Questions Further Paper 3

Find an expression for $v$ in terms of $t$ and an arbitrary constant.
Given that $a = 5$ when $t = 1$, find an expression, in terms of $m$ and $t$, for the horizontal force acting on $P$ at time $t$.

CAIE Further Paper 3 2021 November Q3

3 A light elastic string has natural length $a$ and modulus of elasticity 12 mg . One end of the string is attached to a fixed point $O$. The other end of the string is attached to a particle of mass $m$. The particle hangs in equilibrium vertically below $O$. The particle is pulled vertically down and released from rest with the extension of the string equal to $e$, where $\mathrm { e } > \frac { 1 } { 3 } \mathrm { a }$. In the subsequent motion the particle has speed $\sqrt { 2 \mathrm { ga } }$ when it has ascended a distance $\frac { 1 } { 3 } a$. Find $e$ in terms of $a$.
\includegraphics[max width=\textwidth, alt={}, center]{e4926d36-7246-4cde-b466-44ecc4c30a61-06_488_496_269_781} A uniform lamina $A E C F$ is formed by removing two identical triangles $B C E$ and $C D F$ from a square lamina $A B C D$. The square has side $3 a$ and $E B = D F = h$ (see diagram).

Find the distance of the centre of mass of the lamina $A E C F$ from $A D$ and from $A B$, giving your answers in terms of $a$ and $h$.
The lamina $A E C F$ is placed vertically on its edge $A E$ on a horizontal plane.
Find, in terms of $a$, the set of values of $h$ for which the lamina remains in equilibrium.

CAIE Further Paper 3 2021 November Q5

5 A particle $P$ is projected from a point $O$ on a horizontal plane and moves freely under gravity. Its initial speed is $u \mathrm {~ms} ^ { - 1 }$ and its angle of projection is $\sin ^ { - 1 } \left( \frac { 4 } { 5 } \right)$ above the horizontal. At time 8 s after projection, $P$ is at the point $A$. At time 32 s after projection, $P$ is at the point $B$. The direction of motion of $P$ at $B$ is perpendicular to its direction of motion at $A$. Find the value of $u$.

CAIE Further Paper 3 2021 November Q6

6 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 $P$ moves in complete vertical circles about $O$ with the string taut. The points $A$ and $B$ are on the path of $P$ with $A B$ a diameter of the circle. $O A$ makes an angle $\theta$ with the downward vertical through $O$ and $O B$ makes an angle $\theta$ with the upward vertical through $O$. The speed of $P$ when it is at $A$ is $\sqrt { 5 a g }$. The ratio of the tension in the string when $P$ is at $A$ to the tension in the string when $P$ is at $B$ is $9 : 5$.

Find the value of $\cos \theta$.
Find, in terms of $a$ and $g$, the greatest speed of $P$ during its motion.
\includegraphics[max width=\textwidth, alt={}, center]{e4926d36-7246-4cde-b466-44ecc4c30a61-12_613_718_251_676} The smooth vertical walls $A B$ and $C B$ are at right angles to each other. A particle $P$ is moving with speed $u$ on a smooth horizontal floor and strikes the wall $C B$ at an angle $\alpha$. It rebounds at an angle $\beta$ to the wall $C B$. The particle then strikes the wall $A B$ and rebounds at an angle $\gamma$ to that wall (see diagram). The coefficient of restitution between each wall and $P$ is $e$.
Show that $\tan \beta = e \tan \alpha$.
Express $\gamma$ in terms of $\alpha$ and explain what this result means about the final direction of motion of $P$.
As a result of the two impacts the particle loses $\frac { 8 } { 9 }$ of its initial kinetic energy.
Given that $\alpha + \beta = 90 ^ { \circ }$, find the value of $e$ and the value of $\tan \alpha$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE Further Paper 3 2020 November Q3

3 An object consists of a uniform solid circular cone, of vertical height $4 r$ and radius $3 r$, and a uniform solid cylinder, of height $4 r$ and radius $3 r$. The circular base of the cone and one of the circular faces of the cylinder are joined together so that they coincide. The cone and the cylinder are made of the same material.

Find the distance of the centre of mass of the object from the end of the cylinder that is not attached to the cone.
Show that the object can rest in equilibrium with the curved surface of the cone in contact with a horizontal surface.

CAIE Further Paper 3 2020 November Q4

4 A particle $P$ of mass $m$ is moving in a horizontal circle with angular speed $\omega$ on the smooth inner surface of a hemispherical shell of radius $r$. The angle between the vertical and the normal reaction of the surface on $P$ is $\theta$.

Show that $\cos \theta = \frac { \mathrm { g } } { \omega ^ { 2 } \mathrm { r } }$.
The plane of the circular motion is at a height $x$ above the lowest point of the shell. When the angular speed is doubled, the plane of the motion is at a height $4 x$ above the lowest point of the shell.
Find $x$ in terms of $r$.

CAIE Further Paper 3 2020 November Q5

5 A particle $P$ is projected with speed $u \mathrm {~ms} ^ { - 1 }$ at an angle of $\theta$ 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 $t \mathrm {~s}$ are denoted by $x \mathrm {~m}$ and $y \mathrm {~m}$ respectively.

Starting from the equation of the trajectory given in the List of formulae (MF19), show that $$\mathrm { y } = \mathrm { x } \tan \theta - \frac { \mathrm { gx } ^ { 2 } } { 2 \mathrm { u } ^ { 2 } } \left( 1 + \tan ^ { 2 } \theta \right)$$ When $\theta = \tan ^ { - 1 } 2 , P$ passes through the point with coordinates $( 10,16 )$.
Show that there is no value of $\theta$ for which $P$ can pass through the point with coordinates $( 18,30 )$.

CAIE Further Paper 3 2020 November Q6

6 One end of a light elastic string, of natural length $a$ and modulus of elasticity $k$, is attached to a particle $P$ of mass $m$. The other end of the string is attached to a fixed point $Q$. The particle $P$ is projected vertically upwards from $Q$. When $P$ is moving upwards and at a distance $\frac { 4 } { 3 } a$ directly above $Q$, it has a speed $\sqrt { 2 g a }$. At this point, its acceleration is $\frac { 7 } { 3 } g$ downwards. Show that $\mathrm { k } = 4 \mathrm { mg }$ and find in terms of $a$ the greatest height above $Q$ reached by $P$.

CAIE Further Paper 3 2020 November Q7

7 A particle $P$ of mass $m \mathrm {~kg}$ moves in a horizontal straight line against a resistive force of magnitude $\mathrm { mkv } ^ { 2 } \mathrm {~N}$, where $v \mathrm {~ms} ^ { - 1 }$ is the speed of $P$ after it has moved a distance $x \mathrm {~m}$ and $k$ is a positive constant. The initial speed of $P$ is $u \mathrm {~ms} ^ { - 1 }$.

Show that $\mathrm { x } = \frac { 1 } { \mathrm { k } } \ln 2$ when $\mathrm { v } = \frac { 1 } { 2 } \mathrm { u }$.
Beginning at the instant when the speed of $P$ is $\frac { 1 } { 2 } u$, an additional force acts on $P$. This force has magnitude $\frac { 5 \mathrm {~m} } { \mathrm { v } } \mathrm { N }$ and acts in the direction of increasing $x$.
Show that when the speed of $P$ has increased again to $u \mathrm {~ms} ^ { - 1 }$, the total distance travelled by $P$ is given by an expression of the form $$\frac { 1 } { 3 k } \ln \left( \frac { A - k u ^ { 3 } } { B - k u ^ { 3 } } \right) ,$$ stating the values of the constants $A$ and $B$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE Further Paper 3 2020 June Q1

1 A particle $P$ is projected with speed $u$ at an angle of $30 ^ { \circ }$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The particle reaches its greatest height at time $T$ after projection. Find, in terms of $u$, the speed of $P$ at time $\frac { 2 } { 3 } T$ after projection.
\includegraphics[max width=\textwidth, alt={}, center]{6dcd0997-d7a1-463c-9040-96a5e81623cf-04_362_750_258_653} A light inextensible string of length $a$ is threaded through a fixed smooth ring $R$. One end of the string is attached to a particle $A$ of mass $3 m$. The other end of the string is attached to a particle $B$ of mass $m$. The particle $A$ hangs in equilibrium at a distance $x$ vertically below the ring. The angle between $A R$ and $B R$ is $\theta$ (see diagram). The particle $B$ moves in a horizontal circle with constant angular speed $2 \sqrt { \frac { \mathrm {~g} } { \mathrm { a } } }$. Show that $\cos \theta = \frac { 1 } { 3 }$ and find $x$ in terms of $a$.

CAIE Further Paper 3 2020 June Q3

3 One end of a light elastic spring, of natural length $a$ and modulus of elasticity 5 mg , is attached to a fixed point $A$. The other end of the spring is attached to a particle $P$ of mass $m$. The spring hangs with $P$ vertically below $A$. The particle $P$ is released from rest in the position where the extension of the spring is $\frac { 1 } { 2 } a$.

Show that the initial acceleration of $P$ is $\frac { 3 } { 2 } g$ upwards.
Find the speed of $P$ when the spring first returns to its natural length.
\includegraphics[max width=\textwidth, alt={}, center]{6dcd0997-d7a1-463c-9040-96a5e81623cf-08_581_659_267_708} A uniform square lamina $A B C D$ has sides of length 10 cm . The point $E$ is on $B C$ with $E C = 7.5 \mathrm {~cm}$, and the point $F$ is on $D C$ with $\mathrm { CF } = \mathrm { xcm }$. The triangle $E F C$ is removed from $A B C D$ (see diagram). The centre of mass of the resulting shape $A B E F D$ is a distance $\bar { x } \mathrm {~cm}$ from $C B$ and a distance $\bar { y } \mathrm {~cm}$ from CD.

CAIE Further Paper 3 2020 June Q5

5 A particle $P$ is moving along a straight line with acceleration $3 \mathrm { ku } - \mathrm { kv }$ where $v$ is its velocity at time $t$, $u$ is its initial velocity and $k$ is a constant. The velocity and acceleration of $P$ are both in the direction of increasing displacement from the initial position.

Find the time taken for $P$ to achieve a velocity of $2 u$.
Find an expression for the displacement of $P$ from its initial position when its velocity is $2 u$.

CAIE Further Paper 3 2020 June Q6

6 A particle $P$ of mass $m$ is moving with speed $u$ on a fixed smooth horizontal surface. The particle strikes a fixed vertical barrier. At the instant of impact the direction of motion of $P$ makes an angle $\alpha$ with the barrier. The coefficient of restitution between $P$ and the barrier is $e$. As a result of the impact, the direction of motion of $P$ is turned through $90 ^ { \circ }$.

Show that $\tan ^ { 2 } \alpha = \frac { 1 } { e }$.
The particle $P$ loses two-thirds of its kinetic energy in the impact.
Find the value of $\alpha$ and the value of $e$.

CAIE Further Paper 3 2020 June Q7

7 A hollow cylinder of radius $a$ is fixed with its axis horizontal. A particle $P$, of mass $m$, moves in part of a vertical circle of radius $a$ and centre $O$ on the smooth inner surface of the cylinder. The speed of $P$ when it is at the lowest point $A$ of its motion is $\sqrt { \frac { 7 } { 2 } \mathrm { ga } }$. The particle $P$ loses contact with the surface of the cylinder when $O P$ makes an angle $\theta$ with the upward vertical through $O$.

Show that $\theta = 60 ^ { \circ }$.
Show that in its subsequent motion $P$ strikes the cylinder at the point $A$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE Further Paper 3 2020 June Q1

1 A particle $P$ is projected with speed $u$ at an angle of $30 ^ { \circ }$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The particle reaches its greatest height at time $T$ after projection. Find, in terms of $u$, the speed of $P$ at time $\frac { 2 } { 3 } T$ after projection.
\includegraphics[max width=\textwidth, alt={}, center]{7251b13f-1fae-4138-80ea-e6b8091c94ab-04_362_750_258_653} A light inextensible string of length $a$ is threaded through a fixed smooth ring $R$. One end of the string is attached to a particle $A$ of mass $3 m$. The other end of the string is attached to a particle $B$ of mass $m$. The particle $A$ hangs in equilibrium at a distance $x$ vertically below the ring. The angle between $A R$ and $B R$ is $\theta$ (see diagram). The particle $B$ moves in a horizontal circle with constant angular speed $2 \sqrt { \frac { \mathrm {~g} } { \mathrm { a } } }$. Show that $\cos \theta = \frac { 1 } { 3 }$ and find $x$ in terms of $a$.

CAIE Further Paper 3 2020 June Q3

Show that the initial acceleration of $P$ is $\frac { 3 } { 2 } g$ upwards.
Find the speed of $P$ when the spring first returns to its natural length.
\includegraphics[max width=\textwidth, alt={}, center]{7251b13f-1fae-4138-80ea-e6b8091c94ab-08_581_659_267_708} A uniform square lamina $A B C D$ has sides of length 10 cm . The point $E$ is on $B C$ with $E C = 7.5 \mathrm {~cm}$, and the point $F$ is on $D C$ with $\mathrm { CF } = \mathrm { xcm }$. The triangle $E F C$ is removed from $A B C D$ (see diagram). The centre of mass of the resulting shape $A B E F D$ is a distance $\bar { x } \mathrm {~cm}$ from $C B$ and a distance $\bar { y } \mathrm {~cm}$ from CD.

CAIE Further Paper 3 2020 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 $O$ on a smooth horizontal plane. The particle $P$ moves in horizontal circles about $O$. The tension in the string is $4 m g$. Find, in terms of $a$ and $g$, the time that $P$ takes to make one complete revolution.

CAIE Further Paper 3 2020 June Q2

2 A particle $Q$ of mass $m \mathrm {~kg}$ falls from rest under gravity. The motion of $Q$ is resisted by a force of magnitude $m k v \mathrm {~N}$, where $v \mathrm {~ms} ^ { - 1 }$ is the speed of $Q$ at time $t \mathrm {~s}$ and $k$ is a positive constant. Find an expression for $v$ in terms of $g , k$ and $t$.

CAIE Further Paper 3 2020 June Q3

4 marks

3 A particle $Q$ of mass $m$ is attached to a fixed point $O$ by a light inextensible string of length $a$. The particle moves in complete vertical circles about $O$. The points $A$ and $B$ are on the path of $Q$ with $A B$ a diameter of the circle. $O A$ makes an angle of $60 ^ { \circ }$ with the downward vertical through $O$ and $O B$ makes an angle of $60 ^ { \circ }$ with the upward vertical through $O$. The speed of $Q$ when it is at $A$ is $2 \sqrt { \mathrm { ag } }$. Given that $T _ { A }$ and $T _ { B }$ are the tensions in the string at $A$ and $B$ respectively, find the ratio $T _ { A } : T _ { B }$.
\includegraphics[max width=\textwidth, alt={}, center]{5cc14ffc-e957-4582-b9d0-182fd89b3df5-06_880_428_260_817} A uniform solid circular cone, of vertical height $4 r$ and radius $2 r$, is attached to a uniform solid cylinder, of height $3 r$ and radius $k r$, where $k$ is a constant less than 2 . The base of the cone is joined to one of the circular faces of the cylinder so that the axes of symmetry of the two solids coincide (see diagram). The cone and the cylinder are made of the same material.

Show that the distance of the centre of mass of the combined solid from the vertex of the cone is $\frac { \left( 99 \mathrm { k } ^ { 2 } + 96 \right) \mathrm { r } } { 18 \mathrm { k } ^ { 2 } + 32 }$.
The point $C$ is on the circumference of the base of the cone. When the combined solid is freely suspended from $C$ and hanging in equilibrium, the diameter through $C$ makes an angle $\alpha$ with the downward vertical, where $\tan \alpha = \frac { 1 } { 8 }$.
Given that the centre of mass of the combined solid is within the cylinder, find the value of $k$. [4]

CAIE Further Paper 3 2020 June Q5

5
\includegraphics[max width=\textwidth, alt={}, center]{5cc14ffc-e957-4582-b9d0-182fd89b3df5-08_561_1068_255_500} Two uniform smooth spheres $A$ and $B$ of equal radii each have mass $m$. The two spheres are each moving with speed $u$ on a horizontal surface when they collide. Immediately before the collision A's direction of motion makes an angle of $\alpha ^ { \circ }$ with the line of centres, and $B$ 's direction of motion is perpendicular to that of $A$ (see diagram). The coefficient of restitution between the spheres is $e$. Immediately after the collision, $B$ moves in a direction at right angles to the line of centres.

Show that $\tan \alpha = \frac { 1 + e } { 1 - e }$.
Given that $\tan \alpha = 2$, find the speed of $A$ after the collision.

CAIE Further Paper 3 2020 June Q6

6 A particle $P$ is projected with speed $u$ at an angle $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The direction of motion of $P$ makes an angle $\alpha$ above the horizontal when $P$ first reaches three-quarters of its greatest height.

Show that $\tan \alpha = \frac { 1 } { 2 } \tan \theta$.
Given that $\tan \theta = \frac { 4 } { 3 }$, find the horizontal distance travelled by $P$ when it first reaches three-quarters of its greatest height. Give your answer in terms of $u$ and $g$.
\includegraphics[max width=\textwidth, alt={}, center]{5cc14ffc-e957-4582-b9d0-182fd89b3df5-12_241_1009_269_529} One end of a light spring of natural length $a$ and modulus of elasticity $4 m g$ is attached to a fixed point $O$. The other end of the spring is attached to a particle $A$ of mass $k m$, where $k$ is a constant. Initially the spring lies at rest on a smooth horizontal surface and has length $a$. A second particle $B$, of mass $m$, is moving towards $A$ with speed $\sqrt { \frac { 4 } { 3 } \mathrm { ga } }$ along the line of the spring from the opposite direction to $O$ (see diagram). The particles $A$ and $B$ collide and coalesce. At a point $C$ in the subsequent motion, the length of the spring is $\frac { 3 } { 4 } a$ and the speed of the combined particle is half of its initial speed.
Find the value of $k$.
At the point $C$ the horizontal surface becomes rough, with coefficient of friction $\mu$ between the combined particle and the surface. The deceleration of the combined particle at $C$ is $\frac { 9 } { 20 } \mathrm {~g}$.
Find the value of $\mu$.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

CAIE Further Paper 3 2021 June Q1

1 A particle $P$ of mass 1 kg is moving along a straight line against a resistive force of magnitude $\frac { 10 \sqrt { \mathrm { v } } } { ( \mathrm { t } + 1 ) ^ { 2 } } \mathrm {~N}$, where $\mathrm { vms } ^ { - 1 }$ is the speed of $P$ at time $t \mathrm {~s}$. When $\mathrm { t } = 0 , \mathrm { v } = 25$. Find an expression for $v$ in terms of $t$.

CAIE Further Paper 3 2021 June Q2

2 A hollow hemispherical bowl of radius $a$ has a smooth inner surface and is fixed with its axis vertical. A particle $P$ of mass $m$ moves in horizontal circles on the inner surface of the bowl, at a height $x$ above the lowest point of the bowl. The speed of $P$ is $\sqrt { \frac { 8 } { 3 } } \mathrm { ga }$. Find $x$ in terms of $a$.

CAIE Further Paper 3 2021 June Q3

3 One end of a light elastic string, of natural length $a$ and modulus of elasticity $k m g$, is attached to a fixed point $A$. The other end of the string is attached to a particle $P$ of mass $4 m$. The particle $P$ hangs in equilibrium a distance $x$ vertically below $A$.

Show that $\mathrm { k } = \frac { 4 \mathrm { a } } { \mathrm { x } - \mathrm { a } }$.
An additional particle, of mass $2 m$, is now attached to $P$ and the combined particle is released from rest at the original equilibrium position of $P$. When the combined particle has descended a distance $\frac { 1 } { 3 } a$, its speed is $\frac { 1 } { 3 } \sqrt { \mathrm { ga } }$.
Find $x$ in terms of $a$.
\includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-06_602_520_264_753} A uniform solid circular cone has vertical height $k h$ and radius $r$. A uniform solid cylinder has height $h$ and radius $r$. The base of the cone is joined to one of the circular faces of the cylinder so that the axes of symmetry of the two solids coincide (see diagram, which shows a cross-section). The cone and the cylinder are made of the same material.
Show that the distance of the centre of mass of the combined solid from the base of the cylinder is $\frac { \mathrm { h } \left( \mathrm { k } ^ { 2 } + 4 \mathrm { k } + 6 \right) } { 4 ( 3 + \mathrm { k } ) }$.
The solid is placed on a plane that is inclined to the horizontal at an angle $\theta$. The base of the cylinder is in contact with the plane. The plane is sufficiently rough to prevent sliding. It is given that $3 h = 2 r$ and that the solid is on the point of toppling when $\tan \theta = \frac { 4 } { 3 }$.
Find the value of $k$.

Questions Further Paper 3 (127 questions)

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