Questions - Edexcel M3 - A-Level Maths

Prove that $B$ moves with simple harmonic motion about $O$, with period $\frac { 2 \pi } { 5 } \mathrm {~s}$.
Find the greatest speed of $B$ in the motion.
Find the time when $B$ has first moved a distance 3 m from its initial position.

Edexcel M3 Q5

11 marks Standard +0.8

5. In a "test your strength" game at an amusement park, competitors hit one end of a small lever with a hammer, causing the other end of the lever to strike a ball which then moves in a vertical tube whose total height is adjustable. The ball is attached to one end of an elastic spring of natural length 3 m and modulus of elasticity 120 N . The mass of the ball is 2 kg . The other end of the spring is attached to the top of the tube. The ball is modelled as a particle, the spring as light and the tube is assumed to be smooth. The height of the tube is first set at 3 m . A competitor gives the ball an initial speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the height to which the ball rises before coming to rest. The tube is now adjusted by reducing its height to 2.5 m . The spring and the ball remain unchanged.
Find the initial speed which the ball must now have if it is to rise by the same distance as in part (a).
(5 marks)

Edexcel M3 Q6

14 marks Standard +0.8

6. (a) Show, by integration, that the centre of mass of a uniform right cone, of radius $a$ and height $h$, is a distance $\frac { 3 } { 4 } h$ from the vertex of the cone. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{45c51316-7d58-4c16-9b5f-1d7421060a88-5_789_914_406_486} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} A uniform right cone $C$, of radius $a$ and height $h$, has vertex $A$. A solid $S$ is formed by removing from $C$ another cone, of radius $\frac { 2 } { 3 } a$ and height $\frac { 1 } { 2 } h$, with the same axis as $C$. The plane faces of the two cones coincide, as shown in Fig. 3.
(b) Find the distance of the centre of mass of $S$ from $A$.

Edexcel M3 Q7

17 marks Challenging +1.2

7. A smooth solid hemisphere is fixed with its plane face on a horizontal table and its curved surface uppermost. The plane face of the hemisphere has centre $O$ and radius $a$. The point $A$ is the highest point on the hemisphere. A particle $P$ is placed on the hemisphere at $A$. It is then given an initial horizontal speed $u$, where $u ^ { 2 } = \frac { 1 } { 2 } ( a g )$. When $O P$ makes an angle $\theta$ with $O A$, and while $P$ remains on the hemisphere, the speed of $P$ is $v$.

Find an expression for $v ^ { 2 }$.
Show that, when $\theta = \arccos 0.9 , P$ is still on the hemisphere.
Find the value of $\cos \theta$ when $P$ leaves the hemisphere.
Find the value of $v$ when $P$ leaves the hemisphere. After leaving the hemisphere $P$ strikes the table at $B$.
Find the speed of $P$ at $B$.
Find the angle at which $P$ strikes the table. \section*{Alternative Question 2:}
1. Two light elastic strings $A B$ and $B C$ are joined at $B$. The string $A B$ has natural length 1 m and modulus of elasticity 15 N . The string $B C$ has natural length 1.2 m and modulus of elasticity 30 N . The ends $A$ and $C$ are attached to fixed points 3 m apart and the strings rest in equilibrium with $A B C$ in a straight line.
Find the tension in the combined string $A C$.

Edexcel M3 Specimen Q1

7 marks Standard +0.3

1. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{e256678d-89e8-48eb-aa8a-b8e027b62ef1-2_259_822_367_625}

\end{figure} A car moves round a bend in a road which is banked at an angle $\alpha$ to the horizontal, as shown in Fig. 1. The car is modelled as a particle moving in a horizontal circle of radius 100 m . When the car moves at a constant speed of $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, there is no sideways frictional force on the car. Find, in degrees to one decimal place, the value of $\alpha$.

Edexcel M3 Specimen Q2

7 marks Standard +0.3

2. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{e256678d-89e8-48eb-aa8a-b8e027b62ef1-2_310_1122_1178_466}

\end{figure} Two elastic ropes each have natural length 30 cm and modulus of elasticity $\lambda \mathrm { N }$. One end of each rope is attached to a lead weight $P$ of mass 2 kg and the other ends are attached to two points $A$ and $B$ on a horizontal ceiling, where $A B = 72 \mathrm {~cm}$. The weight hangs in equilibrium 15 cm below the ceiling, as shown in Fig. 2. By modelling $P$ as a particle and the ropes as light elastic strings,

find, to one decimal place, the value of $\lambda$.
State how you have used the fact that $P$ is modelled as a particle.

Edexcel M3 Specimen Q3

8 marks Standard +0.8

3. A particle $P$ of mass 0.5 kg moves away from the origin $O$ along the positive $x$-axis under the action of a force directed towards $O$ of magnitude $\frac { 2 } { x ^ { 2 } } \mathrm {~N}$, where $O P = x$ metres. When $x = 1$, the speed of $P$ is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the distance of $P$ from $O$ when its speed has been reduced to $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
(8)

Edexcel M3 Specimen Q4

10 marks Standard +0.8

4. A man of mass 75 kg is attached to one end of a light elastic rope of natural length 12 m . The other end of the rope is attached to a point on the edge of a horizontal ledge 19 m above the ground. The man steps off the ledge and falls vertically under gravity. The man is modelled as a particle falling from rest. He is brought to instantaneous rest by the rope when he is 1 m above the ground.
Find

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

13 marks Standard +0.3

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

15 marks Challenging +1.2

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

15 marks Challenging +1.2

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 Q5

12 marks Standard +0.8

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

7 marks Standard +0.8

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

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

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-2_341_652_217_621} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A particle of mass 0.6 kg is attached to one end of a light elastic spring of natural length 1 m and modulus of elasticity 30 N . The other end of the spring is fixed to a point $O$ which lies on a smooth plane inclined at an angle $\alpha$ to the horizontal where $\tan \alpha = \frac { 3 } { 4 }$ as shown in Figure 1. The particle is held at rest on the slope at a point 1.2 m from $O$ down the line of greatest slope of the plane.

Find the tension in the spring.
Find the initial acceleration of the particle.

Edexcel M3 Q2

7 marks Standard +0.3

2. A particle $P$ of mass 0.5 kg moves along the positive $x$-axis under the action of a single force directed away from the origin $O$. When $P$ is $x$ metres from $O$, the magnitude of the force is $3 x ^ { \frac { 1 } { 2 } } \mathrm {~N}$ and $P$ has a speed of $v \mathrm {~ms} ^ { - 1 }$. Given that when $x = 1 , P$ is moving away from $O$ with speed $2 \mathrm {~ms} ^ { - 1 }$,

find an expression for $v ^ { 2 }$ in terms of $x$,
show that when $x = 4 , P$ has a speed of $7.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 1 decimal place.

Edexcel M3 Q3

10 marks Standard +0.3

3. A particle is performing simple harmonic motion along a straight line between the points $A$ and $B$ where $A B = 8 \mathrm {~m}$. The period of the motion is 12 seconds.

Find the maximum speed of the particle in terms of $\pi$. The points $P$ and $Q$ are on the line $A B$ at distances of 3 m and 6 m respectively from $A$.
Find, correct to 3 significant figures, the time it takes for the particle to travel directly from $P$ to $Q$.
(6 marks)

Edexcel M3 Q4

11 marks Standard +0.3

4. Whilst in free-fall a parachutist falls vertically such that his velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when he is $x$ metres below his initial position is given by $$v ^ { 2 } = k g \left( 1 - \mathrm { e } ^ { - \frac { 2 x } { k } } \right) ,$$ where $k$ is a constant.
Given that he experiences an acceleration of $\mathrm { f } \mathrm { m } \mathrm { s } ^ { - 2 }$,

show that $f = g \mathrm { e } ^ { - \frac { 2 x } { k } }$. After falling a large distance, his velocity is constant at $49 \mathrm {~ms} ^ { - 1 }$.
Find the value of $k$.
Hence, express $f$ in the form ( $\lambda - \mu v ^ { 2 }$ ) where $\lambda$ and $\mu$ are constants which you should find.
(4 marks)

Edexcel M3 Q5

13 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-3_588_291_1126_662} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A firework is modelled as a uniform solid formed by joining the plane surface of a right circular cone of height $2 r$ and base radius $r$, to one of the plane surfaces of a cylinder of height $h$ and base radius $r$ as shown in Figure 2. Using this model,

show that the distance of the centre of mass of the firework from its plane base is $$\frac { 3 h ^ { 2 } + 4 h r + 2 r ^ { 2 } } { 2 ( 3 h + 2 r ) }$$ The firework is to be launched from rough ground inclined at an angle $\alpha$ to the horizontal. Given that the firework does not slip or topple and that $h = 4 r$,
Find, correct to the nearest degree, the maximum value of $\alpha$.

Edexcel M3 Q6

13 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-4_437_364_196_717} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} The two ends of a light inextensible string of length $3 a$ are attached to fixed points $Q$ and $R$ which are a distance of $a \sqrt { } 3$ apart with $R$ vertically below $Q$. A particle $P$ of mass $m$ is attached to the string at a distance of $2 a$ from $Q$. $P$ is given a horizontal speed, $u$, such that it moves in a horizontal circle with both sections of the string taut as shown in Figure 3.

Show that $\angle P R Q$ is a right angle.
Find $\angle P Q R$ in degrees.
Find, in terms of $a , g , m$ and $u$, the tension in the section of string
1. $P Q$,
2. $P R$.
Show that $u ^ { 2 } \geq \frac { g a } { \sqrt { 3 } }$.

Edexcel M3 Q7

14 marks Challenging +1.2

7. A particle of mass 2 kg is attached to one end of a light elastic string of natural length 1 m and modulus of elasticity 50 N . The other end of the string is attached to a fixed point $O$ on a rough horizontal plane and the coefficient of friction between the particle and the plane is $\frac { 10 } { 49 }$. The particle is projected from $O$ along the plane with an initial speed of $5 \mathrm {~ms} ^ { - 1 }$.

Show that the greatest distance from $O$ which the particle reaches is 1.84 m .
Find, correct to 2 significant figures, the speed at which the particle returns to $O$.

Edexcel M3 Q1

7 marks Moderate -0.3

A student is attempting to model the expansion of an airbag in a car following a collision.

The student considers the displacement from the steering column, $s$ metres, of a point $P$ on the airbag $t$ seconds after a collision and uses the formula $$s = \mathrm { e } ^ { 3 t } - 1 , \quad 0 \leq t \leq 0.1$$ Using this model,

find, correct to the nearest centimetre, the maximum displacement of $P$,
find the initial velocity of $P$,
find the acceleration of $P$ in terms of $t$.
Explain why this model is unlikely to be realistic.

Edexcel M3 Q2

8 marks Standard +0.3

2. A particle $P$ is attached to one end of a light elastic string of modulus of elasticity 80 N . The other end of the string is attached to a fixed point $A$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-2_410_570_1210_735} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} When a horizontal force of magnitude 20 N is applied to $P$, it rests in equilibrium with the string making an angle of $30 ^ { \circ }$ with the vertical and $A P = 1.2 \mathrm {~m}$ as shown in Figure 1.

Find the tension in the string.
Find the elastic potential energy stored in the string.

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