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CAIE Further Paper 3 2022 November Q2

6 marks Standard +0.3

2 A light elastic string has natural length $a$ and modulus of elasticity 4 mg . One end of the string is fixed to a point $O$ 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 along the surface in the direction $O P$. When the length of the string is $\frac { 5 } { 4 } a$, the speed of $P$ is $v$. When the length of the string is $\frac { 3 } { 2 } a$, the speed of $P$ is $\frac { 1 } { 2 } v$.

Find an expression for $v$ in terms of $a$ and $g$.
Find, in terms of $g$, the acceleration of $P$ when the stretched length of the string is $\frac { 3 } { 2 } a$. \includegraphics[max width=\textwidth, alt={}, center]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod $A B$, of length $4 a$ and weight $W$, rests against the surface of the cylinder. The end $A$ of the rod is in contact with the horizontal surface. The vertical plane containing the rod $A B$ is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is $C$, where $A C = 3 a$. The angle between the rod and the horizontal surface is $\theta$ where $\tan \theta = \frac { 3 } { 4 }$ (see diagram). The coefficient of friction between the rod and the horizontal surface is $\frac { 6 } { 7 }$. A particle of weight $k W$ is attached to the rod at $B$. The rod is about to slip. The normal reaction between the rod and the cylinder is $N$.

CAIE Further Paper 3 2022 November Q6

8 marks Challenging +1.8

6 \includegraphics[max width=\textwidth, alt={}, center]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-10_426_1191_267_438} Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $k m$ respectively. The two spheres are moving on a horizontal surface with speeds $u$ and $\frac { 5 } { 8 } u$ respectively. Immediately before the spheres collide, $A$ is travelling along the line of centres, and $B$ 's direction of motion makes an angle $\alpha$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac { 2 } { 3 }$ and $\tan \alpha = \frac { 3 } { 4 }$. After the collision, the direction of motion of $B$ is perpendicular to the line of centres.

Find the value of $k$.
Find the loss in the total kinetic energy as a result of the collision.

CAIE Further Paper 3 2023 November Q2

7 marks Challenging +1.2

2 A ball of mass 2 kg is projected vertically downwards with speed $5 \mathrm {~ms} ^ { - 1 }$ through a liquid. At time $t \mathrm {~s}$ after projection, the velocity of the ball is $v \mathrm {~ms} ^ { - 1 }$ and its displacement from its starting point is $x \mathrm {~m}$. The forces acting on the ball are its weight and a resistive force of magnitude $0.2 v ^ { 2 } \mathrm {~N}$.

Find an expression for $v$ in terms of $t$.
Deduce what happens to $v$ for large values of $t$. \includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-06_803_652_251_703} A uniform square lamina of side $2 a$ and weight $W$ is suspended from a light inextensible string attached to the midpoint $E$ of the side $A B$. The other end of the string is attached to a fixed point $P$ on a rough vertical wall. The vertex $B$ of the lamina is in contact with the wall. The string $E P$ is perpendicular to the side $A B$ and makes an angle $\theta$ with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is $\frac { 1 } { 2 }$. Given that the vertex $B$ is about to slip up the wall, find the value of $\tan \theta$. \includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-08_581_576_269_731} A light elastic string has natural length $8 a$ and modulus of elasticity $5 m g$. A particle $P$ of mass $m$ is attached to the midpoint of the string. The ends of the string are attached to points $A$ and $B$ which are a distance $12 a$ apart on a smooth horizontal table. The particle $P$ is held on the table so that $A P = B P = L$ (see diagram). The particle $P$ is released from rest. When $P$ is at the midpoint of $A B$ it has speed $\sqrt { 80 a g }$.
1. Find $L$ in terms of $a$.
2. Find the initial acceleration of $P$ in terms of $g$.

CAIE Further Paper 3 2022 November Q3

7 marks Challenging +1.2

Show that $\mathrm { N } = \frac { 8 } { 15 } \mathrm {~W} ( 1 + 2 \mathrm { k } )$.
Find the value of $k$.

Show that the initial acceleration of $P$ is $\frac{3}{5}g$ upwards. [3]
Find the speed of $P$ when the spring first returns to its natural length. [4]

CAIE Further Paper 3 2020 June Q4

7 marks Challenging +1.2

\includegraphics{figure_4} A uniform square lamina $ABCD$ has sides of length $10\text{cm}$. The point $E$ is on $BC$ with $EC = 7.5\text{cm}$, and the point $F$ is on $DC$ with $CF = x\text{cm}$. The triangle $EFC$ is removed from $ABCD$ (see diagram). The centre of mass of the resulting shape $ABEFD$ is a distance $\bar{x}\text{cm}$ from $CB$ and a distance $\bar{y}\text{cm}$ from $CD$.

Show that $\bar{x} = \frac{400 - x^2}{80 - 3x}$ and find a corresponding expression for $\bar{y}$. [4]

The shape $ABEFD$ is in equilibrium in a vertical plane with the edge $DF$ resting on a smooth horizontal surface.

Find the greatest possible value of $x$, giving your answer in the form $a + b\sqrt{2}$, where $a$ and $b$ are constants to be determined. [3]

CAIE Further Paper 3 2020 June Q5

8 marks Challenging +1.2

A particle $P$ is moving along a straight line with acceleration $3ku - kv$ 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 $2u$. [3]
Find an expression for the displacement of $P$ from its initial position when its velocity is $2u$. [5]

CAIE Further Paper 3 2020 June Q6

8 marks Challenging +1.2

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 $90°$.

Show that $\tan^2 \alpha = \frac{1}{e}$. [3]

The particle $P$ loses two-thirds of its kinetic energy in the impact.

Find the value of $\alpha$ and the value of $e$. [5]

CAIE Further Paper 3 2020 June Q7

10 marks Challenging +1.8

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 $\sqrt{\frac{7}{2}ga}$. The particle $P$ loses contact with the surface of the cylinder when $OP$ makes an angle $\theta$ with the upward vertical through $O$.

Show that $\theta = 60°$. [5]
Show that in its subsequent motion $P$ strikes the cylinder at the point $A$. [5]

CAIE Further Paper 3 2020 June Q1

2 marks Moderate -0.5

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 $4mg$. Find, in terms of $a$ and $g$, the time that $P$ takes to make one complete revolution. [2]

CAIE Further Paper 3 2020 June Q2

6 marks Standard +0.8

A particle $Q$ of mass $m$ kg falls from rest under gravity. The motion of $Q$ is resisted by a force of magnitude $mkv$ N, where $v$ ms$^{-1}$ is the speed of $Q$ at time $t$ s and $k$ is a positive constant. Find an expression for $v$ in terms of $g$, $k$ and $t$. [6]

CAIE Further Paper 3 2020 June Q3

6 marks Standard +0.8

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 $AB$ a diameter of the circle. $OA$ makes an angle of $60°$ with the downward vertical through $O$ and $OB$ makes an angle of $60°$ with the upward vertical through $O$. The speed of $Q$ when it is at $A$ is $2\sqrt{ag}$. Given that $T_A$ and $T_B$ are the tensions in the string at $A$ and $B$ respectively, find the ratio $T_A : T_B$. [6]

CAIE Further Paper 3 2020 June Q4

4 marks Standard +0.3

\includegraphics{figure_4} A uniform solid circular cone, of vertical height $4r$ and radius $2r$, is attached to a uniform solid cylinder, of height $3r$ and radius $kr$, 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 $$\frac{(99k^2 + 96)r}{18k^2 + 32}.$$ [4]

CAIE Further Paper 3 2020 June Q4

4 marks Challenging +1.2

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 $\tan \alpha = \frac{1}{5}$.

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

4 marks Challenging +1.8

\includegraphics{figure_5} 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 $\alpha°$ 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 $\tan \alpha = \frac{1+e}{1-e}$. [4]

CAIE Further Paper 3 2020 June Q5

4 marks Moderate -0.5

Given that $\tan \alpha = 2$, find the speed of $A$ after the collision. [4]

CAIE Further Paper 3 2020 June Q6

6 marks Challenging +1.2

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 $\tan \alpha = \frac{1}{2}\tan \theta$. [6]

CAIE Further Paper 3 2020 June Q6

4 marks Challenging +1.2

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 $g$. [4]