motion, $P$ is momentarily at rest at a point $1.6\,\mathrm{m}$ above $AB$.

May/June 2012

A particle $P$ with mass $0.4\,\mathrm{kg}$ starts from rest at the top of a smooth plane that is tilted at $30^\circ$ to the horizontal. As $P$ moves down the slope, its motion is resisted by a force of magnitude $0.6x\,\mathrm{N}$, where $x\,\mathrm{m}$ denotes the distance $P$ has moved down the slope. Before it reaches the bottom of the slope, $P$ comes to rest.

May/June 2012

A uniform hemispherical shell of weight $8\,\text{N}$ and a uniform solid hemisphere of weight $12\,\text{N}$ are joined at their circumferences to make a non-uniform sphere of radius $0.2\,\text{m}$. The sphere is resting on a horizontal surface, with its axis of symmetry horizontal. It is kept in equilibrium by a force of magnitude $F\,\text{N}$ acting parallel to the axis of symmetry and applied at the highest point of the sphere.

May/June 2012

The natural length of a light elastic string is $2.2\,\text{m}$, and its modulus of elasticity is $14.3\,\text{N}$. A particle $P$ of mass $m\,\text{kg}$ is fixed to the midpoint of the string. The string ends are fixed at points $A$ and $B$, which are $2.4\,\text{m}$ apart and lie at the same horizontal level. $P$ is released from rest at the midpoint of $AB$. During the motion that follows, $P$ reaches its maximum speed at a point $0.5\,\text{m}$ below $AB$.

May/June 2012

Particles $P$ and $Q$, with masses $0.8\,\text{kg}$ and $0.5\,\text{kg}$ respectively, are fastened to the two ends of a light inextensible string that passes through a small hole in a smooth horizontal table of negligible thickness. $P$ travels on the upper surface of the table in a circular path with constant angular speed $6.25\,\text{rad s}^{-1}$.

May/June 2012

A small sphere $S$ has mass $m\,\text{kg}$ and travels inside a fixed smooth hollow cylinder with a vertical axis. $S$ moves at constant speed around a horizontal circle of radius $0.4\,\text{m}$, while remaining in contact with both the plane base and the curved surface of the cylinder (see diagram).

May/June 2012

A particle $P$ with mass $0.4\,\text{kg}$ is connected to one end of a light elastic string whose natural length is $1.2\,\text{m}$ and modulus of elasticity $19.2\,\text{N}$. The other end of the string is fixed at point $A$. Particle $P$ is released from rest at a point $2.7\,\text{m}$ vertically above $A$. Calculate

May/June 2013

A particle with mass $0.2\,\text{kg}$ is projected vertically downwards at an initial speed of $4\,\text{m s}^{-1}$. While it is descending, a resisting force of magnitude $0.09v\,\text{N}$ acts vertically upwards on the particle, where $v\,\text{m s}^{-1}$ represents the particle's downward velocity at time $t$ after it has been set in motion.

May/June 2013

Particle $P$ is launched from point $O$ with speed $50\,\text{m s}^{-1}$ at an angle of $40^\circ$ above the horizontal. At $2.5\,\text{s}$ after launch, calculate

May/June 2013

A light inextensible string of length $0.2\,\text{m}$ has one end fastened to a fixed point $A$ positioned above a smooth horizontal table. The other end carries a particle $P$ of mass $0.3\,\text{kg}$. $P$ travels in a horizontal circle on the table, with the string kept taut and inclined at an angle of $60^\circ$ to the downward vertical (see diagram).

May/June 2013

A sphere of mass $0.4\,\text{kg}$ travels at constant speed $1.5\,\text{m s}^{-1}$ round a horizontal circle inside a smooth fixed hollow cylinder with diameter $0.6\,\text{m}$. The cylinder’s axis is vertical, and the sphere touches both the horizontal base and the vertical curved surface of the cylinder.

May/June 2013

A particle $P$ with mass $0.2\,\text{kg}$ is fastened to one end of a light elastic string whose natural length is $1.6\,\text{m}$ and modulus of elasticity is $18\,\text{N}$. The opposite end of the string is fixed at point $O$, which lies $1.6\,\text{m}$ vertically above a smooth horizontal surface. $P$ is placed on the surface directly below $O$ and then projected horizontally. $P$ travels in a straight line along the surface with initial speed $1.5\,\text{m s}^{-1}$. Show that, when $OP = 1.8\,\text{m}$,

May/June 2013

A block $B$ with mass $3\,\text{kg}$ is fastened to one end of a light elastic string whose modulus of elasticity is $70\,\text{N}$ and whose natural length is $1.4\,\text{m}$. The opposite end of the string is connected to a particle $P$ of mass $0.3\,\text{kg}$. $B$ is stationary $0.9\,\text{m}$ from the edge of a horizontal table, and the string passes over a small frictionless pulley at the table edge. $P$ is released from rest beside the pulley and moves vertically downwards. At the first moment when $P$ is $0.8\,\text{m}$ below the pulley and still descending, $B$ is in limiting equilibrium with the part of the string attached to $B$ horizontal (see diagram).

May/June 2013

A particle $P$ with mass $0.5\,\text{kg}$ travels along a straight line on a smooth horizontal plane. When the displacement of $P$ from $O$ is $x\,\text{m}$, its velocity is $v\,\text{m s}^{-1}$. A lone horizontal force of size $0.16e^{x}\,\text{N}$ acts on $P$ in the direction $OP$. The speed of $P$ at $O$ is $0.8\,\text{m s}^{-1}$.

May/June 2013

A particle $P$ of mass $0.3\,\text{kg}$ is connected to one end of a light elastic string with natural length $0.6\,\text{m}$ and modulus of elasticity $45\,\text{N}$. The opposite end of the string is fixed at point $O$. The particle $P$ is released from rest at $O$ and drops vertically. Find the extension of the string when $P$ is at its lowest point.

May/June 2013

A smooth hollow cylinder with internal radius $0.3\,\text{m}$ stands fixed with its axis vertical. One end of a light inextensible string of length $0.5\,\text{m}$ is fastened to a point $A$ on the axis. The free end is attached to a particle $P$ of mass $0.2\,\text{kg}$, and $P$ moves in a horizontal circle on the cylinder’s surface (see diagram).

May/June 2013

A light elastic string $S_1$, with modulus of elasticity $20\,\text{N}$ and natural length $0.5\,\text{m}$, has one end fixed at point $O$. Its other end is fastened to a particle $P$ of mass $0.4\,\text{kg}$, and $P$ hangs in equilibrium directly beneath $O$.

May/June 2013

A small ball $B$ of mass $0.2\,\text{kg}$ travels through a narrow fixed smooth cylindrical tube $OA$ of length $1\,\text{m}$, which is sealed at $A$. If the ball is at displacement $x\,\text{m}$ from $O$, its velocity is $v\,\text{m s}^{-1}$ in the direction $OA$ and a resisting force of magnitude $\frac{k}{1-x}\,\text{N}$ acts on it.

May/June 2013

A particle is launched at a speed of $12\text{ m s}^{-1}$ from a point on level ground. It returns to the ground $1.6\text{ s}$ later.

May/June 2014

A non-uniform rod $AB$ has weight $6\text{ N}$ and is in limiting equilibrium, with end $A$ touching a rough vertical wall. The length $AB = 1.2\text{ m}$, the centre of mass of the rod is $0.8\text{ m}$ from $A$, and the angle between $AB$ and the downward vertical is $\theta^\circ$. At $B$, a force of magnitude $10\text{ N}$, acting at an angle of $30^\circ$ to the upwards vertical, is applied to the rod (see diagram). The rod and the line of action of the $10\text{ N}$ force both lie in a vertical plane perpendicular to the wall.

May/June 2014

A light elastic string, with natural length $0.8\text{ m}$ and modulus of elasticity $16\text{ N}$, is fixed at one end to the point $O$. A particle $P$ of mass $0.4\text{ kg}$ is attached to the free end, and it hangs in equilibrium directly beneath $O$.

May/June 2014

Particle $P$ is launched from point $O$ on horizontal ground with speed $20\text{ m s}^{-1}$ at an angle of $40^\circ$ above the horizontal.

May/June 2014

A particle $P$ with mass $0.6\text{ kg}$ is let go from rest at a point above ground level and moves vertically downward. Its motion is resisted by a force of magnitude $3v\text{ N}$, where $v\text{ m s}^{-1}$ denotes the speed of $P$. Just before $P$ arrives at the ground, $v = 1.95$.

May/June 2014

A bead $B$ of mass $m\text{ kg}$ travels with constant speed round a horizontal circle on a smooth fixed wire. The wire is a circle with centre $O$ and radius $0.4\text{ m}$. One end of a light elastic string, whose natural length is $0.4\text{ m}$ and modulus of elasticity is $42m\text{ N}$, is fastened to $B$. The other end of the string is fixed at point $A$, which is $0.3\text{ m}$ vertically above $O$ (see diagram).

May/June 2014

A particle is launched from a point on horizontal ground with speed $12\,\text{m s}^{-1}$. It reaches the ground again $1.6\,\text{s}$ later.

May/June 2014

An elastic string of negligible mass has natural length $0.8\,\text{m}$ and modulus of elasticity $16\,\text{N}$. One end is fixed at point $O$, while a particle $P$ of mass $0.4\,\text{kg}$ is attached to the opposite end. In equilibrium, $P$ hangs vertically beneath $O$.

May/June 2014

A particle $P$ is launched with speed $20\,\text{m s}^{-1}$ at an angle of $40^\circ$ above the horizontal from point $O$ on level ground.

May/June 2014

A particle $P$ of mass $0.6\,\text{kg}$ is let go from rest at a point above ground level and moves vertically downward. The motion of $P$ is resisted by a force of magnitude $3v\,\text{N}$, where $v\,\text{m s}^{-1}$ denotes the speed of $P$. Just before $P$ reaches the ground, $v = 1.95$.

May/June 2014

A bead $B$ of mass $m\,\text{kg}$ travels at constant speed around a horizontal circle on a fixed smooth wire. The wire is a circle of radius $0.4\,\text{m}$ with centre $O$. One end of a light elastic string, whose natural length is $0.4\,\text{m}$ and modulus of elasticity is $42m\,\text{N}$, is fixed to $B$. The other end is fastened to a fixed point $A$ that lies $0.3\,\text{m}$ vertically above $O$ (see diagram).

May/June 2014

A light elastic string has modulus of elasticity $5\,\text{N}$ and natural length $1.5\,\text{m}$. One end of the string is fixed at point $O$, and a particle $P$ of mass $0.1\,\text{kg}$ is fastened to the other end. $P$ is released from rest at a point $2.4\,\text{m}$ vertically below $O$.

May/June 2014

One end of a light inextensible string of length $2.4\,\text{m}$ is fixed at point $A$. Its other end is fastened to a particle $P$ of mass $0.2\,\text{kg}$. $P$ travels at constant speed around a horizontal circle whose centre is vertically beneath $A$, while the string remains taut and inclined at an angle of $60^\circ$ to the vertical.

May/June 2014

A particle $P$ with mass $0.6\,\text{kg}$ is fastened to one end of a light elastic string whose natural length is $1.5\,\text{m}$ and whose modulus of elasticity is $9\,\text{N}$. The string is threaded through a small smooth ring $R$, fixed $0.4\,\text{m}$ above a rough horizontal plane. The other end of the string is attached to a fixed point $O$ that lies $1.5\,\text{m}$ vertically above $R$. Points $A$ and $B$ lie on the horizontal surface, with $B$ directly below $R$. When $P$ is on the surface between $A$ and $B$, $RP$ makes an acute angle $\theta^\circ$ with the horizontal (see diagram).

May/June 2014

A particle $P$ of mass $0.3\,\text{kg}$ is connected to one end of a light elastic string of natural length $0.9\,\text{m}$ and modulus of elasticity $18\,\text{N}$. The other end of the string is secured to a fixed point $O$, which is $3\,\text{m}$ above the ground.

May/June 2015

A particle $P$ of mass $0.1\,\text{kg}$ travels in a straight line on a smooth horizontal surface, with its speed steadily reducing. A horizontal resisting force of magnitude $0.2e^{-x}\,\text{N}$ acts on $P$, where $x\,\text{m}$ is the displacement of $P$ from a fixed point $O$ on the line. If the displacement from $O$ is $x\,\text{m}$, the velocity of $P$ is $v\,\text{m s}^{-1}$. $P$ passes through $O$ with velocity $2.2\,\text{m s}^{-1}$.

May/June 2015

A particle $P$ with mass $0.6\,\text{kg}$ lies on the rough upper surface of a horizontal disc centred at $O$. The distance $OP$ is $0.4\,\text{m}$. The disc and $P$ rotate at angular speed $3\,\text{rad s}^{-1}$ about a vertical axis passing through $O$.

May/June 2015

A light elastic string has natural length $0.5\,\text{m}$ and modulus of elasticity $30\,\text{N}$. One end is fixed at point $O$. The other end is attached to a particle $P$, which hangs in equilibrium vertically beneath $O$, with $OP = 0.8\,\text{m}$.

May/June 2015

A uniform solid cube with edges of length $0.4\,\text{m}$ is in equilibrium on a rough plane that is inclined at $30\degree$ to the horizontal. $ABCD$ shows a cross-section through the cube’s centre of mass, with $AB$ lying along a line of greatest slope. $B$ lies below $A$. One end of a light elastic string, of modulus of elasticity $12\,\text{N}$ and natural length $0.4\,\text{m}$, is fixed to $C$. The other end is fixed to a point on the same line of greatest slope below $B$, so that the string makes an angle of $30\degree$ with the plane (see diagram). The cube is about to topple.

May/June 2015

A force with magnitude $0.4t\,\text{N}$, acting at an angle of $30\degree$ above the horizontal, is applied to a particle $P$, where $t\,\text{s}$ is the time elapsed since the force begins to act. $P$ is stationary on rough horizontal ground when $t = 0$. The mass of $P$ is $0.2\,\text{kg}$ and the coefficient of friction between $P$ and the ground is $\mu$.

May/June 2015

A light elastic string with natural length $0.4\text{ m}$ and modulus of elasticity $20\text{ N}$ has one end fixed at point $A$ on a smooth plane inclined at $30^\circ$ to the horizontal. Its other end is fastened to a particle $P$ of mass $0.5\text{ kg}$, which is initially in equilibrium on the plane. From that equilibrium position, $P$ is projected down the plane with speed $5\text{ m s}^{-1}$. When the particle is moving at $2\text{ m s}^{-1}$ for the first time, the string has extension $e\text{ m}$.

May/June 2015

Particle P, with mass $0.7\text{ kg}$, is connected to one end of a light inextensible string of length $0.5\text{ m}$. The other end is fixed at point $A$, which lies $h\text{ m}$ above a smooth horizontal surface. $P$ remains in contact with the surface and undergoes uniform circular motion about the point on the surface directly below $A$.

May/June 2015

A particle $P$ with mass $0.4\text{ kg}$ is released from rest at point $O$ on a smooth plane inclined at $30^{\circ}$ to the horizontal. If the displacement of $P$ from $O$ is $x\text{ m}$ down the plane, its velocity is $v\text{ m s}^{-1}$. A force of magnitude $0.8\mathrm{e}^{-x}\text{ N}$ acts on $P$ up the plane along the line of greatest slope through $O$.

May/June 2016

A particle is launched at an angle of $\theta^{\circ}$ below the horizontal from the top of a vertical cliff that is $26\text{ m}$ high. It lands on horizontal ground $8\text{ m}$ from the foot of the cliff $2\text{ s}$ after it is projected.

May/June 2016

A light inextensible string is threaded through a small smooth bead $B$ of mass $0.4\text{ kg}$. One end of the string is fastened to a fixed point $A$ $0.4\text{ m}$ above a fixed point $O$ on a smooth horizontal surface, and the other end is fastened to a fixed point $C$ which lies vertically below $A$ and $0.3\text{ m}$ above the surface. The bead moves at constant speed on the surface in a circle of centre $O$ and radius $0.3\text{ m}$ (see diagram).

May/June 2016

A particle $P$ is fastened to one end of a light elastic string whose natural length is $1.2\text{ m}$ and modulus of elasticity is $12\text{ N}$. The other end of the string is fixed at point $O$ on a smooth plane inclined at $30^{\circ}$ to the horizontal. $P$ is in equilibrium on the plane, $1.6\text{ m}$ from $O$.

May/June 2016

A light elastic string with natural length $0.4\,\text{m}$ has one end fastened to the fixed point $O$. Its other end is attached to a particle of weight $5\,\text{N}$, and the particle is in equilibrium $0.6\,\text{m}$ vertically below $O$.

May/June 2016

A uniform solid cone has a base radius of $0.4\,\text{m}$ and a height of $4.4\,\text{m}$. A uniform solid cylinder has radius $0.4\,\text{m}$ and a weight equal to that of the cone. The cylinder is fixed to the cone so that the cone’s base and one circular face of the cylinder touch, with their circumferences matching exactly. The resulting object is in equilibrium with its circular base resting on a plane that is inclined at $20^\circ$ to the horizontal (see diagram).

May/June 2016

A particle is launched from the top of a vertical cliff of height $26\,\text{m}$ at an angle of $\theta^\circ$ below the horizontal. It lands on horizontal ground $8\,\text{m}$ from the base of the cliff $2\,\text{s}$ after being projected.

May/June 2016

A light inextensible string is threaded through a small smooth bead $B$ of mass $0.4\,\text{kg}$. One end is fastened to a fixed point $A$ $0.4\,\text{m}$ above a fixed point $O$ on a smooth horizontal surface. The other end is fastened to a fixed point $C$, which is vertically below $A$ and $0.3\,\text{m}$ above the surface. The bead moves with constant speed on the surface in a circle centred at $O$ with radius $0.3\,\text{m}$ (see diagram).

May/June 2016

A particle $P$ is connected to one end of a light elastic string whose natural length is $1.2\,\text{m}$ and modulus of elasticity is $12\,\text{N}$. The opposite end of the string is fixed at point $O$ on a smooth plane that is inclined at $30^\circ$ to the horizontal. In equilibrium, $P$ lies on the plane $1.6\,\text{m}$ from $O$.

May/June 2016

A particle is projected at speed $20\text{ m s}^{-1}$ at an angle of $60^\circ$ above the horizontal. Calculate the time after projection when the particle is descending at an angle of $40^\circ$ below the horizontal.

May/June 2017

A particle $P$ with mass $0.15\,\text{kg}$ is attached to one end of a light elastic string whose natural length is $0.4\,\text{m}$ and modulus of elasticity is $12\,\text{N}$. The other end of the string is fixed at point $A$. Particle $P$ travels in a horizontal circle with its centre directly below $A$, while the string makes an angle of $\theta^\circ$ with the vertical and $AP = 0.5\,\text{m}$.

May/June 2017

A particle $P$ of mass $0.2\,\mathrm{kg}$ travels with speed $4\,\mathrm{m\,s^{-1}}$ and angular speed $5\,\mathrm{rad\,s^{-1}}$ in a horizontal circle on a smooth surface. $P$ is connected to one end of a light elastic string of natural length $0.6\,\mathrm{m}$. The other end of the string is fixed to the point on the surface that is the centre of the circular motion of $P$.

May/June 2017

The two ends of a pair of light inextensible strings, each of length $0.7\,\mathrm{m}$, are fixed to a particle $P$. Their other ends are fastened to two fixed points $A$ and $B$, which are in the same vertical line with $A$ higher than $B$. The particle $P$ undergoes horizontal circular motion with its centre at the midpoint of $AB$. Both strings make an angle of $60^\circ$ with the vertical. The tension in the string attached to $A$ is $6\,\mathrm{N}$ and the tension in the string attached to $B$ is $4\,\mathrm{N}$ (see diagram).

May/June 2017

An object of mass $0.4\,\text{kg}$ is dropped from rest from a point $8\,\text{m}$ above the ground. It falls vertically, and when its downward displacement from the starting position is $x\,\text{m}$ the speed is $v\,\text{m s}^{-1}$. As the object moves, a force of magnitude $0.2v^2\,\text{N}$ acts opposite to the motion.

May/June 2017

A particle of mass $0.3\,\text{kg}$ is fastened to one end of a light elastic string with natural length $0.8\,\text{m}$ and modulus of elasticity $6\,\text{N}$. The other end of the string is fixed at point $O$. The particle is projected vertically downwards from $O$ with initial speed $2\,\text{m s}^{-1}$.

May/June 2017

At rest on a rough horizontal plane is the end $A$ of a non-uniform rod $AB$ with length $0.6\,\text{m}$ and weight $8\,\text{N}$, where $AB$ is set at $60^\circ$ to the horizontal. The rod is kept in equilibrium by a force of magnitude $3\,\text{N}$ applied at $B$. This force acts $30^\circ$ above the horizontal in the vertical plane containing the rod (see diagram).

May/June 2017

A particle is projected at speed $20\,\text{m s}^{-1}$ at an angle of $60^\circ$ above the horizontal. Calculate the time after projection when the particle is descending at an angle of $40^\circ$ below the horizontal.

May/June 2017

Particle $P$, with mass $0.7\,\text{kg}$, is connected by a light elastic string to a fixed point $O$ on a smooth plane inclined at $30^{\circ}$ to the horizontal. The string has natural length $0.5\,\text{m}$ and modulus of elasticity $20\,\text{N}$. The particle $P$ is projected up the line of greatest slope through $O$ from a point $A$ below the level of $O$. The initial kinetic energy of $P$ is $1.8\,\text{J}$ and the initial elastic potential energy in the string is also $1.8\,\text{J$.

May/June 2018

Particle $P$ has mass $0.2\,\text{kg}$ and is fastened to one end of a light inextensible string of length $0.6\,\text{m}$. At the other end, the string is joined to a particle $Q$ of mass $0.3\,\text{kg}$. It then passes through a small hole $H$ in a smooth horizontal surface. A light elastic string with natural length $0.3\,\text{m}$ and modulus of elasticity $15\,\text{N}$ connects $Q$ to a fixed point $A$, which lies $0.4\,\text{m}$ vertically below $H$. Particle $P$ travels on the surface in a horizontal circle centred at $H$ (see diagram).

May/June 2018

A small ball $B$ is launched from point $O$ on level ground in the direction of point $A$, which is $12\,\text{m}$ above the ground. After $0.9\,\text{s}$, $B$ has moved $20\,\text{m}$ horizontally and lies directly beneath $A$ (see diagram).

May/June 2018

A particle $P$ with mass $0.2\,\text{kg}$ is let go from rest at a point $O$ above horizontal ground. At time $t\,\text{s}$ after release, the downward velocity of $P$ is $v\,\text{m s}^{-1}$. A force of magnitude $0.6t\,\text{N}$ acts vertically downwards on $P$. A vertically upwards force of magnitude $ke^{-t}\,\text{N}$, where $k$ is a constant, also acts on $P$.

May/June 2018