Questions - Edexcel M3 - A-Level Maths

the modulus of elasticity of the rope,
(5)
the speed of the man when he is 2 m above the ground, giving your answer in $\mathrm { m } \mathrm { s } ^ { - 1 }$ to 3 significant figures.
(5)

Edexcel M3 Specimen Q5

5.
\includegraphics[max width=\textwidth, alt={}, center]{e256678d-89e8-48eb-aa8a-b8e027b62ef1-3_423_357_918_847} A uniform solid, $S$, is placed with its plane face on horizontal ground. The solid consists of a right circular cylinder, of radius $r$ and height $r$, joined to a right circular cone, of radius $r$ and height $h$. The plane face of the cone coincides with one of the plane faces of the cylinder, as shown in Fig. 3.

Show that the distance of the centre of mass of $S$ from the ground is $$\frac { 6 r ^ { 2 } + 4 r h + h ^ { 2 } } { 4 ( 3 r + h ) }$$ (8) The solid is now placed with its plane face on a rough plane which is inclined at an angle $\alpha$ to the horizontal. The plane is rough enough to prevent $S$ from sliding. Given that $h = 2 r$, and that $S$ is on the point of toppling,
find, to the nearest degree, the value of $\alpha$.
(5)

Edexcel M3 Specimen Q6

6. A particle $P$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle is hanging in equilibrium below $O$ when it receives a horizontal impulse giving it a speed $u$, where $u ^ { 2 } = 3 g a$. The string becomes slack when $P$ is at the point $B$. The line $O B$ makes an angle $\theta$ with the upward vertical.

Show that $\cos \theta = \frac { 1 } { 3 }$.
(9)
Show that the greatest height of $P$ above $B$ in the subsequent motion is $\frac { 4 a } { 27 }$.
(6)

Edexcel M3 Specimen Q7

7. A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity 6 mg . The other end of the string is attached to a fixed point $O$. When the particle hangs in equilibrium with the string vertical, the extension of the string is $e$.

Find $e$.
(2) The particle is now pulled down a vertical distance $\frac { 1 } { 3 } a$ below its equilibrium position and released from rest. At time $t$ after being released, during the time when the string remains taut, the extension of the string is $e + x$. By forming a differential equation for the motion of $P$ while the string remains taut,
show that during this time $P$ moves with simple harmonic motion of period $2 \pi \sqrt { \frac { a } { 6 g } }$.
(6)
Show that, while the string remains taut, the greatest speed of $P$ is $\frac { 1 } { 3 } \sqrt { } ( 6 g a )$.
Find $t$ when the string becomes slack for the first time. \section*{END}

Edexcel M3 Specimen Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-02_438_492_328_735} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A garden game is played with a small ball $B$ of mass $m$ attached to one end of a light inextensible string of length 13l. The other end of the string is fixed to a point $A$ on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius $5 l$ and centre $C$, where $C$ is vertically below $A$. Modelling the ball as a particle, find

the tension in the string,
the speed of the ball.

Edexcel M3 Specimen Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-08_325_684_306_639} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.9 m and modulus of elasticity $\lambda$ newtons. The other end of the spring is attached to a fixed point $O$ on a rough plane which is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 3 } { 5 }$. The coefficient of friction between the particle and the plane is 0.15 . The particle is held on the plane at a point which is 1.5 m down the line of greatest slope from $O$, as shown in Figure 2. The particle is released from rest and first comes to rest again after moving 0.7 m up the plane. Find the value of $\lambda$.

Edexcel M3 Specimen Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-12_410_579_312_689} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A container is formed by removing a right circular solid cone of height $4 l$ from a uniform solid right circular cylinder of height $6 l$. The centre $O$ of the plane face of the cone coincides with the centre of a plane face of the cylinder and the axis of the cone coincides with the axis of the cylinder, as shown in Figure 3. The cylinder has radius $2 l$ and the base of the cone has radius $l$.

Find the distance of the centre of mass of the container from $O$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-12_435_560_1274_699} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The container is placed on a plane which is inclined at an angle $\theta ^ { \circ }$ to the horizontal. The open face is uppermost, as shown in Figure 4. The plane is sufficiently rough to prevent the container from sliding. The container is on the point of toppling.
Find the value of $\theta$.
\includegraphics[max width=\textwidth, alt={}, center]{689d4bd3-db24-4159-986b-40496213321a-15_78_28_2588_1889}

Edexcel M3 Specimen Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-16_446_437_324_758} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} 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 fixed at the point $O$. The particle is initially held with $O P$ horizontal and the string taut. It is then projected vertically upwards with speed $u$, where $u ^ { 2 } = 5 a g$. When $O P$ has turned through an angle $\theta$ the speed of $P$ is $v$ and the tension in the string is $T$, as shown in Figure 5.

Find, in terms of $a , g$ and $\theta$, an expression for $v ^ { 2 }$.
Find, in terms of $m , g$ and $\theta$, an expression for $T$.
Prove that $P$ moves in a complete circle.
Find the maximum speed of $P$.

Edexcel M3 Q5

5. A cyclist is travelling around a circular track which is banked at $25 ^ { \circ }$ to the horizontal. The coefficient of friction between the cycle's tyres and the track is 0.6 . The cyclist moves with constant speed in a horizontal circle of radius 40 m , without the tyres slipping. Find the maximum speed of the cyclist.

Edexcel M3 Q1

A bird of mass 0.5 kg , flying around a vertical feeding post at a constant speed of $6 \mathrm {~ms} ^ { - 1 }$, banks its wings to move in a horizontal circle of radius 2 m . The aerodynamic lift $L$ newtons is perpendicular to the bird's wings, as shown.
Modelling the bird as a particle, find, to the nearest degree, the
\includegraphics[max width=\textwidth, alt={}, center]{430c3b75-57aa-42ff-867e-304b85e7d521-1_303_472_349_1505}
angle that its wings make with the vertical.
A thin elastic string, of modulus $\lambda \mathrm { N }$ and natural length 20 cm , passes round two small, smooth pegs $A$ and $B$ on the same horizontal level to form a closed loop. $A B = 10 \mathrm {~cm}$. The ends of the string are attached to a weight $P$ of mass 0.7 kg .
When $P$ rests in equilibrium, $A P B$ forms an equilateral triangle.
\includegraphics[max width=\textwidth, alt={}, center]{430c3b75-57aa-42ff-867e-304b85e7d521-1_346_371_836_1560}
1. Find the value of $\lambda$.
2. State one assumption that you have made about the weight $P$, explaining how you have used this assumption in your solution.
3. A particle $P$ of mass 0.5 kg moves along a straight line. When $P$ is at a distance $x \mathrm {~m}$ from a fixed point $O$ on the line, the force acting on it is directed towards $O$ and has magnitude $\frac { 8 } { x ^ { 2 } } \mathrm {~N}$. When $x = 2$, the speed of $P$ is $4 \mathrm {~ms} ^ { - 1 }$.
  Find the speed of $P$ when it is 0.5 m from $O$.
4. A particle $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light 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 $O . P$ is released from rest at $O$ and falls vertically downwards under gravity. The greatest distance below $O$ reached by $P$ is $2 l \mathrm {~m}$.
5. Show that $\lambda = 4 m g$.
6. Find, in terms of $l$ and $g$, the speed with which $P$ passes through the point $A$, where
$$O A = \frac { 5 l } { 4 } \mathrm {~m} .$$ (6 marks) \section*{MECHANICS 3 (A) TEST PAPER 1 Page 2}

Edexcel M3 Q5

\includegraphics[max width=\textwidth, alt={}]{430c3b75-57aa-42ff-867e-304b85e7d521-2_389_412_265_386}

A uniform solid right circular cone has height $h$ and base radius $r$. The top part of the cone is removed by cutting through the cone parallel to the base at a height $\frac { h } { 2 }$.

Show that the centre of mass of the remaining solid is at a height
$\frac { 11 h } { 56 }$ above the base, along its axis of symmetry. The remaining part of the solid is suspended from the point $D$ on the circumference of its smaller circular face, and the axis of symmetry then makes an angle $\alpha$ with the vertical, where $\tan \alpha = \frac { 1 } { 2 }$.
Find the value of the ratio $h : r$.

Edexcel M3 Q6

6. A light elastic string, of natural length $l \mathrm {~m}$ and modulus of elasticity $\frac { m g } { 2 }$ newtons, has one end fastened to a fixed point $O$. A particle $P$, of mass $m \mathrm {~kg}$, is attached to the other end of the string. $P$ hangs in equilibrium at the point $E$, vertically below $O$, where $O E = ( l + e ) \mathrm { m }$

Find the numerical value of the ratio $e : l$.
$P$ is now pulled down a further distance $\frac { 3 l } { 2 } \mathrm {~m}$ from $E$ and is released from rest.
In the subsequent motion, the string remains taut. At time $t \mathrm {~s}$ after being released, $P$ is at a distance $x \mathrm {~m}$ below $E$.
Write down a differential equation for the motion of $P$ and show that the motion is simple harmonic.
Write down the period of the motion.
Find the speed with which $P$ first passes through $E$ again.
Show that the time taken by $P$ after it is released to reach the point $A$ above $E$, where $$A E = \frac { 3 l } { 4 } \mathrm {~m} , \text { is } \frac { 2 \pi } { 3 } \sqrt { \frac { 2 l } { g } } \mathrm {~s} .$$

Edexcel M3 Q7

A particle $P$ is attached to one end of a light inextensible string of length $l \mathrm {~m}$. The other end of the string is attached to a fixed point $O$. When $P$ is hanging at rest vertically below $O$, it is given a horizontal speed $u \mathrm {~ms} ^ { - 1 }$ and starts to move in a vertical circle.

Given that the string becomes slack when it makes an angle of $120 ^ { \circ }$ with the downward vertical through $O$,

show that $u ^ { 2 } = \frac { 7 g l } { 2 }$.
Find, in terms of $l$, the greatest height above $O$ reached by $P$ in the subsequent motion.
(7 marks)

Edexcel M3 Q1

A particle of mass $m \mathrm {~kg}$ moves in a horizontal straight line. Its initial speed is $u \mathrm {~ms} ^ { - 1 }$ and the only force acting on it is a variable resistance of magnitude $m k v \mathrm {~N}$, where $v \mathrm {~ms} ^ { - 1 }$ is the speed of the particle after $t$ seconds and $k$ is a constant.
Show that $v = u e ^ { - k t }$.
A particle $P$ of mass $m \mathrm {~kg}$ moves in a horizontal circle at one end of a light inextensible string of length 40 cm , as shown. The other end of the string is attached to a fixed point $O$.
The angular velocity of $P$ is $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$.
If the angle $\theta$ which the string makes with the vertical must not
\includegraphics[max width=\textwidth, alt={}, center]{e756c147-2711-4ee4-9ec6-7680fe84a0c5-1_314_401_722_1576}
(7 marks)
exceed $60 ^ { \circ }$, calculate the greatest possible value of $\omega$.
A particle $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light elastic string of natural length 0.5 m and modulus of elasticity $\frac { m g } { 2 } \mathrm {~N}$. The other end of the string is attached to a fixed point $O$ and $P$ hangs vertically below $O$.
1. Find the stretched length of the string when $P$ rests in equilibrium.
2. Find the elastic potential energy stored in the string in the equilibrium position. $P$, which is still attached to the string, is now held at rest at $O$ and then lowered gently into its equilibrium position.
3. Find the work done by the weight of the particle as it moves from $O$ to the equilibrium position.
4. Explain the discrepancy between your answers to parts (b) and (c).
5. A particle $P$, of mass $m \mathrm {~kg}$, is attached to two light elastic strings, each of natural length $l \mathrm {~m}$ and modulus of elasticity 3 mg N . The other ends of the strings are attached to the fixed points $A$ and $B$, where $A B$ is horizontal and $A B = 2 l \mathrm {~m}$. If $P$ rests in equilibrium vertically below the mid-point of $A B$, with each string making an angle $\theta$ with the vertical, show that
  \includegraphics[max width=\textwidth, alt={}, center]{e756c147-2711-4ee4-9ec6-7680fe84a0c5-1_410_474_2052_1510}
$$\cot \theta - \cos \theta = \frac { 1 } { 6 } .$$ \section*{MECHANICS 3 (A) TEST PAPER 2 Page 2}

Edexcel M3 Q5

A small bead $P$, of mass $m \mathrm {~kg}$, can slide on a smooth circular ring, with centre $O$ and radius $r \mathrm {~m}$, which is fixed in a vertical plane. $P$ is projected from the lowest point $L$ of the ring with speed $\sqrt { } ( 3 g r ) \mathrm { ms } ^ { - 1 }$. When $P$ has reached a position such that $O P$ makes an angle $\theta$ with the downward vertical, as shown, its speed is $v \mathrm {~ms} ^ { - 1 }$.
\includegraphics[max width=\textwidth, alt={}, center]{e756c147-2711-4ee4-9ec6-7680fe84a0c5-2_355_337_262_1590}
1. Show that $v ^ { 2 } = g r ( 1 + 2 \cos \theta )$.
2. Show that the magnitude of the reaction $R N$ of the ring on the bead is given by
$$R = m g ( 1 + 3 \cos \theta ) .$$
Find the values of $\cos \theta$ when
1. $P$ is instantaneously at rest, (ii) the reaction $R$ is instantaneously zero.
Hence show that the ratio of the heights of $P$ above $L$ in cases (i) and (ii) is $9 : 8$.

Edexcel M3 Q6

6. A light elastic string, of natural length 0.8 m , has one end fastened to a fixed point $O$. The other end of the string is attached to a particle $P$ of mass 0.5 kg . When $P$ hangs in equilibrium, the length of the string is 1.5 m .

Calculate the modulus of elasticity of the string.
$P$ is displaced to a point 0.5 m vertically below its equilibrium position and released from rest.
Show that the subsequent motion of $P$ is simple harmonic, with period 1.68 s .
Calculate the maximum speed of $P$ during its motion.
Show that the time taken for $P$ to first reach a distance 0.25 m from the point of release is 0.28 s , to 2 significant figures.

Edexcel M3 Q7

7. (a) Show that the centre of mass of a uniform solid hemisphere of radius $r$ is at a distance $\frac { 3 r } { 8 }$ from the centre $O$ of the plane face. The figure shows the vertical cross-section of a rough solid hemisphere at rest on a rough inclined plane inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 3 } { 10 }$.
(b) Indicate on a copy of the figure the three forces acting on the hemisphere, clearly stating what they are, and paying
\includegraphics[max width=\textwidth, alt={}, center]{e756c147-2711-4ee4-9ec6-7680fe84a0c5-2_356_520_2042_1457}
(c) Given that the plane face containing the diameter $A B$ makes an angle $\alpha$ with the vertical, show that $\cos \alpha = \frac { 4 } { 5 }$.

Edexcel M3 Q1

One end of a light inextensible string of length $2 r \mathrm {~m}$ is attached to a fixed point $O$. A particle of mass $m \mathrm {~kg}$ is attached to the other end $Q$ of the string, so that it can move in a vertical plane. The string is held taut and horizontal and the particle is projected vertically downwards with a speed $\sqrt { } ( g r ) \mathrm { ms } ^ { - 1 }$. When the string is vertical it begins to wrap round a small, smooth peg $X$ at a distance $r \mathrm {~m}$ vertically below $O$. The particle continues to move.
1. Find the speed of the particle when it reaches $O$, in terms of $g$ and $r$.
2. Show that, when $Q X$ is horizontal, the tension in the string is 3 mgN .
3. A particle moving along the $x$-axis describes simple harmonic motion about the origin $O$. The period of its motion is $\frac { \pi } { 2 }$ seconds. When it is at a distance 1 m from $O$, its speed is $3 \mathrm {~ms} ^ { - 1 }$. Calculate
4. the amplitude of its motion,
5. the maximum acceleration of the particle,
6. the least time that it takes to move from $O$ to a point 0.25 m from $O$.
7. A particle $P$ of mass $m \mathrm {~kg}$ is attached to the mid-point of a light elastic string of natural length $8 l \mathrm {~m}$ and modulus of elasticity $\lambda \mathrm { N }$. The two ends of the string are attached to fixed points $A$ and $B$ on the same horizontal level, where $A B = 81 \mathrm {~m} . P$ is released from rest at the mid-point of $A B$.
8. If $P$ comes to instantaneous rest at a depth $3 / \mathrm { m }$ below $A B$, find an expression for $\lambda$ in terms of $m$ and $g$.
9. Using this value of $\lambda$, show that the speed $v \mathrm {~ms} ^ { - 1 }$ of $P$ when it passes through the point $2 l \mathrm {~m}$ below $A B$ is given by $v ^ { 2 } = 4 ( 24 \sqrt { 5 } - 53 ) g l$.
10. A particle $P$ of mass 0.8 kg moves along a straight line $O L$ and is acted on by a resistive force of magnitude $R \mathrm {~N}$ directed towards the fixed point $O$. When the displacement of $P$ from $O$ is $x \mathrm {~m} , R = \frac { 0 \cdot 8 x v ^ { 2 } } { 1 + x ^ { 2 } }$, where $v \mathrm {~ms} ^ { - 1 }$ is the speed of $P$ at that instant.
11. Write down a differential equation for the motion of $P$.
Given that $v = 2$ when $x = 0$,
find the speed with which $P$ passes through the point $A$, where $O A = 1 \mathrm {~m}$. \section*{MECHANICS 3 (A) TEST PAPER 3 Page 2}

Edexcel M3 Q5

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

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

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

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

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

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)

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