Questions - CAIE Further Paper 3

Find the value of $\sin \theta$.
Find, in terms of $m$ and $g$, the tension in the string when $P$ is at $A$.
\includegraphics[max width=\textwidth, alt={}, center]{454be64a-204f-4fa4-a5fc-72fd88e1289f-06_846_767_258_689} An object is formed from a solid hemisphere, of radius $2 a$, and a solid cylinder, of radius $a$ and height $d$. The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane face of the hemisphere. The line $O C$ forms a diameter of the base of the cylinder, where $C$ is the centre of the plane face of the hemisphere and $O$ is common to both circumferences (see diagram). Relative to axes through $O$, parallel and perpendicular to $O C$ as shown, the centre of mass of the object is ( $\mathrm { x } , \mathrm { y }$ ).

CAIE Further Paper 3 2023 June Q1

1 A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle $P$ is held at the point $A$, where $O A$ makes an angle $\alpha$ with the downward vertical through $O$, and with the string taut. The particle $P$ is projected perpendicular to $O A$ in an upwards direction with speed $\sqrt { 3 a g }$. It then starts to move along a circular path in a vertical plane. The string goes slack when $P$ is at $B$, where $O B$ makes an angle $\theta$ with the upward vertical. Given that $\cos \alpha = \frac { 4 } { 5 }$, find the value of $\cos \theta$.

CAIE Further Paper 3 2023 June Q2

2 One end of a light elastic string, of natural length $a$ and modulus of elasticity $\lambda \mathrm { mg }$, is attached to a fixed point $O$. The string lies on a smooth horizontal surface. A particle $P$ of mass $m$ is attached to the other end of the string. The particle $P$ is projected in the direction $O P$. When the length of the string is $\frac { 4 } { 3 } a$, the speed of $P$ is $\sqrt { 2 \mathrm { ag } }$. When the length of the string is $\frac { 5 } { 3 } a$, the speed of $P$ is $\frac { 1 } { 2 } \sqrt { 2 \mathrm { ag } }$. Find the value of $\lambda$.
\includegraphics[max width=\textwidth, alt={}, center]{8a62b72b-cffc-4d6d-b09a-8b0cb8f8eea3-04_654_502_267_817} A uniform lamina is in the form of a triangle $A B C$, with $A C = 8 a , B C = 6 a$ and angle $A C B = 90 ^ { \circ }$. The point $D$ on $A C$ is such that $A D = 3 a$. The point $E$ on $C B$ is such that $C E = x$ (see diagram). The triangle $C D E$ is removed from the lamina.

Find, in terms of $a$ and $x$, the distance of the centre of mass of the remaining object $A D E B$ from $A C$.
The object $A D E B$ is on the point of toppling about the point $E$ when the object is in the vertical plane with its edge $E B$ on a smooth horizontal surface.
Find $x$ in terms of $a$.
\includegraphics[max width=\textwidth, alt={}, center]{8a62b72b-cffc-4d6d-b09a-8b0cb8f8eea3-06_419_1160_274_488} Two identical smooth uniform spheres $A$ and $B$ each have mass $m$. The two spheres are moving on a smooth horizontal surface when they collide with speeds $u$ and $2 u$ respectively. Immediately before the collision, $A$ 's direction of motion makes an angle of $30 ^ { \circ }$ with the line of centres, and $B$ 's direction of motion is perpendicular to the line of centres (see diagram). After the collision, $A$ and $B$ are moving in the same direction. The coefficient of restitution between the spheres is $e$.

CAIE Further Paper 3 2023 June Q5

5 One end of a light elastic string, of natural length $12 a$ and modulus of elasticity $k m g$, is attached to a fixed point $O$. The other end of the string is attached to a particle of mass $m$. The particle moves with constant speed $\frac { 3 } { 2 } \sqrt { 3 a g }$ in a horizontal circle with centre at a distance $12 a$ below $O$. The string is inclined at an angle $\theta$ to the downward vertical through $O$.

Find, in terms of $a$, the extension of the string.
Find the value of $k$.

CAIE Further Paper 3 2023 June Q6

6 A particle of mass $m \mathrm {~kg}$ falls vertically under gravity, from rest. At time $t \mathrm {~s} , P$ has fallen $x \mathrm {~m}$ and has velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The only forces acting on $P$ are its weight and a resistance of magnitude $k m g v \mathrm {~N}$, where $k$ is a constant.

Find an expression for $v$ in terms of $t , g$ and $k$.
Given that $k = 0.05$, find, in metres, how far $P$ has fallen when its speed is $12 \mathrm {~ms} ^ { - 1 }$.

CAIE Further Paper 3 2023 June Q7

7 The points $O$ and $P$ are on a horizontal plane, a distance 8 m apart. A ball is thrown from $O$ with speed $u \mathrm {~ms} ^ { - 1 }$ at an angle $\theta$ above the horizontal, where $\tan \theta = \frac { 4 } { 3 }$. At the same instant, a model aircraft is launched with speed $5 \mathrm {~ms} ^ { - 1 }$ parallel to the horizontal plane from a point 4 m vertically above $P$. The model aircraft moves in the same vertical plane as the ball and in the same horizontal direction as the ball. The model aircraft moves horizontally with a constant speed of $5 \mathrm {~ms} ^ { - 1 }$. After $T \mathrm {~s}$, the ball and the model aircraft collide.

Find the value of $T$.
Find the direction in which the ball is moving immediately before the collision.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE Further Paper 3 2024 June Q2

2 The points $A$ and $B$ are at the same horizontal level a distance 4a apart. The ends of a light elastic string, of natural length $4 a$ and modulus of elasticity $\lambda$, are attached to $A$ and $B$. A particle $P$ of mass $m$ is attached to the midpoint of the string. The system is in equilibrium with $P$ at a distance $\frac { 3 } { 2 } a$ below $M$, the midpoint of $A B$.

Find $\lambda$ in terms of $m$ and $g$.
The particle $P$ is pulled down vertically and released from rest at a distance $\frac { 8 } { 3 } a$ below $M$.
Find, in terms of $a$ and $g$, the speed of $P$ as it passes through $M$ in the subsequent motion.

CAIE Further Paper 3 2024 June Q3

3 At time $t = 0$ seconds, a particle $P$ is projected with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $60 ^ { \circ }$ above the horizontal from a point $O$. In the subsequent motion $P$ moves freely under gravity. The direction of motion of $P$ when $t = 5$ is perpendicular to its direction of motion when $t = 15$. Find the value of $u$.
\includegraphics[max width=\textwidth, alt={}, center]{b57762bf-7a4f-486d-b9f2-8ae727bfb630-08_419_876_255_596} A ring of weight $W$, with radius $a$ and centre $O$, is at rest on a rough surface that is inclined to the horizontal at an angle $\alpha$ where $\tan \alpha = \frac { 1 } { 2 }$. The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point $P$ on the circumference of the ring is such that $O P$ is parallel to the surface. A light inextensible string is attached to $P$ and to the point $Q$, which is on the surface, such that $P Q$ is horizontal (see diagram). The points $O , P$ and $Q$ are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is $\mu$.

Find, in terms of $W$, the tension in the string $P Q$.
Find the value of $\mu$.

CAIE Further Paper 3 2024 June Q5

5 Two particles $A$ and $B$ of masses $m$ and $k m$ respectively are connected by a light inextensible string of length $a$. The particles are placed on a rough horizontal circular turntable with the string taut and lying along a radius of the turntable. Particle $A$ is at a distance $a$ from the centre of the turntable and particle $B$ is at a distance $2 a$ from the centre of the turntable. The coefficient of friction between each particle and the turntable is $\frac { 1 } { 5 }$.
When the turntable is made to rotate with angular speed $\frac { 2 } { 5 } \sqrt { \frac { g } { a } }$, the system is in limiting equilibrium.

Find the tension in the string, in terms of $m$ and $g$.
Find the value of $k$.

CAIE Further Paper 3 2024 June Q6

6 A particle $P$ of mass 2 kg moving on a horizontal straight line has displacement $x \mathrm {~m}$ from a fixed point $O$ on the line and velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t \mathrm {~s}$. The only horizontal force acting on $P$ has magnitude $\frac { 1 } { 10 } ( 2 \mathrm { v } - 1 ) ^ { 2 } \mathrm { e } ^ { - \mathrm { t } } \mathrm { N }$ and acts towards $O$. When $t = 0 , x = 1$ and $v = 3$.

Find an expression for $v$ in terms of $t$.
\includegraphics[max width=\textwidth, alt={}, center]{b57762bf-7a4f-486d-b9f2-8ae727bfb630-12_69_1569_466_328}
Find an expression for $x$ in terms of $t$.

CAIE Further Paper 3 2024 June Q7

7 A smooth sphere with centre $O$ and of radius $a$ is fixed to a horizontal plane. A particle $P$ of mass $m$ is projected horizontally from the highest point of the sphere with speed $u$, so that it begins to move along the surface of the sphere. The particle $P$ loses contact with the sphere at the point $Q$ on the sphere, where $O Q$ makes an angle $\theta$ with the upward vertical through $O$.

Show that $\cos \theta = \frac { \mathrm { u } ^ { 2 } + 2 \mathrm { ag } } { 3 \mathrm { ag } }$.
It is given that $\cos \theta = \frac { 5 } { 6 }$.
Find, in terms of $a$ and $g$, an expression for the vertical component of the velocity of $P$ just before it hits the horizontal plane to which the sphere is fixed.
Find an expression for the time taken by $P$ to fall from $Q$ to the plane. Give your answer in the form $\mathrm { k } \sqrt { \frac { \mathrm { a } } { \mathrm { g } } }$, stating the value of $k$ correct to 3 significant figures.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE Further Paper 3 2024 June Q3

3 At time $t = 0$ seconds, a particle $P$ is projected with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $60 ^ { \circ }$ above the horizontal from a point $O$. In the subsequent motion $P$ moves freely under gravity. The direction of motion of $P$ when $t = 5$ is perpendicular to its direction of motion when $t = 15$. Find the value of $u$.
\includegraphics[max width=\textwidth, alt={}, center]{c1a3340d-158d-4c37-9577-96074e59ac3d-08_419_876_255_596} A ring of weight $W$, with radius $a$ and centre $O$, is at rest on a rough surface that is inclined to the horizontal at an angle $\alpha$ where $\tan \alpha = \frac { 1 } { 2 }$. The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point $P$ on the circumference of the ring is such that $O P$ is parallel to the surface. A light inextensible string is attached to $P$ and to the point $Q$, which is on the surface, such that $P Q$ is horizontal (see diagram). The points $O , P$ and $Q$ are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is $\mu$.

Find, in terms of $W$, the tension in the string $P Q$.
Find the value of $\mu$.

CAIE Further Paper 3 2024 June Q5

5 Two particles $A$ and $B$ of masses $m$ and $k m$ respectively are connected by a light inextensible string of length $a$. The particles are placed on a rough horizontal circular turntable with the string taut and lying along a radius of the turntable. Particle $A$ is at a distance $a$ from the centre of the turntable and particle $B$ is at a distance $2 a$ from the centre of the turntable. The coefficient of friction between each particle and the turntable is $\frac { 1 } { 5 }$.
When the turntable is made to rotate with angular speed $\frac { 2 } { 5 } \sqrt { \frac { \mathrm {~g} } { \mathrm { a } } }$, the system is in limiting equilibrium.

Find the tension in the string, in terms of $m$ and $g$.
Find the value of $k$.

CAIE Further Paper 3 2024 June Q6

Find an expression for $v$ in terms of $t$.
\includegraphics[max width=\textwidth, alt={}, center]{c1a3340d-158d-4c37-9577-96074e59ac3d-12_69_1569_466_328}
Find an expression for $x$ in terms of $t$.

CAIE Further Paper 3 2024 June Q1

1
\includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-02_358_1019_287_523} Two smooth uniform spheres $A$ and $B$ of equal radii have masses $m$ and $2 m$ respectively. The two spheres are moving on a smooth horizontal surface when they collide with speeds $u$ and $\frac { 1 } { 2 } u$ respectively. Immediately before the collision, $A$ 's direction of motion is along the line of centres, and $B$ 's direction of motion makes an angle $\theta$ with the line of centres (see diagram). As a result of the collision, the direction of motion of $A$ is reversed and its speed is reduced to $\frac { 1 } { 4 } u$. The direction of motion of $B$ again makes an angle $\theta$ with the line of centres, but on the opposite side of the line of centres. The speed of $B$ is unchanged. Find the value of the coefficient of restitution between the spheres.
\includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-02_2716_37_141_2012}

CAIE Further Paper 3 2024 June Q2

2 A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $2 m g$. A particle $Q$ of mass $k m$ is attached to the other end of the string. Particle $P$ lies on a smooth horizontal table. The string has part of its length in contact with the table and then passes through a small smooth hole $H$ in the table. Particle $P$ moves in a horizontal circle on the surface of the table with constant speed $\sqrt { \frac { 1 } { 2 } g a }$. Particle $Q$ hangs in equilibrium vertically below the hole with $H Q = \frac { 1 } { 4 } a$.

Find, in terms of $a$, the extension in the string.
Find the value of $k$.

CAIE Further Paper 3 2024 June Q3

3 A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. When the particle is hanging vertically below $O$, it is projected horizontally with speed $u$ so that it begins to move along a circular path. When $P$ is at the lowest point of its motion, the tension in the string is $T$. When $O P$ makes an angle $\theta$ with the upward vertical, the tension in the string is $S$.

Show that $S = T - 3 m g ( 1 + \cos \theta )$.
\includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-04_2718_38_141_2010}
Given that $u = \sqrt { 4 a g }$, find the value of $\cos \theta$ when the string goes slack.
\includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-06_348_707_285_680} A light spring of natural length $a$ and modulus of elasticity $k m g$ is attached to a fixed point $O$ on a smooth plane inclined to the horizontal at an angle $\theta$, where $\sin \theta = \frac { 3 } { 4 }$. A particle of mass $m$ is attached to the lower end of the spring and is held at the point $A$ on the plane, where $O A = 2 a$ and $O A$ is along a line of greatest slope of the plane (see diagram). The particle is released from rest and is moving with speed $V$ when it passes through the point $B$ on the plane, where $O B = \frac { 3 } { 2 } a$. The speed of the particle is $\frac { 1 } { 2 } V$ when it passes through the point $C$ on the plane, where $O C = \frac { 3 } { 4 } a$. Find the value of $k$.
\includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-06_2718_35_141_2012}

CAIE Further Paper 3 2024 June Q5

3 marks

5
\includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-08_579_987_292_539} A uniform lamina is in the form of a triangle $O B C$, with $O C = 18 a , O B = 24 a$ and angle $C O B = 90 ^ { \circ }$. The point $A$ on $O B$ is such that $O A = x$ (see diagram). The triangle $O A C$ is removed from the lamina.

Find, in terms of $a$ and $x$, the distance of the centre of mass of the remaining object $A B C$ from $O C$. [3]
\includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-08_2718_40_141_2010}
The object $A B C$ is suspended from $C$ .In its equilibrium position,the side $A B$ makes an angle $\theta$ with the vertical,where $\tan \theta = \frac { 6 } { 5 }$ .
Find $x$ in terms of $a$ .

CAIE Further Paper 3 2024 June Q6

6 A particle $P$ is projected with speed $u \mathrm {~ms} ^ { - 1 }$ at an angle $\theta$ above the horizontal from a point $O$ and moves freely under gravity. After 5 seconds the speed of $P$ is $\frac { 3 } { 4 } u$.

Show that $\frac { 7 } { 16 } u ^ { 2 } - 100 u \sin \theta + 2500 = 0$.
\includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-10_2713_31_145_2014}
It is given that the velocity of $P$ after 5 seconds is perpendicular to the initial velocity. Find, in either order, the value of $u$ and the value of $\sin \theta$.

CAIE Further Paper 3 2024 June Q7

7 A parachutist of mass $m \mathrm {~kg}$ opens his parachute when he is moving vertically downwards with a speed of $50 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At time $t \mathrm {~s}$ after opening his parachute, he has fallen a distance $x \mathrm {~m}$ from the point where he opened his parachute, and his speed is $v \mathrm {~ms} ^ { - 1 }$. The forces acting on him are his weight and a resistive force of magnitude $m v \mathrm {~N}$.

Find an expression for $v$ in terms of $t$.
\includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-12_2715_40_144_2007}
Find an expression for $x$ in terms of $t$.
Find the distance that the parachutist has fallen, since opening his parachute, when his speed is $15 \mathrm {~ms} ^ { - 1 }$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\includegraphics[max width=\textwidth, alt={}, center]{73f73a7a-79d0-40fc-8c6d-1e46dacda788-14_2715_35_143_2012}

CAIE Further Paper 3 2020 November Q1

1 A particle $P$ of mass $m$ is placed on a fixed smooth plane which is inclined at an angle $\theta$ to the horizontal. A light spring, of natural length $a$ and modulus of elasticity $3 m g$, has one end attached to $P$ and the other end attached to a fixed point $O$ at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length. Find, in terms of $a$ and $\theta$, an expression for the greatest extension of the spring in the subsequent motion.
\includegraphics[max width=\textwidth, alt={}, center]{0581e302-2fc5-46f0-b597-e5cae1f664a2-04_515_707_267_685} A particle $P$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle $P$ is held with the string taut and making an angle $\theta$ with the downward vertical. The particle $P$ is then projected with speed $\frac { 4 } { 5 } \sqrt { 5 a g }$ perpendicular to the string and just completes a vertical circle (see diagram). Find the value of $\cos \theta$.

CAIE Further Paper 3 2020 November Q3

3 One end of a light elastic string, of natural length $a$ and modulus of elasticity $4 m g$, is attached to a fixed point $O$. The other end of the string is attached to a particle of mass $m$. The particle moves in a horizontal circle with a constant angular speed $\sqrt { \frac { \mathrm { g } } { \mathrm { a } } }$ with the string inclined at an angle $\theta$ to the downward vertical through $O$. The length of the string during this motion is $( \mathrm { k } + 1 ) \mathrm { a }$.

Find the value of $k$.
Find the value of $\cos \theta$.
\includegraphics[max width=\textwidth, alt={}, center]{0581e302-2fc5-46f0-b597-e5cae1f664a2-06_581_695_267_667} The diagram shows the cross-section $A B C D$ of a uniform solid object which is formed by removing a cone with cross-section $D C E$ from the top of a larger cone with cross-section $A B E$. The perpendicular distance between $A B$ and $D C$ is $h$, the diameter $A B$ is $6 r$ and the diameter $D C$ is $2 r$.
Find an expression, in terms of $h$, for the distance of the centre of mass of the solid object from $A B$.
The object is freely suspended from the point $B$ and hangs in equilibrium. The angle between $A B$ and the downward vertical through $B$ is $\theta$.
Given that $h = \frac { 13 } { 4 } r$, find the value of $\tan \theta$.

CAIE Further Paper 3 2020 November Q5

5 A particle $P$ is projected with speed $u$ at an angle $\alpha$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of $P$ from $O$ at a subsequent time $t$ are denoted by $x$ and $y$ respectively.

Derive the equation of the trajectory of $P$ in the form $$\mathrm { y } = \mathrm { x } \tan \alpha - \frac { \mathrm { gx } ^ { 2 } } { 2 \mathrm { u } ^ { 2 } } \sec ^ { 2 } \alpha$$ The point $Q$ is the highest point on the trajectory of $P$ in the case where $\alpha = 45 ^ { \circ }$.
Show that the $x$-coordinate of $Q$ is $\frac { \mathrm { u } ^ { 2 } } { 2 \mathrm {~g} }$.
Find the other value of $\alpha$ for which $P$ would pass through the point $Q$.

CAIE Further Paper 3 2020 November Q6

6 Two smooth spheres $A$ and $B$ have equal radii and masses $m$ and $2 m$ respectively. Sphere $B$ is at rest on a smooth horizontal floor. Sphere $A$ is moving on the floor with velocity $u$ and collides directly with $B$. The coefficient of restitution between the spheres is $e$.

Find, in terms of $u$ and $e$, the velocities of $A$ and $B$ after the collision.
Subsequently, $B$ collides with a fixed vertical wall which makes an angle $\theta$ with the direction of motion of $B$, where $\tan \theta = \frac { 3 } { 4 }$. The coefficient of restitution between $B$ and the wall is $\frac { 2 } { 3 }$. Immediately after $B$ collides with the wall, the kinetic energy of $A$ is $\frac { 5 } { 32 }$ of the kinetic energy of $B$.
Find the possible values of $e$.

CAIE Further Paper 3 2020 November Q7

7 A particle $P$ moving in a straight line has displacement $x \mathrm {~m}$ from a fixed point $O$ on the line at time $t \mathrm {~s}$. The acceleration of $P$, in $\mathrm { ms } ^ { - 2 }$, is given by $\frac { 200 } { x ^ { 2 } } - \frac { 100 } { x ^ { 3 } }$ for $x > 0$. When $t = 0 , x = 1$ and $P$ has velocity $10 \mathrm {~ms} ^ { - 1 }$ directed towards $O$.

Show that the velocity $v \mathrm {~ms} ^ { - 1 }$ of $P$ is given by $\mathrm { v } = \frac { 10 ( 1 - 2 \mathrm { x } ) } { \mathrm { x } }$.
Show that $x$ and $t$ are related by the equation $\mathrm { e } ^ { - 40 \mathrm { t } } = ( 2 \mathrm { x } - 1 ) \mathrm { e } ^ { 2 \mathrm { x } - 2 }$ and deduce what happens to $x$ as $t$ becomes large.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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