Questions M3 - A-Level Maths

1. Show that $P$ moves with simple harmonic motion with centre $B$.
2. Find the period of this motion.
Find the speed of $P$ when it reaches $B$. The point $D$ is the midpoint of $A B$.
Find the time taken for $P$ to move directly from $C$ to $D$. When $P$ first comes to instantaneous rest a particle $Q$ of mass 0.3 kg is placed at $B$. When $P$ reaches $B$ again, $P$ strikes and adheres to $Q$ to form a single particle $R$.
Show that $R$ also moves with simple harmonic motion.
Find the amplitude of this motion.

Edexcel M3 2017 June Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-02_672_732_226_589} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform lamina is in the shape of the region $R$. Region $R$ is bounded by the curve with equation $y = 4 - x ^ { 2 }$, the positive $x$-axis and the positive $y$-axis, as shown shaded in Figure 1. Use algebraic integration to find the $x$ coordinate of the centre of mass of the lamina.

Edexcel M3 2017 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-04_723_636_219_733} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass $m$ is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point $A$. The particle moves in a horizontal circle with constant angular speed $\sqrt { 58.8 } \mathrm { rad } \mathrm { s } ^ { - 1 }$. The centre $O$ of the circle is vertically below $A$ and the string makes a constant angle $\theta ^ { \circ }$ with the downward vertical, as shown in Figure 2. Given that the tension in the string is 1.2 mg , find

the value of $\theta$
the length of the string.

Edexcel M3 2017 June Q3

A particle $P$ of mass $m \mathrm {~kg}$ is initially held at rest at the point $O$ on a smooth plane which is inclined at $30 ^ { \circ }$ to the horizontal. The particle is released from rest and slides down the plane against a force of magnitude $\frac { 1 } { 2 } m x ^ { 2 }$ newtons acting towards $O$, where $x$ metres is the distance of $P$ from $O$.
1. Find the speed of $P$ when $x = 3$
2. Find the distance $P$ has moved when it first comes to instantaneous rest.

Edexcel M3 2017 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-10_570_410_237_826} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A thin uniform right hollow cylinder, of radius $a$ and height $4 a$, has a base but no top. A thin uniform hemispherical shell, also of radius $a$, is made of the same material as the cylinder. The hemispherical shell is attached to the open end of the cylinder forming a container $C$. The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of $C$ is $O$, as shown in Figure 3.

Find the distance from $O$ to the centre of mass of $C$. The container is placed with its circular base on a plane which is inclined at $\theta ^ { \circ }$ to the horizontal. The plane is sufficiently rough to prevent $C$ from sliding. The container is on the point of toppling.
Find the value of $\theta$.

Edexcel M3 2017 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-14_565_696_219_721} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A hollow cylinder is fixed with its axis horizontal. A particle $P$ moves in a vertical circle, with centre $O$ and radius $a$, on the smooth inner surface of the cylinder. The particle moves in a vertical plane which is perpendicular to the axis of the cylinder. The particle is projected vertically downwards with speed $\sqrt { 7 a g }$ from the point $A$, where $O A$ is horizontal and $O A = a$. When angle $A O P = \theta$, the speed of $P$ is $v$, as shown in Figure 4.

Show that $v ^ { 2 } = a g ( 7 + 2 \sin \theta )$
Verify that $P$ will move in a complete circle.
Find the maximum value of $v$.

Edexcel M3 2017 June Q6

The ends of a light elastic string, of natural length 0.4 m and modulus of elasticity $\lambda$ newtons, are attached to two fixed points $A$ and $B$ which are 0.6 m apart on a smooth horizontal table. The tension in the string is 8 N .
1. Show that $\lambda = 16$
A particle $P$ is attached to the midpoint of the string. The particle $P$ is now pulled horizontally in a direction perpendicular to $A B$ to a point 0.4 m from the midpoint of $A B$. The particle is held at rest by a horizontal force of magnitude $F$ newtons acting in a direction perpendicular to $A B$, as shown in Figure 5 below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-18_623_796_792_573} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure}
Find the value of $F$. The particle is released from rest. Given that the mass of $P$ is 0.3 kg ,
find the speed of $P$ as it crosses the line $A B$.

Edexcel M3 2017 June Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-24_173_968_223_488} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} The fixed points $A$ and $B$ are 4 m apart on a smooth horizontal floor. One end of a light elastic string, of natural length 1.8 m and modulus of elasticity 45 N , is attached to a particle $P$ and the other end is attached to $A$. One end of another light elastic string, of natural length 1.2 m and modulus of elasticity 20 N , is attached to $P$ and the other end is attached to $B$. The particle $P$ rests in equilibrium at the point $O$, where $A O B$ is a straight line, as shown in Figure 6.

Show that $A O = 2.2 \mathrm {~m}$. The point $C$ lies on the straight line $A O B$ with $A C = 2.7 \mathrm {~m}$. The mass of $P$ is 0.6 kg . The particle $P$ is held at $C$ and then released from rest.
Show that, while both strings are taut, $P$ moves with simple harmonic motion with centre $O$. The point $D$ lies on the straight line $A O B$ with $A D = 1.8 \mathrm {~m}$. When $P$ reaches $D$ the string $P B$ breaks.
Find the time taken by $P$ to move directly from $C$ to $A$.

Edexcel M3 2018 June Q1

A rough disc is rotating in a horizontal plane with constant angular speed $\omega$ about a vertical axis through the centre of the disc. A particle $P$ is placed on the disc at a distance $r$ from the axis. The coefficient of friction between $P$ and the disc is $\mu$.

Given that $P$ does not slip on the disc, show that $$\omega \leqslant \sqrt { \frac { \mu g } { r } }$$

Edexcel M3 2018 June Q2

2. A light elastic string has natural length 1.2 m and modulus of elasticity $\lambda$ newtons. One end of the string is attached to a fixed point $O$. A particle of mass 0.5 kg is attached to the other end of the string. The particle is moving with constant angular speed $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$ in a horizontal circle with the string stretched. The circle has radius 0.9 m and its centre is vertically below $O$. The string is inclined at $60 ^ { \circ }$ to the horizontal. Find

the value of $\lambda$,
the value of $\omega$.

Edexcel M3 2018 June Q3

3. A particle $P$ of mass $m$ moves in a straight line away from the centre of the Earth. The Earth is modelled as a sphere of radius $R$. When $P$ is at a distance $x , x \geqslant R$, from the centre of the Earth, the force exerted by the Earth on $P$ is directed towards the centre of the Earth and has magnitude $\frac { m g R ^ { 2 } } { x ^ { 2 } }$. When $P$ is at a distance $2 R$ from the surface of the Earth, the speed of $P$ is $\sqrt { \frac { g R } { 3 } }$. Assuming that air resistance can be ignored, find the distance of $P$ from the surface of the Earth when the speed of $P$ is $2 \sqrt { \frac { g R } { 3 } }$.

Edexcel M3 2018 June Q4

4. One end of a light elastic string, of modulus of elasticity $2 m g$ and natural length $l$, is fixed to a point $O$ on a rough plane. The plane is inclined at angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 3 } { 5 }$. The other end of the string is attached to a particle $P$ of mass $m$ which is held at rest on the plane at the point $O$. The coefficient of friction between $P$ and the plane is $\frac { 1 } { 4 }$. The particle is released from rest and slides down the plane, coming to instantaneous rest at the point $A$, where $O A = k l$. Given that $k > 1$, find, to 3 significant figures, the value of $k$.

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$\_\_\_\_$ T

Edexcel M3 2018 June Q5

5. A uniform solid hemisphere has radius $r$. The centre of the plane face of the hemisphere is $O$.

Use algebraic integration to show that the distance from $O$ to the centre of mass of the hemisphere is $\frac { 3 } { 8 } r$.
[0pt] [You may assume that the volume of a sphere of radius $r$ is $\frac { 4 } { 3 } \pi r ^ { 3 }$ ] \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2cf74ba3-857a-4ce9-ab5b-e6203b279161-14_378_602_740_671} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A solid $S$ is formed by joining a uniform solid hemisphere of radius $a$ to a uniform solid hemisphere of radius $\frac { 1 } { 2 } a$. The plane faces of the hemispheres are joined together so that their centres coincide at $O$, as shown in Figure 1. The mass per unit volume of the smaller hemisphere is $k$ times the mass per unit volume of the larger hemisphere.
Find the distance from $O$ to the centre of mass of $S$. When $S$ is placed on a horizontal plane with any point on the curved surface of the larger hemisphere in contact with the plane, $S$ remains in equilibrium.
Find the value of $k$.

Edexcel M3 2018 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2cf74ba3-857a-4ce9-ab5b-e6203b279161-18_481_606_246_667} \captionsetup{labelformat=empty} \caption{Figure 2}

\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 attached to a fixed point $O$. The particle is held at the point $A$, where $O A = a$ and $O A$ is horizontal. The particle is projected vertically upwards with speed $u$, as shown in Figure 2. When the string makes an angle $\theta$ with the horizontal through $O$ and the string is still taut, the tension in the string is $T$.

Show that $T = \frac { m } { a } \left( u ^ { 2 } - 3 a g \sin \theta \right)$ The particle moves in complete circles.
Find, in terms of $a$ and $g$, the minimum value of $u$. Given that the least tension in the string is $S$ and the greatest tension in the string is $4 S$,
find, in terms of $a$ and $g$, an expression for $u$.

Edexcel M3 2018 June Q7

7. A particle $P$ of mass 0.5 kg is attached to one end of a light elastic string. The string has natural length $l$ metres and modulus of elasticity 29.4 N . The other end of the string is attached to a fixed point $A$. The particle hangs freely in equilibrium at the point $B$, where $B$ is vertically below $A$ and $A B = 1.4 \mathrm {~m}$.

Show that $l = 1.2$ The point $C$ is vertically below $A$ and $A C = 1.8 \mathrm {~m}$. The particle is pulled down to $C$ and released from rest.
Show that, while the string is taut, $P$ moves with simple harmonic motion.
Calculate the speed of $P$ at the instant when the string first becomes slack. The particle first comes to instantaneous rest at the point $D$.
Find the time taken by $P$ to return directly from $D$ to $C$.

Edexcel M3 Q1

A particle $P$ moves on the positive $x$-axis. When the displacement of $P$ from $O$ is $x$ metres, its acceleration is $( 6 - 4 x ) \mathrm { m } \mathrm { s } ^ { - 2 }$, measured in the direction of $x$ increasing. Initially $P$ is at $O$ and the velocity of $P$ is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the direction $O x$.

Find the distance of $P$ from $O$ when $P$ is instantaneously at rest.

Edexcel M3 Q2

2. A light elastic string $A B$ has one end $A$ attached to a fixed point on a ceiling. A particle $P$ of mass 0.3 kg is attached to $B$. When $P$ hangs in equilibrium with $A B$ vertical, $A B = 100 \mathrm {~cm}$. The particle $P$ is replaced by another particle $Q$ of mass 0.5 kg . When $Q$ hangs in equilibrium with $A B$ vertical, $A B = 110 \mathrm {~cm}$. Find

the natural length of the string,
the modulus of elasticity of the string.

Edexcel M3 Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{45c51316-7d58-4c16-9b5f-1d7421060a88-3_485_855_1073_584} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $3 a$. The other end of the string is attached to a fixed point $A$ which is a vertical distance $a$ above a smooth horizontal table. The particle moves on the table in a circle whose centre $O$ is vertically below $A$, as shown in Fig. 1. The string is taut and the speed of $P$ is $2 \sqrt { } ( a g )$. Find

the tension in the string,
the normal reaction of the table on $P$.

Edexcel M3 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{45c51316-7d58-4c16-9b5f-1d7421060a88-4_332_1056_251_459} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A small smooth bead $B$ of mass 0.2 kg is threaded on a smooth horizontal wire. The point $A$ is on the same horizontal level as the wire and at a perpendicular distance $d$ from the wire. The point $O$ is the point on the wire nearest to $A$, as shown in Fig. 2. The bead experiences a force of magnitude $5 ( A B )$ newtons in the direction $B A$ towards $A$. Initially $B$ is at rest with $O B = 2 \mathrm {~m}$.

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

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

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

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

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

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.

Questions M3 (745 questions)

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