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Edexcel M3 Q5

10 marks Standard +0.3

The diagram shows a uniform solid right circular cone of mass $m \mathrm {~kg}$, height $h \mathrm {~m}$ and base radius $r \mathrm {~m}$ suspended by two vertical strings attached to the points $P$ and $Q$ on the circumference of the base. The vertex $O$ of the cone is vertically below $P$.
1. Show that the tension in the string attached at $Q$ is $\frac { 3 m g } { 8 } \mathrm {~N}$. \includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_296_277_269_1668}
2. Find, in terms of $m$ and $g$, the tension in the other string.
3. Two identical particles $P$ and $Q$ are connected by a light inextensible string passing through a small smooth-edged hole in a smooth table, as shown. $P$ moves on the table in a horizontal circle of radius 0.2 m and $Q$ hangs at rest. \includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_309_430_859_1476}
4. Calculate the number of revolutions made per minute by $P$.
  (5 marks) $Q$ is now also made to move in a horizontal circle of radius 0.2 m below the table. The part of the string between $Q$ and the table makes an angle of $45 ^ { \circ }$ with the vertical.
5. Show that the numbers of revolutions per minute made by $P$ and $Q$ respectively are in the ratio $2 ^ { 1 / 4 } : 1$. \includegraphics[max width=\textwidth, alt={}, center]{309da227-759c-475e-b12e-dcd9e338a417-2_293_428_1213_1499}
6. A particle $P$ of mass $m \mathrm {~kg}$ is fixed to one end of a light elastic string of natural length $l \mathrm {~m}$ and modulus of elasticity $k m g \mathrm {~N}$. The other end of the string is fixed to a point $X$ on a horizontal plane. $P$ rests at $O$, where $O X = l \mathrm {~m}$, with the string just taut. It is then pulled away from $X$ through a distance $\frac { 3 l } { 4 } \mathrm {~m}$ and released from rest. On this side of $O$, the plane is smooth.
7. Show that, as long as the string is taut, $P$ performs simple harmonic motion.
8. Given that $P$ first returns to $O$ with speed $\sqrt { } ( g l ) \mathrm { ms } ^ { - 1 }$, find the value of $k$.
9. On the other side of $O$ the plane is rough, the coefficient of friction between $P$ and the plane being $\mu$. If $P$ does not reach $X$ in the subsequent motion, show that $\mu > \frac { 1 } { 2 }$. ( 4 marks)
10. If, further, $\mu = \frac { 3 } { 4 }$, show that the time which elapses after $P$ is released and before it comes to rest is $\frac { 1 } { 24 } ( 9 \pi + 32 ) \sqrt { \frac { l } { g } }$ s.
  (6 marks)

Edexcel M3 Q1

7 marks Standard +0.3

A motorcyclist rides in a cylindrical well of radius 5 m . He maintains a horizontal circular path at a constant speed of $10 \mathrm {~ms} ^ { - 1 }$. The coefficient of friction between the wall and the wheels of the cycle is $\mu$.
Modelling the cyclist and his machine as a particle in contact \includegraphics[max width=\textwidth, alt={}, center]{d1acee30-26a7-44d0-b4f5-ab721451b8b8-1_359_263_370_1595}
with the wall, show that he will not slip downwards provided that $\mu \geq 0.49$.
A particle $P$ moves with simple harmonic motion in a straight line. The centre of oscillation is $O$. When $P$ is at a distance 1 m from $O$, its speed is $8 \mathrm {~ms} ^ { - 1 }$. When it is at a distance 2 m from $O$, its speed is $4 \mathrm {~ms} ^ { - 1 }$.
1. Find the amplitude of the motion.
2. Show that the period of motion is $\frac { \pi } { 2 } \mathrm {~s}$.
3. A particle of mass $m \mathrm {~kg}$ is attached to the end $B$ of a light elastic string $A B$. The string has natural length $l \mathrm {~m}$ and modulus of elasticity $\lambda . \mathrm { N }$.
The end $A$ is attached to a fixed point on a smooth plane \includegraphics[max width=\textwidth, alt={}, center]{d1acee30-26a7-44d0-b4f5-ab721451b8b8-1_289_543_1329_1425}
inclined at an angle $\alpha$ to the horizontal, as shown, and the particle rests in equilibrium with the length $A B = \frac { 5 l } { 4 } \mathrm {~m}$.

Edexcel M3 Q4

9 marks Standard +0.3

4. The acceleration $a \mathrm {~ms} ^ { - 2 }$ of a particle $P$ moving in a straight line away from a fixed point $O$ is given by $a = \frac { k } { 1 + t }$, where $t \mathrm {~s}$ is the time that has elapsed since $P$ left $O$, and $k$ is a constant.

By solving a suitable differential equation, find an expression for the velocity $v \mathrm {~ms} ^ { - 1 }$ of $P$ in terms of $t , k$ and another constant $c$. Given that $v = 0$ when $t = 0$ and that $v = 4$ when $t = 2$,
show that $v \ln 3 = 4 \ln ( 1 + t )$.
Calculate the time when $P$ has a speed of $8 \mathrm {~ms} ^ { - 1 }$. \section*{MECHANICS 3 (A)TEST PAPER 4 Page 2}

Edexcel M3 Q5

13 marks Standard +0.3

A particle of mass $m \mathrm {~kg}$, at a distance $x \mathrm {~m}$ from the centre of the Earth, experiences a force of magnitude $\frac { k m } { x ^ { 2 } } \mathrm {~N}$ towards the centre of the Earth, where $k$ is a constant. Given that the radius of the Earth is $6.37 \times 10 ^ { 6 } \mathrm {~m}$, and that a 3 kg mass experiences a force of 30 N at the surface of the Earth,
1. calculate the value of $k$, stating the units of your answer.
The 3 kg mass falls from rest at a distance $x = 12.74 \times 10 ^ { 6 } \mathrm {~m}$ from the centre of the Earth. Ignoring air resistance,
show that it reaches the surface of the Earth with speed $7.98 \times 10 ^ { 3 } \mathrm {~ms} ^ { - 1 }$. In a simplified model, the particle is assumed to fall with a constant acceleration $10 \mathrm {~ms} ^ { - 2 }$. According to this model it attains the same speed as in (b), $7.98 \times 10 ^ { 3 } \mathrm {~ms} ^ { - 1 }$, at a distance $( 12 \cdot 74 - d ) \times 10 ^ { 6 } \mathrm {~m}$ from the centre of the Earth.
Find the value of $d$.

Edexcel M3 Q6

15 marks Challenging +1.2

6. A particle $P$ of mass 0.4 kg hangs by a light, inextensible string of length 20 cm whose other end is attached to a fixed point $O$. It is given a horizontal velocity of $1.4 \mathrm {~ms} ^ { - 1 }$ so that it begins to move in a vertical circle. If in the ensuing motion the string makes an angle of $\theta$ with the downward vertical through $O$, show that

$\theta$ cannot exceed $60 ^ { \circ }$,
the tension, $T \mathrm {~N}$, in the string is given by $T = 3.92 ( 3 \cos \theta - 1 )$. If the string breaks when $\cos \theta = \frac { 3 } { 5 }$ and $P$ is ascending,
find the greatest height reached by $P$ above the initial point of projection.

Edexcel M3 Q7

15 marks Challenging +1.8

7. A uniform solid sphere, of radius $a$, is divided into two sections by a plane at a distance $\frac { a } { 2 }$ from the centre and parallel to a diameter.

Show that the centre of gravity of the smaller cap from its plane face is $\frac { 7 a } { 40 }$. This smaller cap is now placed on an inclined plane whose angle of inclination to the horizontal is $\theta$. The plane is rough enough to prevent slipping and the cap rests with its curved surface in contact with the plane.
If the maximum value of $\theta$ for which this is possible without the cap turning over is $30 ^ { \circ }$, find the corresponding maximum inclination of the axis of symmetry of the cap to the vertical.
(6 marks)

Edexcel M3 Q1

7 marks Standard +0.3

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

11 marks Challenging +1.2

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

13 marks Standard +0.8

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

15 marks Challenging +1.2

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

7 marks Standard +0.3

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

10 marks Challenging +1.2

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

13 marks Challenging +1.2

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

16 marks Challenging +1.8

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

3 marks Moderate -0.3

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

8 marks Standard +0.3

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

14 marks Challenging +1.2

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

16 marks Challenging +1.8

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

17 marks Challenging +1.8

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

8 marks Challenging +1.2

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 $\frac { 1 } { 2 } m ( 1 - \ln 2 ) \mathrm { J }$.

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

7 marks Standard +0.3

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

9 marks Standard +0.8

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 .

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