Questions - Edexcel M3 - A-Level Maths

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Edexcel M3 Q1

A light spring, of natural length 30 cm , is fixed in a vertical position. When a small ball of mass 0.4 kg rests on top of it, the spring is compressed by 10 cm . The ball is then held at a height of 15 cm vertically above the top of the spring and released from rest.
Calculate the maximum compression of the string in the resulting motion.
Aliya, whose mass is $m \mathrm {~kg}$, is playing rounders. She rounds the first base at a speed of $v \mathrm {~ms} ^ { - 1 }$, making the turn on a horizontal circular path of radius $r \mathrm {~m}$.
1. Write down, in terms of $m , v$ and $r$, the magnitude of the horizontal force acting on her.
2. Show that if she continues on the same circular path, the reaction force exerted on her by the ground must act at an angle $\theta$ to the vertical, where $\tan \theta = \frac { v ^ { 2 } } { g r }$.
3. A particle $P$ of mass 0.2 kg is suspended by two identical light inelastic strings, with one end of each string attached to $P$ and the other ends fixed to points $O$ and $X$ on the same horizontal level. Both strings are inclined at $30 ^ { \circ }$ to the horizontal.
4. Find the tension in the strings when $P$ is at rest.
The string $X P$ is suddenly cut, so that $P$ begins to move in a vertical circle with centre $O$.
Find the tension in the string $O P$ when it makes an angle of $60 ^ { \circ }$ with the horizontal.

Edexcel M3 Q4

4. The radius of the Earth is $R \mathrm {~m}$. The force of attraction towards the centre of the Earth experienced by a body of mass $m \mathrm {~kg}$ at a distance $x \mathrm {~m}$ from the centre is of magnitude $\frac { k m } { x ^ { 2 } } \mathrm {~N}$, where $k$ is a constant.

Show that $k = g R ^ { 2 }$. Two satellites $A$ and $B$, each of mass $m \mathrm {~kg}$, are moving in circular orbits around the Earth at distances $3 R \mathrm {~m}$ and $4 R \mathrm {~m}$ respectively from the centre of the Earth. Given that the satellites move in the same plane and that they lie along the same radial line from the centre at any time,
show that the ratio of the speed of $B$ to that of $A$ is $4 : 3$. If, in addition, the satellites are linked with a taut, straight wire in the same plane and along the same radial line,
find, in terms of $m$ and $g$, the magnitude of the force in the wire.
(8 marks) \section*{MECHANICS 3 (A) TEST PAPER 5 Page 2}

Edexcel M3 Q5

A light inelastic string of length $l \mathrm {~m}$ passes through a small smooth ring which is fixed at a point $O$ and is free to rotate about a vertical axis through $O$. Particles $P$ and $Q$, of masses 0.06 kg and 0.04 kg respectively, are attached to the ends of the string.
1. $Q$ describes a horizontal circle with centre $P$, while $P$ hangs at rest at a depth $d \mathrm {~m}$ below $O$. Show that $d = \frac { 2 l } { 5 }$.
  (6 marks)
  \includegraphics[max width=\textwidth, alt={}, center]{d8a0a9aa-a0af-48e6-a94c-12caaf0870db-2_273_365_406_1448}
2. $P$ and $Q$ now both move in horizontal circles with the same angular velocity $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$ about a vertical axis through $O$. Show that $O Q = \frac { 3 l } { 5 } \mathrm {~m}$. (7 marks)
  \includegraphics[max width=\textwidth, alt={}, center]{d8a0a9aa-a0af-48e6-a94c-12caaf0870db-2_250_433_694_1489}
3. The figure show a wine glass consisting of a hemispherical cup of radius $r$, a cylindrical solid stem of height $r$ and a circular base of radius $r$. The cup has mass $M$ and the stem has mass $m$. Modelling the cup as a thin, uniform hemispherical shell, the base as a uniform lamina made of the same thin
  \includegraphics[max width=\textwidth, alt={}, center]{d8a0a9aa-a0af-48e6-a94c-12caaf0870db-2_268_241_1005_1704}
  material as the cup, and the stem as a uniform solid cylinder,
4. show that the mass of the circular base is $\frac { 1 } { 2 } M$.
Given that the centre of mass of the glass is at a distance $\frac { 13 r } { 14 }$ from the base along the vertical axis of symmetry,
express $M$ in terms of $m$. If the cup is now filled with liquid whose mass is $2 M$,
show that the position of the centre of mass rises through a distance $\frac { 13 r } { 35 }$.
State an assumption that you have made about the liquid.

Edexcel M3 Q7

7. A particle of mass $m \mathrm {~kg}$ is attached to one end of an elastic string of natural length $l \mathrm {~m}$ and modulus of elasticity $\lambda \mathrm { N }$. The other end of the string is attached to a fixed point $O$. The particle hangs in equilibrium at a point $C$.

1. Prove that if the particle is slightly displaced in a vertical direction, it will perform simple harmonic motion about $C$.
2. Find the period, $T \mathrm {~s}$, of the motion in terms of $l , m$ and $\lambda$.
3. Explain the significance of the term 'slightly' as used in (i) above. When an additional mass $M$ is attached to the particle, it is found that the system oscillates about a point $D$, at a distance $d$ below $C$, with period $T _ { 1 } \mathrm {~s}$.
1. Write down an expression for $T _ { 1 }$ in terms of $l , m , M$ and $\lambda$.
2. Hence show that $T _ { 1 } ^ { 2 } - T ^ { 2 } = \frac { 4 \pi ^ { 2 } d } { g }$.

Edexcel M3 Q1

A particle of mass $m \mathrm {~kg}$ is attached to one end of a light inextensible string of length $l \mathrm {~m}$ whose other end is fixed to a point $O$. The particle is made to move in a vertical circle with centre $O$, with constant angular velocity $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$. At a certain instant it is in the position shown, where the string makes an angle $\theta$ radians with the downward vertical through $O$.
\includegraphics[max width=\textwidth, alt={}, center]{3321c06a-29c3-430a-99a8-ec3a245abf10-1_341_328_347_1631}
1. Find an expression, in terms of $m , l$ and $\omega$, for the kinetic energy of the particle at this instant.
2. Find an expression, in terms of $m , g , l$ and $\theta$, for the potential energy of the particle relative to the horizontal plane through the lowest point $A$.
3. Determine the position of the particle when the rate of increase of its total energy, with respect to time, is a maximum.
4. A particle moves along a straight line in such a way that its displacement $x \mathrm {~m}$ from a fixed point $O$ on the line, at time $t$ seconds after it leaves $O$, is given by $x = p \sin \omega t + q \cos \omega t$ where $p , q$ and $\omega$ are constants.
5. Show that the motion of the particle is simple harmonic.
6. If the particle leaves $O$ with speed $15 \mathrm {~ms} ^ { - 1 }$, and $\omega = 3$, find the amplitude of the motion.
7. A particle $P$ of mass 0.2 kg moves in a horizontal circle on one end of an elastic string whose other end is attached to a fixed point $O$. The angular velocity of $P$ is $\pi \mathrm { rad } \mathrm { s } ^ { - 1 }$. The natural length of the string is 1 m and, while $P$ is in motion, the distance $O P = 1.15 \mathrm {~m}$.
8. Calculate, to 3 significant figures, the modulus of elasticity of the string.
The motion now ceases and $P$ hangs at rest vertically below $O$.
Show that the extension in the string in this position is about 13 cm .

Edexcel M3 Q4

4. A small stone $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light elastic string of modulus 3 mg N and natural length $2 l \mathrm {~m}$. The other end of the string is fixed to a point $O$ at a height $3 l \mathrm {~m}$ above a horizontal surface. $P$ is released from rest at $O$; it hits the surface and rebounds to a height of $2 l \mathrm {~m}$. The coefficient of restitution between $P$ and the surface is $e$.
Calculate the value of $e$. State one assumption (other than the string being light) that you have used in your solution.
(1 mark) \section*{MECHANICS 3 (A) TEST PAPER 6 Page 2}

Edexcel M3 Q5

A small sphere $S$, of mass $m \mathrm {~kg}$ is released from rest at the surface of a liquid in a right circular cylinder whose axis is vertical. When $S$ is moving downwards with speed $v \mathrm {~ms} ^ { - 1 }$, the viscous resistive force acting upwards on it has magnitude $v ^ { 2 } \mathrm {~N}$.
1. Write down a differential equation for the motion of $S$, clearly defining any symbol(s) that you introduce.
2. Find, in terms of $m$, the distance $S$ has fallen when its speed is $\sqrt { \frac { m g } { 2 } } \mathrm {~ms} ^ { - 1 }$.
3. The diagram shows two identical particles, each of mass $m \mathrm {~kg}$, connected by a thin, light inextensible string. $P$ slides on the surface of a smooth right circular cylinder fixed with its axis, through $O$, horizontal. $Q$ moves vertically. $O P$ makes an angle $\theta$ radians with the horizontal.
The system is released from rest in the position where $\theta = 0$.
\includegraphics[max width=\textwidth, alt={}, center]{3321c06a-29c3-430a-99a8-ec3a245abf10-2_355_307_818_1629}
Show that the vertical distance moved by $Q$ is $\frac { \theta } { \sin \theta }$ times the vertical distance moved by $P$.
In the position where $\theta = \frac { \pi } { 6 }$, prove that the reaction of the cylinder on $P$ has magnitude $\left( 1 - \frac { \pi } { 6 } \right) m g \mathrm {~N}$.

Edexcel M3 Q7

7. A container consists of two sections made from the same material : a hollow portion formed by removing a cone (shaded in the figure) from a solid cylinder of radius $r$ and height $h$, and a solid hemisphere of radius $r$. The vertex of the removed cone coincides with the centre $O$ of the horizontal plane face of the hemisphere. $C D$ is a
\includegraphics[max width=\textwidth, alt={}, center]{3321c06a-29c3-430a-99a8-ec3a245abf10-2_332_350_1581_1604}
diameter of this plane face.

Show that the distance of the centre of mass of the container from the plane face of the hemisphere is $\left| \frac { 3 } { 8 } ( h - r ) \right|$ Explain why the modulus sign is necessary.
Find the ratio $h : r$ in each of the following cases :
1. When the container is suspended from the point $C$, the angle made by $C D$ with the vertical is equal to the angle which $C D$ would make with the vertical if the hemisphere alone were suspended from $C$.
2. The container is able to stand without toppling in any position when it is placed with the surface of the hemispherical part in contact with a smooth horizontal table.
  (3 marks)

Edexcel M3 Q1

A particle of mass 0.6 kg moves in a horizontal circle with constant angular speed 1.5 radians per second. If the force directed towards the centre of the circle has magnitude 0.27 N , find the radius of the circular path.
The diagram shows a particle of mass 0.7 kg resting on a rough horizontal table. The coefficient of friction between the particle and the table is 0.25 . A light elastic string, of natural length 50 cm and modulus of elasticity 6.86 N , is attached to the particle. The string is kept at an angle of $60 ^ { \circ }$ to the horizontal and is gradually extended by pulling on it until the particle moves. Show that the particle starts to move when the extension in the string is 17 cm .
A smooth circular hoop of radius 1 m , with centre $O$, is fixed in a vertical plane. A small ring $Q$, of mass 0.1 kg , is threaded onto the hoop and held so that the angle $Q O H = 30 ^ { \circ }$, where $H$ is the highest point of the hoop. $Q$ is released from rest at this position. Find, in terms of $g$,
1. the horizontal and vertical components of the acceleration of $Q$ when it reaches the lowest
2. A particle $P$ moves with simple harmonic motion in a straight line, with the centre of motion at the point $O$ on the line. $A$ and $B$ are on opposite sides of $O$, with $O A = 4 \mathrm {~m} , O B = 6 \mathrm {~m}$.
  When passing through $A$ and $B , P$ has speed $6 \mathrm {~ms} ^ { - 1 }$ and $4 \mathrm {~ms} ^ { - 1 }$ respectively.
3. Find the amplitude of the motion.
4. Show that the period of motion is $2 \pi \mathrm {~s}$.
5. (a) Prove that the centre of mass of a uniform solid hemisphere of radius $r$ is at a distance $\frac { 3 r } { 8 }$ from its plane face.
A solid cylinder of radius $\frac { 3 r } { 4 }$ and height $k r$, where $k < 1$, is welded to a uniform hemisphere of radius $r$ made of the same material, so that their axes of symmetry coincide. The figure shows the cross section of the resulting solid. If the centre of mass of this solid is at $O$, the centre of the plane face of the hemisphere,
\includegraphics[max width=\textwidth, alt={}, center]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-1_191_172_2298_1805}
find the value of $k$.

Edexcel M3 Q3

3. A smooth circular hoop of radius 1 m , with centre $O$, is fixed in a vertical plane. A small ring $Q$, of mass 0.1 kg , is threaded onto the hoop and held so that the angle $Q O H = 30 ^ { \circ }$, where $H$ is the highest point of the hoop. $Q$ is released from rest at this position. Find, in terms of $g$,

the horizontal and vertical components of the acceleration of $Q$ when it reaches the lowest
\includegraphics[max width=\textwidth, alt={}, center]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-1_271_458_620_1528}
horzonal and graduals moves. Show that the particle starts to move when the extension in the string is 17 cm . point of the hoop;
the magnitude of the reaction between $Q$ and the hoop at this lowest point. a
point of the hoop; the magnitude of the reaction between $Q$ and the hoop at this lowest point.\includegraphics[max width=\textwidth, alt={}]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-1_46_1257_1469_468} $$\sum \cos + 20 + 2$$ ,
\includegraphics[max width=\textwidth, alt={}, center]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-1_90_327_1622_1604}
\includegraphics[max width=\textwidth, alt={}, center]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-1_200_81_1512_1850} \section*{MECHANICS 3 (A)TEST PAPER 7 Page 2}

Edexcel M3 Q5

continued...

The solid now stands on the base of the cylindrical portion, whose diameter is $A B$, and is gently tilted about $A$.
\includegraphics[max width=\textwidth, alt={}, center]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-2_227_237_219_1745}
(c) Find the angle between $A B$ and the horizontal when it is on the point of toppling.

Edexcel M3 Q6

6. The gravitational attraction $F \mathrm {~N}$ between two point masses $m _ { 1 } \mathrm {~kg}$ and $m _ { 2 } \mathrm {~kg}$ at a distance $x \mathrm {~m}$ apart is given by $F = \frac { k m _ { 1 } m _ { 2 } } { x ^ { 2 } }$, where $k$ is a constant. Given that a small body of mass 1 kg experiences a force of $g \mathrm {~N}$ at the surface of the Earth, which has radius $R \mathrm {~m}$ and mass $M \mathrm {~kg}$,

show that $k = \frac { g R ^ { 2 } } { M }$. A small communications satellite of mass $m \mathrm {~kg}$ is put into a circular orbit of radius $r \mathrm {~m}$ around the Earth. Modelling the Earth as a particle of mass $M \mathrm {~kg}$, and using the value of $k$ from (a), (b) prove that the period of rotation, $T \mathrm {~s}$, of the satellite is given by $T = \frac { 2 \pi } { R } \sqrt { \frac { r ^ { 3 } } { g } }$. (4 marks) To cover transmission to any point on the Earth, three small satellites $X , Y$ and $Z$, each of mass $m \mathrm {~kg}$, are placed in a common circular orbit of radius $r$ and form an equilateral triangle as shown.
\includegraphics[max width=\textwidth, alt={}, center]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-2_223_238_1215_1709}
Show that the period of rotation of $X$ is given by $T \sqrt { \frac { 3 M } { 3 M + m \sqrt { 3 } } } \mathrm {~s}$, where $T \mathrm {~s}$ is the period found in (b).

Edexcel M3 Q7

7. One end of a light elastic string, of natural length $3 l \mathrm {~m}$, is attached to a fixed point $O$. A particle of mass $m \mathrm {~kg}$ is attached to the other end of the string. When the particle hangs freely in equilibrium, the string is extended by a length of $l \mathrm {~m}$. The particle is then pulled down through a further distance $2 l \mathrm {~m}$ and released from rest.

Prove that as long as the string is taut, the particle performs simple harmonic motion about its equilibrium position.
Show that the time between the release of the particle and the instant when the string becomes slack is $\frac { 2 } { 3 } \pi \sqrt { \frac { l } { g } } \mathrm {~s}$.
Find the greatest height reached by the particle above its point of release.
Show that the time $T$ s taken to reach this greatest height from the moment of release is given by $T = \left( \frac { 2 \pi } { 3 } + \sqrt { 3 } \right) \sqrt { \frac { l } { g } }$.
(4 marks)

Edexcel M3 Q1

A particle $P$ of mass $m \mathrm {~kg}$ moves in a horizontal circle at one end of a light elastic string of natural length $l \mathrm {~m}$ and modulus of elasticity $m g \mathrm {~N}$. The other end of the string is attached to a fixed point $O$. Given that the string makes an angle of $60 ^ { \circ }$ with the vertical,
1. show that $O P = 31 \mathrm {~m}$.
2. Find, in terms of $l$ and $g$, the angular speed of $P$.
3. A particle $P$ of mass $m \mathrm {~kg}$ moves vertically upwards under gravity, starting from ground level. It is acted on by a resistive force of magnitude $m \mathrm { f } ( x ) \mathrm { N }$, where $\mathrm { f } ( x )$ is a function of the height $x \mathrm {~m}$ of $P$ above the ground. When $P$ is at this height, its upward speed $v \mathrm {~ms} ^ { - 1 }$ is given by $v ^ { 2 } = 2 \mathrm { e } ^ { - 2 g x } - 1$.
4. Write down a differential equation for the motion of $P$ and hence determine $\mathrm { f } ( x )$ in terms of $g$ and $x$.
5. Show that the greatest height reached by $P$ above the ground is $\frac { 1 } { 2 g } \ln 2 \mathrm {~m}$.
Given that the work, in J , done by $P$ against the resisting force as it moves from ground level to a point $H \mathrm {~m}$ above the ground is equal to $\int _ { 0 } ^ { H } m \mathrm { f } ( x ) \mathrm { d } x$,
show that the total work done by $P$ against the resistance during its upward motion is $\frac { 1 } { 2 } m ( 1 - \ln 2 ) \mathrm { J }$.

Edexcel M3 Q3

3. A car of mass $m \mathrm {~kg}$ moves round a curve of radius $r \mathrm {~m}$ on a road which is banked at an angle $\theta$ to the horizontal. When the speed of the car is $u \mathrm {~ms} ^ { - 1 }$, the car experiences no sideways frictional force. Given that $\tan \theta = \frac { u ^ { 2 } } { g r }$, show that the sideways frictional force on the car when its speed is $\frac { u } { 2 } \mathrm {~ms} ^ { - 1 }$ has magnitude $\frac { 3 } { 4 } m g \sin \theta \mathrm {~N}$.

Edexcel M3 Q4

4. Two light elastic strings, each of length $l \mathrm {~m}$ and modulus of elasticity $\lambda \mathrm { N }$,are attached to a particle $P$ of mass $m \mathrm {~kg}$. The other ends of the strings are attached to fixed points $A$ and $B$ on the same horizontal level,
\includegraphics[max width=\textwidth, alt={}, center]{45b4918e-5d4c-470d-b725-e3b46900d190-1_202_362_2068_1633}
where $A B = 2 l \mathrm {~m} . P$ is held vertically below the mid-point of $A B$, with each string taut and inclined at $30 ^ { \circ }$ to the horizontal, and released from rest. Given that $P$ comes to instantaneous rest when each string makes an angle of $60 ^ { \circ }$ with the horizontal, show that $\lambda = \frac { 3 m g } { 6 - 2 \sqrt { } 3 }$. \section*{MECHANICS 3 (A) TEST PAPER 8 Page 2}

Edexcel M3 Q5

A particle $P$ is projected horizontally with speed $u \mathrm {~ms} ^ { - 1 }$ from the highest point of a smooth sphere of radius $r \mathrm {~m}$ and centre $O$. It moves on the surface in a vertical plane, and at a particular instant the radius $O P$ makes an angle $\theta$ with the upward vertical, as shown. At this instant $P$ has speed $v \mathrm {~ms} ^ { - 1 }$ and
\includegraphics[max width=\textwidth, alt={}, center]{45b4918e-5d4c-470d-b725-e3b46900d190-2_264_261_294_1693}
the magnitude of the reaction between $P$ and the sphere is $X \mathrm {~N}$.
1. Assuming that $u ^ { 2 } < g r$, show that
  1. $v ^ { 2 } = u ^ { 2 } + 2 g r ( 1 - \cos \theta )$,
  2. $X = m g \left( 3 \cos \theta - 2 - \frac { y ^ { 2 } } { g r } \right)$.
    (2 marks)
    (4 marks)
2. Show that $P$ leaves the surface of the sphere when $\cos \theta = \frac { u ^ { 2 } + 2 g r } { 3 g r }$.
3. Discuss what happens if $u ^ { 2 } \geq g r$.
4. A particle $P$ of mass $m \mathrm {~kg}$ hangs in equilibrium at one end of a light spring, of natural length $l \mathrm {~m}$ and modulus of elasticity $\lambda \mathrm { N }$, whose other end is fixed at a point vertically above $P$. In this position the length of the spring is $( l + e ) \mathrm { m }$. When $P$ is displaced vertically through a small distance and released, it performs simple harmonic motion with 5 oscillations per second.
5. Show that $\frac { \lambda } { l } = 100 \pi ^ { 2 } \mathrm {~m}$.
6. Express $e$ in terms of $g$.
7. Determine, in terms of $m$ and $l$, the magnitude of the tension in the spring when it is stretched to twice its natural length.
8. (a) Prove that the centre of mass of a uniform solid right circular cone of height $h$ and base radius $r$ is at a distance $\frac { 3 h } { 4 }$ from the vertex.
An item of confectionery consists of a thin wafer in the form of a hollow right circular cone of height $h$ and mass $m$, filled with solid chocolate, also of mass $m$, to a depth of $k h$ as shown. The centre of mass of the item is at $O$, the centre of the horizontal plane face
\includegraphics[max width=\textwidth, alt={}, center]{45b4918e-5d4c-470d-b725-e3b46900d190-2_314_332_1896_1636}
of the chocolate.
Show that $k = \frac { 8 h } { 15 }$. (3 marks) In the packaging process, the cone has to move on a conveyor belt inclined at an angle $\alpha$ to the horizontal as shown. If the belt is rough enough to prevent sliding, and the maximum value of $\alpha$ for which
\includegraphics[max width=\textwidth, alt={}, center]{45b4918e-5d4c-470d-b725-e3b46900d190-2_284_445_2271_1560}
the cone does not topple is $45 ^ { \circ }$,
find the radius of the base of the cone in terms of $h$.

Edexcel M3 Q1

A small bead is threaded onto a smooth circular hoop, of radius $r \mathrm {~m}$, fixed in a vertical plane. It is projected with speed $u \mathrm {~ms} ^ { - 1 }$ from the lowest point of the hoop. Find $u$ in terms of $g$ and $r$ if
1. the bead just reaches the highest point of the hoop,
2. the reaction on the bead is zero when it is at the highest point of the hoop.
3. An ornamental tower is made from a solid right circular cylinder of mass $M$ and height $h$ by removing three identical cylindrical sections, each of height $\frac { h } { 8 }$, equally spaced above a base of height $\frac { h } { 4 }$, as shown. The tower is held in position by light, thin vertical strips $A B$ and $C D$.
  \includegraphics[max width=\textwidth, alt={}, center]{9699f53e-366b-4064-af9f-89992c5e93b7-1_312_328_703_1622}
Find the distance of the centre of mass of the tower from its horizontal base.

Edexcel M3 Q3

3. Two particles $A$ and $B$, of masses $M \mathrm {~kg}$ and $m \mathrm {~kg}$ respectively, are connected by a light inextensible string passing over a smooth fixed pulley. $B$ is placed on a smooth horizontal table and $A$ hangs freely, as shown. $B$ is attached to a spring of natural length $l \mathrm {~m}$ and modulus of elasticity $\lambda \mathrm { N }$, whose other end is fixed to a vertical wall.
\includegraphics[max width=\textwidth, alt={}, center]{9699f53e-366b-4064-af9f-89992c5e93b7-1_333_405_1160_1572}
The system starts to move from rest when the string is taut and the spring neither extended nor compressed. $A$ does not reach the ground, nor does $B$ reach the pulley, during the motion.

Show that the maximum extension of the spring is $\frac { 2 M g l } { \lambda } \mathrm {~m}$.
If $M = 3 , m = 1.5$ and $\lambda = 35 l$, find the speed of $A$ when the extension in the spring is 0.5 m .

Edexcel M3 Q4

4. A particle $P$ of mass $m \mathrm {~kg}$ moves along a straight line under the action of a force of magnitude $\frac { k m } { x ^ { 2 } } \mathrm {~N}$, where $k$ is a constant, directed towards a fixed point $O$ on the line, where $O P = x \mathrm {~m} P$ starts from rest at $A$, at a distance $a \mathrm {~m}$ from $O$. When $O P = x \mathrm {~m}$, the speed of $P$ is $v \mathrm {~ms} ^ { - 1 }$.

Show that $v = \sqrt { \frac { 2 k ( a - x ) } { a x } }$.
$B$ is the point half-way between $O$ and $A$. When $k = \frac { 1 } { 2 }$ and $a = 1$, the time taken by $P$ to travel from $A$ to $B$ is $T$ seconds
Assuming the result that, for $0 \leq x \leq 1 , \int \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x = \arcsin ( \sqrt { } x ) - \sqrt { } \left( x - x ^ { 2 } \right) +$ constant,
find the value of $T$. \section*{MECHANICS 3 (A) TEST PAPER 9 Page 2}

Edexcel M3 Q5

A car moves round a circular racing track of radius 100 m , which is banked at an angle of $4 ^ { \circ }$ to the horizontal.
1. Show that when its speed is $8.28 \mathrm {~ms} ^ { - 1 }$, there is no sideways force acting on the car.
  (4 marks)
2. When the speed of the car is $12.5 \mathrm {~ms} ^ { - 1 }$, find the smallest value of the coefficient of friction between the car and the track which will prevent side-slip.
3. The diagram shows a particle $P$ of mass $m \mathrm {~kg}$ moving on the inner surface of a smooth fixed hemispherical bowl of radius $r \mathrm {~m}$ which is fixed with its axis vertical. $P$ moves at a constant speed in a horizontal circle, at a depth $h \mathrm {~m}$ below the top of the bowl.
  \includegraphics[max width=\textwidth, alt={}, center]{9699f53e-366b-4064-af9f-89992c5e93b7-2_254_431_786_1528}
4. Show that the force $R$ exerted on $P$ by the bowl has magnitude $\frac { m g r } { h } \mathrm {~N}$.
5. Find, in terms of $g , h$ and $r$, the constant speed of $P$.
The bowl is now inverted and $P$ moves on the smooth outer surface at a height $h$ above the plane face under the action of a force of magnitude $m g$ applied tangentially as shown. The reaction of the
\includegraphics[max width=\textwidth, alt={}, center]{9699f53e-366b-4064-af9f-89992c5e93b7-2_209_355_1224_1636}
surface of the sphere on $P$ now has magnitude $S \mathrm {~N}$.
Given that $r = 2 h$, prove that $S < \frac { 1 } { 6 } R$.

Edexcel M3 Q7

7. A particle $P$ of mass $m \mathrm {~kg}$ is fixed to one end of a light elastic string of modulus $m g \mathrm {~N}$ and natural length $l \mathrm {~m}$. The other end of the string is attached to a fixed point $O$ on a rough horizontal table. Initially $P$ is at rest in limiting equilibrium on the table at the point $X$ where $O X = \frac { 5 l } { 4 } \mathrm {~m}$.

Find the coefficient of friction between $P$ and the table.
$P$ is now given a small displacement $x \mathrm {~m}$ horizontally along $O X$, away from $O$. While $P$ is in motion, the frictional resistance remains constant at its limiting value.
Show that as long as the string remains taut, $P$ performs simple harmonic motion with $X$ as the centre. If $P$ is held at the point where the extension in the string is $l m$ and then released,
show that the string becomes slack after a time $\left( \frac { \pi } { 2 } + \arcsin \left( \frac { 1 } { 3 } \right) \right) \sqrt { \frac { l } { g } } \mathrm {~s}$.
Determine the speed of $P$ when it reaches $O$.

Edexcel M3 Q1

A cyclist travels on a banked track inclined at $8 ^ { \circ }$ to the horizontal. He moves in a horizontal circle of radius 10 m at a constant speed of $v \mathrm {~ms} ^ { - 1 }$. If there is no sideways frictional force on the cycle, calculate the value of $v$.
The figure shows a particle $P$, of mass 0.8 kg , attached to the ends of two light elastic strings. $A P$ has natural length 20 cm and modulus of elasticity $\lambda \mathrm { N } . B P$ has natural length 20 cm and modulus of of elasticity $\mu \mathrm { N } . A$ and $B$ are fixed to points on the same horizontal
\includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-1_211_287_726_1704}
level so that $A B = 50 \mathrm {~cm}$. When $P$ is suspended in equilibrium, $A P =$ 30 cm and $B P = 40 \mathrm {~cm}$. Calculate the values of $\lambda$ and $\mu$.
Suraiya, whose mass is $m \mathrm {~kg}$, takes a running jump into a swimming pool so that she begins to swim in a straight line with speed $0.2 \mathrm {~ms} ^ { - 1 }$. She continues to move in the same straight line, the only force acting on her being a resistance of magnitude $m v ^ { 2 } \sin \left( \frac { t } { 100 } \right) \mathrm { N }$, where $v \mathrm {~ms} ^ { - 1 }$ is her speed at time $t$ seconds after entering the pool and $0 \leq t \leq 50 \pi$.
1. Find an expression for $v$ in terms of $t$.
2. Calculate her greatest and least speeds during her motion.
3. A uniform lamina is in the shape of the region enclosed by the coordinate axes and the curve with equation $y = 1 + \cos x$, as shown.
4. Show by integration that the centre of mass of the lamina is at a distance $\frac { \pi ^ { 2 } - 4 } { 2 \pi }$ from the $y$-axis.
  (9 marks)
  \includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-1_314_410_1677_1512}
Given that the centre of mass is at a distance 0.75 units from the $x$-axis, and that $P$ is the point $( 0,2 )$ and $O$ is the origin $( 0,0 )$,
find, to the nearest degree, the angle between the line $O P$ and the vertical when the lamina is freely suspended from $P$.

Edexcel M3 Q5

5. A particle $P$, of mass 0.5 kg , rests on the surface of a rough horizontal table. The coefficient of friction between $P$ and the table is $0.5 . P$ is connected to a particle $Q$, of mass 0.2 kg , by a light inextensible string passing through
\includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-1_280_519_2367_1499}
[0pt] [Turn over ... \section*{MECHANICS 3 (A) TEST PAPER 10 Page 2}

continued ...
a small smooth hole at a point $O$ on the table, such that the distance $O Q$ is $0.4 \mathrm {~m} . Q$ moves in a horizontal circle while $P$ remains in limiting equilibrium.
1. Calculate the angle $\theta$ which $O Q$ makes with the vertical.
2. Show that the speed of $Q$ is $1.33 \mathrm {~ms} ^ { - 1 }$.
The motion is altered so that $Q$ hangs at rest below $O$ and $P$ moves in a horizontal circle on the table with speed $0.84 \mathrm {~ms} ^ { - 1 }$, at a constant distance $r \mathrm {~m}$ from $O$ but tending to slip away from $O$.
Find the value of $r$.

Edexcel M3 Q6

6. The figure shows a swing consisting of two identical vertical light springs attached symmetrically to a light horizontal cross-bar and supported from a strong fixed horizontal beam. When a mass of 24 kg is placed at the mid-
\includegraphics[max width=\textwidth, alt={}, center]{627b3411-07ba-4ee4-a672-93a64eeb90b3-2_200_318_923_1668}
point of the cross-bar, both springs extend by 30 cm to the position $A$, as shown. Each spring has natural length $l \mathrm {~m}$ and modulus of elasticity $\lambda \mathrm { N }$.

Show that $\lambda = 392 l$. The 24 kg mass is left on the bar and the bar is then displaced downwards by a further 20 cm .
Prove that the system comprising the bar and the mass now performs simple harmonic motion with the centre of oscillation at the level $A$.
Calculate the number of oscillations made per second in this motion.
Find the maximum acceleration which the mass experiences during the motion.

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