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WJEC Further Unit 6 2019 June Q1

A large aeroplane, of mass 360 tonnes, starts from rest at the beginning of a straight horizontal runway. The aeroplane produces a constant thrust of 980 kN and experiences a variable resistance to motion of magnitude $\left( 80 + 0 \cdot 1 v ^ { 2 } \right) \mathrm { kN }$, where $v \mathrm {~ms} ^ { - 1 }$ is the speed of the aeroplane after it has travelled $x$ metres.
1. (i) Find the maximum speed that the aeroplane can attain.
  (ii) Show that $v$ satisfies the differential equation
$$3600 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 9000 - v ^ { 2 } .$$
Find an expression for $v ^ { 2 }$ in terms of $x$.
Given that the aeroplane must achieve a speed of at least $85 \mathrm {~ms} ^ { - 1 }$ to take off, determine the minimum length of the runway.
Explain why, according to this model, the aeroplane will not reach the speed found in (a)(i).

WJEC Further Unit 6 2019 June Q2

2. A metal sign is formed by removing triangle $B C D$ from a rectangular lamina $A C E F$ made of uniform material, and adding a quarter circle XYZ, made of the same uniform material, with a particle attached to its vertex at $Y$. The sign is supported by two light chains fixed at $E$ and $F$. The quarter circle has radius 24 cm and the particle at $Y$ has a mass equal to half of that of the removed triangle. $X D$ is parallel to $A C$ and $B Z$ is parallel to $A F$. The dimensions, in cm , are as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-3_885_636_712_715}

Calculate the distance of the centre of mass of the sign from
1. $A F$,
2. $A C$.
The support at $F$ comes loose so that the sign is freely suspended at $E$ by one chain alone. Given that it then hangs in equilibrium, calculate the angle that $E F$ makes with the vertical.

WJEC Further Unit 6 2019 June Q3

3. A light elastic string, of natural length $l \mathrm {~m}$ and modulus of elasticity 14 N , is hanging vertically with its upper end fixed and a particle of mass $m \mathrm {~kg}$ attached to the lower end. The particle is initially in equilibrium and air resistance is to be neglected.

Find, in terms of $m , g$ and $l$, the extension, $e$, of the string when the particle is in equilibrium. The particle is pulled vertically downwards a further distance from its equilibrium position and released. In its subsequent motion, the string remains taut. Let $x \mathrm {~m}$ denote the extension of the string from the equilibrium position at time $t \mathrm {~s}$.
1. Write down, in terms of $x , m , g$ and $l$, an expression for the tension in the string.
2. Hence, show that the particle is moving with Simple Harmonic Motion which satisfies the differential equation, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 14 } { m l } x$$
3. State the maximum distance that the particle could be pulled vertically downwards from its equilibrium position and still move with Simple Harmonic Motion. Give a reason for your answer.
Given that $m = 0.5 , l = 0.7$ and that the particle is pulled to the position where $x = 0.2$ before being released,
1. find the maximum speed of the particle,
2. determine the time taken for the particle to reach $x = 0.15$ for the first time.

WJEC Further Unit 6 2019 June Q4

4. Ryan is playing a game of snooker. The horizontal table is modelled as the horizontal $x - y$ plane with the point $O$ as the origin and unit vectors parallel to the $x$-axis and the $y$-axis denoted by $\mathbf { i }$ and $\mathbf { j }$ respectively. All balls on the table have a common mass $m \mathrm {~kg}$. The table and the four sides, called cushions, are modelled as smooth surfaces. The dimensions of the table, in metres, are as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-5_663_1138_667_482} Initially, all balls are stationary. Ryan strikes ball $A$ so that it collides with ball $B$. Before the collision, $A$ has velocity $( - \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ and, after the collision, it has velocity $( 2 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }$.

Show that the velocity of ball $B$ after the collision is $( - 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$. After the collision with ball $A$, ball $B$ hits the cushion at point $C$ before rebounding and moving towards the pocket at $P$. The cushion is parallel to the vector $\mathbf { i }$ and the coefficient of restitution between the cushion and ball $B$ is $\frac { 5 } { 7 }$.
Calculate the velocity of ball $B$ after impact with the cushion.
Find, in terms of $m$, the magnitude of the impulse exerted on ball $B$ by the cushion at $C$, stating your units clearly.
Given that $C$ has position vector $( x \mathbf { i } + 1 \cdot 75 \mathbf { j } ) \mathrm { m }$,
1. determine the time taken between the ball hitting the cushion at $C$ and entering the pocket at $P$,
2. find the value of $x$.
Describe one way in which the model used could be refined. Explain how your refinement would affect your answer to (d)(i).

WJEC Further Unit 6 2019 June Q5

5. (a) Show, by integration, that the centre of mass of a uniform solid hemisphere of radius $r$ is at a distance of $\frac { 3 r } { 8 }$ from the plane face.
(b) The diagram shows a composite solid body which consists of a uniform right circular cylinder capped by a uniform hemisphere. The total height of the solid is $3 r \mathrm {~cm}$, where $r$ represents the common radius of the hemisphere and the cylinder.
\includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-6_397_340_762_858} Given that the density of the hemisphere is $50 \%$ more than that of the cylinder, find the distance of the centre of mass of the solid from its base along the axis of symmetry.

WJEC Further Unit 6 2019 June Q6

6.
\includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-7_606_506_365_781} A uniform ladder $A B$, of mass 10 kg and length 5 m , rests with one end $A$ against a smooth vertical wall and the other end $B$ on rough horizontal ground. The ladder is inclined at an angle $\theta$ to the horizontal. A woman of mass 75 kg stands on the ladder so that her weight acts at a distance $x \mathrm {~m}$ from $B$.

Show that the frictional force, $F \mathrm {~N}$, between the ladder and the horizontal ground is given by $$F = 5 g \cot \theta ( 1 + 3 x ) .$$ For safety reasons, it is recommended that $\theta$ is chosen such that the ratio $C B : C A$ is $1 : 4$.
Determine the least value of the coefficient of friction such that the ladder will not slip however high the woman climbs.
State one modelling assumption that you have made in your solution.

WJEC Further Unit 6 2022 June Q1

A particle is moving along the $x$-axis. At time $t$ seconds the particle is $x$ metres from the origin, $O$, and its velocity $v \mathrm {~ms} ^ { - 1 }$ is given by

$$v = \frac { 24 } { 4 x + 9 }$$

Find, in terms of $x$, an expression for the acceleration of the particle at time $t \mathrm {~s}$.
At $t = T$ the acceleration of the particle is $- \frac { 4 } { 3 } \mathrm {~ms} ^ { - 2 }$.
1. Determine the value of $x$ when $t = T$.
2. Given that $x = - 2$ when $t = 0$, find an expression for $t$ in terms of $x$ and hence find the value of $T$.

WJEC Further Unit 6 2022 June Q2

2. A particle $P$ moves along the $x$-axis such that its position $x$ metres, after $t$ seconds, is given by $$x = \sin ( \pi t ) + \sqrt { 3 } \cos ( \pi t )$$

1. Show that the motion of the particle $P$ is Simple Harmonic. State the value of $x$ at the centre of motion.
2. Show that the period of the motion of $P$ is 2 s and determine the amplitude. Suppose that another particle $Q$ is introduced so that it also moves along the $x$-axis with Simple Harmonic Motion with centre of motion, $O$, and period equal to that of particle $P$. When $t = 0$, the particle $Q$ is at $O$ and when it is $2 \sqrt { 3 } \mathrm {~m}$ from $O$ its speed is $2 \pi \mathrm {~ms} ^ { - 1 }$.
Find the amplitude of particle $Q$.
Determine the time when particles $P$ and $Q$ first meet.

WJEC Further Unit 6 2022 June Q3

3. The diagram below shows a lamina $A B C D E$ which is made of a uniform material. It consists of a rectangle $A B D E$ with $A B = 6 a$ and $A E = 8 a$, together with an isosceles triangle $B C D$ with $B C = D C = 5 a$. A semicircle, with its centre at the midpoint of $A E$ and radius $3 a$, is removed from $A B D E$.
\includegraphics[max width=\textwidth, alt={}, center]{b9c63cb4-d446-4548-be42-e30b10cb4b99-3_606_703_603_680}

Write down the distance of the centre of mass of the lamina $A B C D E$ from $A B$.
Show that the distance of the centre of mass of the lamina $A B C D E$ from $A E$ is $\frac { 140 } { 40 - 3 \pi } a$.
The lamina $A B C D E$ is freely suspended from the point $D$ and hangs in equilibrium.
1. Calculate the angle that $B D$ makes with the vertical.
2. The mass of the lamina is $M$. When a particle of mass $k M$ is attached at the point $C$, the lamina hangs in equilibrium with $A B$ horizontal. Determine the value of $k$.

WJEC Further Unit 6 2022 June Q4

4. The diagram below shows a uniform rod $A B$, of weight 10 N , hinged to a vertical wall at $A$. The rod is held in a horizontal position by means of a light inextensible string. One end of the string is attached to a point $C$ on the rod and the other end is attached to a point $D$ on the wall. The point $D$ is 0.6 m vertically above $A$ and the length of $A C$ is 0.8 m . A particle of weight 25 N is attached to the rod at $B$ and the tension in the string is 75 N .
\includegraphics[max width=\textwidth, alt={}, center]{b9c63cb4-d446-4548-be42-e30b10cb4b99-4_572_808_612_625}

Find the length of the rod $A B$.
Calculate the magnitude and direction of the reaction at the hinge at $A$.

WJEC Further Unit 6 2022 June Q5

5. Two smooth spheres $A$ and $B$, of equal radii, are moving on a smooth horizontal plane when they collide. Immediately after the collision sphere $A$ has velocity ( $- 2 \mathbf { i } - 5 \mathbf { j }$ ) $\mathrm { ms } ^ { - 1 }$ and sphere $B$ has velocity $( \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$. When the spheres collide, their line of centres is parallel to the vector $\mathbf { i }$ and the coefficient of restitution between the spheres is $\frac { 2 } { 5 }$. Sphere $A$ has mass 4 kg and sphere $B$ has mass 2 kg .

Find the velocity of $A$ and the velocity of $B$ immediately before the collision. After the collision, sphere $A$ continues to move with velocity ( $- 2 \mathbf { i } - 5 \mathbf { j }$ ) $\mathrm { ms } ^ { - 1 }$ until it collides with a smooth vertical wall. The impulse exerted by the wall on $A$ is $32 \mathbf { j }$ Ns.
State whether the wall is parallel to the vector $\mathbf { i }$ or to the vector $\mathbf { j }$. Give a reason for your answer.
Find the speed of $A$ after the collision with the wall.
Calculate the loss of kinetic energy caused by the collision of sphere $A$ with the wall.

WJEC Further Unit 6 2022 June Q6

6. The diagram shows a particle $P$, of mass 4 kg , lying on a smooth horizontal surface. It is attached by two light springs to fixed points $A$ and $B$, where $A B = 2.8 \mathrm {~m}$.
Spring $A P$ has natural length 0.8 m and modulus of elasticity 60 N .
Spring $P B$ has natural length 1.2 m and modulus of elasticity 30 N .
\includegraphics[max width=\textwidth, alt={}, center]{b9c63cb4-d446-4548-be42-e30b10cb4b99-5_231_1253_612_404} When $P$ is in equilibrium, it is at the point $C$.

Show that $A C = 1 \mathrm {~m}$.
The particle $P$ is pulled horizontally and is initially held at rest at the midpoint of $A B$. The system is then released.
1. Show that $P$ performs Simple Harmonic Motion about centre $C$ and find the period of its motion.
2. Determine the shortest time taken for $P$ to reach a position where there is no tension in the spring $A P$. \section*{END OF PAPER}

WJEC Further Unit 6 2023 June Q1

The diagram shows a uniform rod $A B$, of length 8 m and mass 23 kg , in limiting equilibrium with its end $A$ on rough horizontal ground and point $C$ resting against a smooth fixed cylinder. The rod is inclined at an angle of $30 ^ { \circ }$ to the ground.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_240_869_603_598}

The coefficient of friction between the ground and the rod is $\frac { 2 } { 3 }$.

Calculate the magnitude of the normal reaction at $C$ and the magnitude of the normal reaction to the ground at $A$.
Find the length $A C$.
Suppose instead that the rod is non-uniform with its centre of mass closer to $A$ than to $B$. Without carrying out any further calculations, state whether or not this will affect your answers in part (a). Give a reason for your answer.

WJEC Further Unit 6 2023 June Q2

2. You are given that the centre of mass of a uniform solid cone of height $h$ and base radius $r$ is at a height of $\frac { 1 } { 4 } h$ above its base. The diagram shows a solid conical frustum. It is formed by taking a uniform right circular cone, of base radius $3 x$ and height $6 y$, and removing a smaller cone, of base radius $x$, with the same vertex.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_490_903_1937_575} Show that the distance of the centre of mass of the frustum from its base along the axis of symmetry is $\frac { 18 } { 13 } y$.

WJEC Further Unit 6 2023 June Q3

3. The vertical motion of a point on the surface of the water in a certain harbour may be modelled as Simple Harmonic Motion about a mean level. The diagram shows that, on a particular day, the depth of water in the harbour at low tide is 2 m and the depth of the water in the harbour at high tide is 10 m . The table below shows the times of high and low tides on this day.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-4_405_912_621_233}

Write down the period and amplitude of the motion.
Let $x \mathrm {~m}$ denote the height of water above mean level $t$ hours after 5a.m. Find an expression for $x$ in terms of $t$.
The depth of water must be at least 4 m for boats to safely use the harbour. Determine the earliest time, after low tide at 5 a.m., at which boats can safely leave the harbour and hence find the latest possible time of return before the next low tide.
Calculate the rate at which the level of water is falling at 2 p.m.

WJEC Further Unit 6 2023 June Q4

4. The diagram shows three light rods $A B , B C$ and $C A$ rigidly joined together so that $A B C$ is a right-angled triangle with $A B = 45 \mathrm {~cm} , A C = 28 \mathrm {~cm}$ and $\widehat { A B } = 90 ^ { \circ }$. The rods support a uniform lamina, of density $2 m \mathrm {~kg} / \mathrm { cm } ^ { 2 }$, in the shape of a quarter circle $A D E$ with radius 12 cm and centre at the vertex $A$. Three particles are attached to $B C$ : one at $B$, one at $C$ and one at $F$, the midpoint of $B C$. The masses at $C , F$ and $B$ are $50 m \mathrm {~kg} , 30 m \mathrm {~kg}$ and $20 m \mathrm {~kg}$ respectively.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-5_604_908_756_575}

Calculate the distance of the centre of mass of the system from
1. $A C$,
2. $A B$.
When the system is freely suspended from a point $P$ on $A C$, it hangs in equilibrium with $A B$ vertical. Write down the length $A P$.
When the system is freely suspended from a point $Q$ on $A D$, it hangs in equilibrium with $Q B$ making an angle of $60 ^ { \circ }$ with the vertical. Find the distance $A Q$.

WJEC Further Unit 6 2023 June Q5

5. In this question, $\mathbf { i }$ and $\mathbf { j }$ represent unit vectors due east and due north respectively. Two smooth spheres $P$ and $Q$, of equal radii, are moving on a smooth horizontal surface. The mass of $P$ is 2 kg and the mass of $Q$ is 6 kg . The velocity of $P$ is $( 8 \mathbf { i } - 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ and the velocity of $Q$ is $( 4 \mathbf { i } + 10 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$. At a particular instant, $Q$ is positioned 12 m east and 48 m south of $P$.

Prove that $P$ and $Q$ will collide. At the instant the spheres collide, the line joining their centres is parallel to the vector $\mathbf { j }$. Immediately after the collision, sphere $Q$ has speed $5 \mathrm {~ms} ^ { - 1 }$.
Determine the coefficient of restitution between the spheres and hence calculate the velocity of $P$ immediately after the collision.
Find the magnitude of the impulse required to stop sphere $P$ after the collision.

WJEC Further Unit 6 2023 June Q6

6. The diagram on the left shows a train of mass 50 tonnes approaching a buffer at the end of a straight horizontal railway track. The buffer is designed to prevent the train from running off the end of the track. The buffer may be modelled as a light horizontal spring $A B$, as shown in the diagram on the right, which is fixed at the end $A$. The train strikes the buffer so that $P$ makes contact with $B$ at $t = 0$ seconds. While $P$ is in contact with $B$, an additional resistive force of $250000 v \mathrm {~N}$ will oppose the motion of the train, where $v \mathrm {~ms} ^ { - 1 }$ is the speed of the train at time $t$ seconds. The spring has natural length 1 m and modulus of elasticity 312500 N . At time $t$ seconds, the compression of the spring is $x$ metres.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-7_358_1506_824_283}

Show that, while $P$ is in contact with $B$, $x$ satisfies the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 20 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 25 x = 0$$
Given that, when $P$ first makes contact with $B$, the speed of the train is $U \mathrm {~ms} ^ { - 1 }$, find an expression for $x$ in terms of $U$ and $t$.
When the train comes to rest, the compression of the buffer is 0.3 m . Determine the speed of the train when it strikes the buffer.
State which type of damping is described by the motion of $P$. Give a reason for your answer.

WJEC Further Unit 6 Specimen Q1

A ball of mass 0.4 kg is thrown vertically upwards from a point $O$ with initial speed $17 \mathrm {~ms} ^ { - 1 }$. When the ball is at a height of $x \mathrm {~m}$ above $O$ and its speed is $v \mathrm {~ms} ^ { - 1 }$, the air resistance acting on the ball has magnitude $0.01 v ^ { 2 } \mathrm {~N}$.
1. Show that, as the ball is ascending, $v$ satisfies the differential equation
$$40 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \left( 392 + v ^ { 2 } \right)$$
Find an expression for $v$ in terms of $x$.
Calculate, correct to two decimal places, the greatest height of the ball.
State, with a reason, whether the speed of the ball when it returns to $O$ is greater than $17 \mathrm {~ms} ^ { - 1 }$, less than $17 \mathrm {~ms} ^ { - 1 }$ or equal to $17 \mathrm {~ms} ^ { - 1 }$.

WJEC Further Unit 6 Specimen Q2

2. (a) Prove that the centre of mass of a uniform solid cone of height $h$ and base radius $b$ is at a height of $\frac { 1 } { 4 } h$ above its base.
(b) A uniform solid cone $C _ { 1 }$ has height 3 m and base radius 2 m . A smaller cone $C _ { 2 }$ of height 2 m and base radius 1 m is contained symmetrically inside $C _ { 1 }$. The bases of $C _ { 1 }$ and $C _ { 2 }$ have a common centre and the axis of $C _ { 2 }$ is part of the axis of $C _ { 1 }$. If $C _ { 2 }$ is removed from $C _ { 1 }$, show that the centre of mass of the remaining solid is at a distance of $\frac { 11 } { 5 } \mathrm {~m}$ from the vertex of $C _ { 1 }$.
(c) The remaining solid is suspended from a string which is attached to a point on the outer curved surface at a distance of $\frac { 1 } { 3 } \sqrt { 13 } \mathrm {~m}$ from the vertex of $C _ { 1 }$. Given that the axis of symmetry is inclined at an angle of $\alpha$ to the vertical, find $\tan \alpha$.

WJEC Further Unit 6 Specimen Q3

3. A body, of mass 9 kg , is projected along a straight horizontal track with an initial speed of $20 \mathrm {~ms} ^ { - 1 }$. At time $t \mathrm {~s}$ the body experiences a resistance of magnitude $( 0.2 + 0.03 v ) \mathrm { N }$ where $v \mathrm {~ms} ^ { - 1 }$ is its speed.

Show that $v$ satisfies the differential equation $$900 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( 20 + 3 v )$$
Find an expression for $t$ in terms of $v$.
Calculate, to the nearest second, the time taken for the body to come to rest.

WJEC Further Unit 6 Specimen Q4

4. The diagram shows a uniform lamina consisting of a rectangular section $G P Q E$ with a semi-circular section EFG of radius 4 cm . Quadrants $A P B$ and $C Q D$ each with radius 2 cm are removed. Dimensions in cm are as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-3_758_604_497_651}

Write down the distance of the centre of mass of the lamina $A B C D E F G$ from $A G$.
Determine the distance of the centre of mass of the lamina $A B C D E F G$ from $B C$.
The lamina $A B C D E F G$ is suspended freely from the point $E$ and hangs in equilibrium. Calculate the angle EG makes with the vertical.

WJEC Further Unit 6 Specimen Q5

5. A particle $A$, of mass $m \mathrm {~kg}$, has position vector $11 \mathbf { i } + 6 \mathbf { j }$ and a velocity $2 \mathbf { i } + 7 \mathbf { j }$. At the same moment, second particle $B$, of mass $2 m \mathrm {~kg}$, has position vector $7 \mathbf { i } + 10 \mathbf { j }$ and a velocity $5 \mathbf { i } + 4 \mathbf { j }$.

If the particles continue to move with these velocities, prove that the particles will collide. Given that the particles coalesce after collision, find the common velocity of the particles after collision.
Determine the impulse exerted by $A$ on $B$.
Calculate the loss of kinetic energy caused by the collision.

WJEC Further Unit 6 Specimen Q6

3 marks

6. The diagram shows a playground ride consisting of a seat $P$, of mass 12 kg , attached to a vertical spring, which is fixed to a horizontal board. When the ride is at rest with nobody on it, the compression of the spring is 0.05 m .
\includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-4_305_654_1032_667} The spring is of natural length 0.75 m and modulus of elasticity $\lambda$.

Find the value of $\lambda$. The seat $P$ is now pushed vertically downwards a further 0.05 m and is then released from rest.
Show that $P$ makes Simple Harmonic oscillations of period $\frac { \pi } { 7 }$ and write down the amplitude of the motion.
Find the maximum speed of $P$.
Calculate the speed of $P$ when it is at a distance 0.03 m from the equilibrium position.
Find the distance of $P$ from the equilibrium position 1.6 s after it is released.[3]
State one modelling assumption you have made about the seat and one modelling assumption you have made about the spring.

Questions Further Unit 6 (24 questions)

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WJEC Further Unit 6 2019 June Q1

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