Questions - Edexcel - A-Level Maths

Show that the distance from $O$ to the centre of mass of $S$ is $\frac { 73 } { 72 } r$ The solid $S$ is placed with its plane face on a rough horizontal plane. The coefficient of friction between $S$ and the plane is $\mu$. A horizontal force $P$ is applied to the highest point of $S$. The magnitude of $P$ is gradually increased.
Find the range of values of $\mu$ for which $S$ will slide before it starts to tilt.

Edexcel FM2 2020 June Q5

11 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{962c2b40-3c45-4eed-a0af-a59068bda0e1-16_501_606_244_731} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A particle $P$ of mass 0.75 kg is attached to one end of a light inextensible string of length 60 cm . The other end of the string is attached to a fixed point $A$ that is vertically above the point $O$ on a smooth horizontal table, such that $O A = 40 \mathrm {~cm}$. The particle remains in contact with the table, with the string taut, and moves in a horizontal circle with centre $O$, as shown in Figure 4. The particle is moving with a constant angular speed of 3 radians per second.

Find (i) the tension in the string,
(ii) the normal reaction between $P$ and the table. The angular speed of $P$ is now gradually increased.
Find the angular speed of $P$ at the instant $P$ loses contact with the table.

Edexcel FM2 2020 June Q6

13 marks Standard +0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{962c2b40-3c45-4eed-a0af-a59068bda0e1-20_533_543_242_760} \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 $l$. The other end of the string is attached to a fixed point $O$. The particle is held with the string taut and $O P$ horizontal. The particle is then projected vertically downwards with speed $u$, where $u ^ { 2 } = \frac { 9 } { 5 } \mathrm { gl }$. When $O P$ has turned through an angle $\alpha$ and the string is still taut, the speed of $P$ is $v$, as shown in Figure 5. At this instant the tension in the string is $T$.

Show that $T = 3 m g \sin \alpha + \frac { 9 } { 5 } m g$
Find, in terms of $g$ and $l$, the speed of $P$ at the instant when the string goes slack.
Find, in terms of $l$, the greatest vertical height reached by $P$ above the level of $O$.

Edexcel FM2 2020 June Q7

15 marks Challenging +1.2

A light elastic spring has natural length $l$ and modulus of elasticity $4 m g$. A particle $P$ of mass $m$ is attached to one end of the spring. The other end of the spring is attached to a fixed point $A$. The point $B$ is vertically below $A$ with $A B = \frac { 7 } { 4 } l$. The particle $P$ is released from rest at $B$.
1. Show that $P$ moves with simple harmonic motion with period $\pi \sqrt { \frac { l } { g } }$
2. Find, in terms of $m , l$ and $g$, the maximum kinetic energy of $P$ during the motion.
3. Find the time within each complete oscillation for which the length of the spring is less than $l$.

Edexcel FM2 2021 June Q1

8 marks Standard +0.3

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d4fc2ea6-3ffc-42f2-b462-9694adfe2ec1-02_826_649_244_708} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A letter P from a shop sign is modelled as a uniform plane lamina which consists of a rectangular lamina, $O A B D E$, joined to a semicircular lamina, $B C D$, along its diameter $B D$. $$O A = E D = a , A B = 2 a , O E = 4 a \text {, and the diameter } B D = 2 a \text {, as shown in Figure } 1 .$$ Using the model,

find, in terms of $\pi$ and $a$, the distance of the centre of mass of the letter P ,
from (i) $O E$
(ii) $O A$ The letter P is freely suspended from $O$ and hangs in equilibrium. The angle between $O E$ and the downward vertical is $\alpha$. Using the model,
find the exact value of $\tan \alpha$

Edexcel FM2 2021 June Q2

10 marks Challenging +1.2

At time $t = 0$, a small stone $P$ of mass $m$ is released from rest and falls vertically through the air. At time $t$, the speed of $P$ is $v$ and the resistance to the motion of $P$ from the air is modelled as a force of magnitude $k v ^ { 2 }$, where $k$ is a constant.
1. Show that $t = \frac { V } { 2 g } \ln \left( \frac { V + v } { V - v } \right)$ where $V ^ { 2 } = \frac { m g } { k }$
2. Give an interpretation of the value of $V$, justifying your answer.
At time $t , P$ has fallen a distance $s$.
Show that $s = \frac { V ^ { 2 } } { 2 g } \ln \left( \frac { V ^ { 2 } } { V ^ { 2 } - v ^ { 2 } } \right)$

Edexcel FM2 2021 June Q3

6 marks Challenging +1.2

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d4fc2ea6-3ffc-42f2-b462-9694adfe2ec1-10_552_807_246_630} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A uniform solid hemisphere $H$ has radius $2 a$. A solid hemisphere of radius $a$ is removed from the hemisphere $H$ to form a bowl. The plane faces of the hemispheres coincide and the centres of the two hemispheres coincide at the point $O$, as shown in Figure 2. The centre of mass of the bowl is at the point $G$.

Show that $O G = \frac { 45 a } { 56 }$ Figure 3 below shows a cross-section of the bowl which is resting in equilibrium with a point $P$ on its curved surface in contact with a rough plane. The plane is inclined to the horizontal at an angle $\alpha$ and is sufficiently rough to prevent the bowl from slipping. The line $O G$ is horizontal and the points $O , G$ and $P$ lie in a vertical plane which passes through a line of greatest slope of the inclined plane. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d4fc2ea6-3ffc-42f2-b462-9694adfe2ec1-10_812_1086_1667_493} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
Find the size of $\alpha$, giving your answer in degrees to 3 significant figures.

Edexcel FM2 2021 June Q4

10 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d4fc2ea6-3ffc-42f2-b462-9694adfe2ec1-14_682_817_246_625} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} One end of a light inextensible string of length $2 l$ is attached to a fixed point $A$. A small smooth ring $R$ of mass $m$ is threaded on the string and the other end of the string is attached to a fixed point $B$. The point $B$ is vertically below $A$, with $A B = l$. The ring is then made to move with constant speed $V$ in a horizontal circle with centre $B$. The string is taut and $B R$ is horizontal, as shown in Figure 4.

Show that $B R = \frac { 31 } { 4 }$ Given that air resistance is negligible,
find, in terms of $m$ and $g$, the tension in the string,
find $V$ in terms of $g$ and $l$.

Edexcel FM2 2021 June Q5

16 marks Challenging +1.2

A light inextensible string of length $a$ has one end attached to a fixed point $O$. The other end of the string is attached to a small stone of mass $m$. The stone is held with the string taut and horizontal. The stone is then projected vertically upwards with speed $U$.

The stone is modelled as a particle and air resistance is modelled as being negligible.
Assuming that the string does not break, use the model to

find the least value of $U$ so that the stone will move in complete vertical circles. The string will break if the tension in it is equal to $\frac { 11 m g } { 2 }$
Given that $U = 2 \sqrt { a g }$, use the model to
find the total angle that the string has turned through, from when the stone is projected vertically upwards, to when the string breaks,
find the magnitude of the acceleration of the stone at the instant just before the string breaks.

Edexcel FM2 2021 June Q6

16 marks Challenging +1.2

A light elastic string, of natural length $l$ and modulus of elasticity $2 m g$, has one end attached to a fixed point $A$ and the other end attached to a particle $P$ of mass $m$. The particle $P$ hangs in equilibrium at the point $O$.
1. Show that $A O = \frac { 3 l } { 2 }$
The particle $P$ is pulled down vertically from $O$ to the point $B$, where $O B = l$, and released from rest. Air resistance is modelled as being negligible.
Using the model,
prove that $P$ begins to move with simple harmonic motion about $O$ with period $\pi \sqrt { \frac { 2 l } { g } }$ The particle $P$ first comes to instantaneous rest at the point $C$.
Using the model,
find the length $B C$ in terms of $l$,
find, in terms of $l$ and $g$, the exact time it takes $P$ to move directly from $B$ to $C$.

Edexcel FM2 2021 June Q7

9 marks Challenging +1.2

\hspace{0pt} [In this question, you may assume that the centre of mass of a circular arc, radius $r$, with angle at centre $2 \alpha$, is a distance $\frac { r \sin \alpha } { \alpha }$ from the centre.]

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d4fc2ea6-3ffc-42f2-b462-9694adfe2ec1-26_828_561_422_753} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} A thin non-uniform metal plate is in the shape of a sector $O A B$ of a circle with centre $O$ and radius $a$. The angle $A O B = \frac { \pi } { 2 }$, as shown in Figure 5. The plate is modelled as a non-uniform lamina.
The mass per unit area of the lamina, at any point $P$ of the lamina, is modelled as $k ( O P ) ^ { 2 }$, where $k = \frac { 4 \lambda } { \pi a ^ { 4 } }$ and $\lambda$ is a constant. Using the model,

find the mass of the plate in terms of $\lambda$,
find, in terms of $a$, the distance of the centre of mass of the plate from $O$.

Edexcel FM2 2022 June Q1

6 marks Moderate -0.5

Three particles of masses $2 m , 3 m$ and $k m$ are placed at the points with coordinates (3a, 2a), (a, -4a) and (-3a, 4a) respectively.

The centre of mass of the three particles lies at the point with coordinates $( \bar { x } , \bar { y } )$.

1. Find $\bar { x }$ in terms of $a$ and $k$
2. Find $\bar { y }$ in terms of $a$ and $k$ Given that the distance of the centre of mass of the three particles from the point ( 0,0 ) is $\frac { 1 } { 3 } a$
find the possible values of $k$

Edexcel FM2 2022 June Q2

7 marks Standard +0.8

A cyclist and her cycle have a combined mass of 60 kg . The cyclist is moving along a straight horizontal road and is working at a constant rate of 200 W .

When she has travelled a distance $x$ metres, her speed is $v \mathrm {~ms} ^ { - 1 }$ and the magnitude of the resistance to motion is $3 v ^ { 2 } \mathrm {~N}$.

Show that $\frac { \mathrm { d } v } { \mathrm {~d} x } = \frac { 200 - 3 v ^ { 3 } } { 60 v ^ { 2 } }$ The distance travelled by the cyclist as her speed increases from $2 \mathrm {~ms} ^ { - 1 }$ to $4 \mathrm {~ms} ^ { - 1 }$ is $D$ metres.
Find the exact value of $D$

Edexcel FM2 2022 June Q3

7 marks

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1f39620e-c10f-4344-89f1-626fff36d187-08_517_753_258_657} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Nine uniform rods are joined together to form the rigid framework $A B C D E F A$, with $A B = B C = D F = 3 a , B F = C D = D E = 4 a$ and $A F = F E = C F = 5 a$, as shown in Figure 1. All nine rods lie in the same plane. The mass per unit length of each of the rods $B F , C F$ and $D F$ is twice the mass per unit length of each of the other six rods.

Find the distance of the centre of mass of the framework from $A C$ The mass of the framework is $M$. A particle of mass $k M$ is attached to the framework at $E$ to form a loaded framework. When the loaded framework is freely suspended from $F$, it hangs in equilibrium with $C E$ horizontal.
Find the exact value of $k$

Edexcel FM2 2022 June Q4

8 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1f39620e-c10f-4344-89f1-626fff36d187-12_640_645_258_699} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A small smooth ring $R$ of mass $m$ is threaded onto a light inextensible string. One end of the string is attached to a fixed point $A$ and the other end of the string is attached to the fixed point $B$ such that $B$ is vertically above $A$ and $A B = 6 a$ The ring moves with constant angular speed $\omega$ in a horizontal circle with centre $A$. The string is taut and $B R$ makes a constant angle $\theta$ with the downward vertical, as shown in Figure 2. The ring is modelled as a particle.
Given that $\tan \theta = \frac { 8 } { 15 }$

find, in terms of $m$ and $g$, the magnitude of the tension in the string,
find $\omega$ in terms of $a$ and $g$

Edexcel FM2 2022 June Q5

11 marks Standard +0.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1f39620e-c10f-4344-89f1-626fff36d187-16_567_602_260_735} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The uniform plane lamina shown in Figure 3 is formed from two squares, $A B C O$ and $O D E F$, and a sector $O D C$ of a circle with centre $O$. Both squares have sides of length $3 a$ and $A O$ is perpendicular to $O F$. The radius of the sector is $3 a$
[0pt] [In part (a) you may use, without proof, any of the centre of mass formulae given in the formulae booklet.]

Show that the distance of the centre of mass of the sector $O D C$ from $O C$ is $\frac { 4 a } { \pi }$
Find the distance of the centre of mass of the lamina from $F C$ The lamina is freely suspended from $F$ and hangs in equilibrium with $F C$ at an angle $\theta ^ { \circ }$ to the downward vertical.
Find the value of $\theta$

Edexcel FM2 2022 June Q6

10 marks Challenging +1.2

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1f39620e-c10f-4344-89f1-626fff36d187-20_369_815_255_632} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} The shaded region shown in Figure 4 is bounded by the $x$-axis, the line with equation $x = 9$ and the line with equation $y = \frac { 1 } { 3 } x$. This shaded region is rotated through $360 ^ { \circ }$ about the $x$-axis to form a solid of revolution. This solid of revolution is used to model a solid right circular cone of height 9 cm and base radius 3 cm . The cone is non-uniform and the mass per unit volume of the cone at the point ( $x , y , z$ ) is $\lambda x \mathrm {~kg} \mathrm {~cm} ^ { - 3 }$, where $0 \leqslant x \leqslant 9$ and $\lambda$ is constant.

Find the distance of the centre of mass of the cone from its vertex. A toy is made by joining the circular plane face of the cone to the circular plane face of a uniform solid hemisphere of radius 3 cm , so that the centres of the two plane surfaces coincide. The weight of the cone is $W$ newtons and the weight of the hemisphere is $k W$ newtons.
When the toy is placed on a smooth horizontal plane with any point of the curved surface of the hemisphere in contact with the plane, the toy will remain at rest.
Find the value of $k$

Edexcel FM2 2022 June Q7

12 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1f39620e-c10f-4344-89f1-626fff36d187-24_639_593_246_737} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} A package $P$ of mass $m$ is attached to one end of a string of length $\frac { 2 a } { 5 }$. The other end of the string is attached to a fixed point $O$. The package hangs at rest vertically below $O$ with the string taut and is then projected horizontally with speed $u$, as shown in Figure 5. When $O P$ has turned through an angle $\theta$ and the string is still taut, the tension in the string is $T$ The package is modelled as a particle and the string as being light and inextensible.

Show that $T = 3 m g \cos \theta - 2 m g + \frac { 5 m u ^ { 2 } } { 2 a }$ Given that $P$ moves in a complete vertical circle with centre $O$
find, in terms of $a$ and $g$, the minimum possible value of $u$ Given that $u = 2 \sqrt { a g }$
find, in terms of $g$, the magnitude of the acceleration of $P$ at the instant when $O P$ is horizontal.
Apart from including air resistance, suggest one way in which the model could be refined to make it more realistic.

Edexcel FM2 2022 June Q8

14 marks Challenging +1.2

Throughout this question, use $\boldsymbol { g } = \mathbf { 1 0 m ~ s } ^ { \mathbf { - 2 } }$

A light elastic string has natural length 1.25 m and modulus of elasticity 25 N .
A particle $P$ of mass 0.5 kg is attached to one end of the string. The other end of the string is attached to a fixed point $A$. Particle $P$ hangs freely in equilibrium with $P$ vertically below $A$ The particle is then pulled vertically down to a point $B$ and released from rest.

Show that, while the string is taut, $P$ moves with simple harmonic motion with period $\frac { \pi } { \sqrt { 10 } }$ seconds. The maximum kinetic energy of $P$ during the subsequent motion is 2.5 J .
Show that $A B = 2 \mathrm {~m}$ The particle returns to $B$ for the first time $T$ seconds after it was released from rest at $B$
Find the value of $T$

Edexcel FM2 2023 June Q1

6 marks Moderate -0.5

Three particles of masses $3 m , 4 m$ and $k m$ are positioned at the points with coordinates ( $2 a , 3 a$ ), ( $a , 5 a$ ) and ( $2 \mu a , \mu a$ ) respectively, where $k$ and $\mu$ are constants.

The centre of mass of the three particles is at the point with coordinates $( 2 a , 4 a )$.
Find (i) the value of $k$
(ii) the value of $\mu$

Edexcel FM2 2023 June Q2

8 marks Standard +0.8

A particle of mass 2 kg is moving in a straight line on a smooth horizontal surface under the action of a horizontal force of magnitude $F$ newtons.

At time $t$ seconds $( t > 0 )$,

the particle is moving with speed $v \mathrm {~ms} ^ { - 1 }$
$F = 2 + v$

The time taken for the speed of the particle to increase from $5 \mathrm {~ms} ^ { - 1 }$ to $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is $T$ seconds.

Show that $T = 2 \ln \frac { 12 } { 7 }$ The distance moved by the particle as its speed increases from $5 \mathrm {~ms} ^ { - 1 }$ to $10 \mathrm {~ms} ^ { - 1 }$ is $D$ metres.
Find the exact value of $D$.

Edexcel FM2 2023 June Q3

9 marks Standard +0.8

\hspace{0pt} [In this question you may quote, without proof, the formula for the distance of the centre of mass of a uniform circular arc from its centre.]

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-08_816_483_338_790} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Five pieces of a uniform wire are joined together to form the rigid framework $O A B C O$ shown in Figure 1, where

$O A , O B$ and $B C$ are straight, with $O A = O B = B C = r$
arc $A B$ is one quarter of a circle with centre $O$ and radius $r$
arc $O C$ is one quarter of a circle of radius $r$
all five pieces of wire lie in the same plane
1. Show that the centre of mass of arc $A B$ is a distance $\frac { 2 r } { \pi }$ from $O A$.

Given that the distance of the centre of mass of the framework from $O A$ is $d$,

show that $\mathrm { d } = \frac { 7 r } { 2 ( 3 + ) }$ The framework is freely pivoted at $A$.
The framework is held in equilibrium, with $A O$ vertical, by a horizontal force of magnitude $F$ which is applied to the framework at $C$. Given that the weight of the framework is $W$

find $F$ in terms of $W$

Edexcel FM2 2023 June Q4

9 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-12_490_1177_219_507} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A smooth hemisphere of radius $a$ is fixed on a horizontal surface with its plane face in contact with the surface. The centre of the plane face of the hemisphere is $O$. A particle $P$ of mass $M$ is disturbed from rest at the highest point of the hemisphere.
When $P$ is still on the surface of the hemisphere and the radius from $O$ to $P$ is at an angle $\theta$ to the vertical,

the speed of $P$ is $v$
the normal reaction between the hemisphere and the particle is $R$, as shown in Figure 2.
1. Show that $\mathrm { R } = \mathrm { Mg } ( 3 \cos \theta - 2 )$
2. Find, in terms of $a$ and $g$, the speed of the particle at the instant when the particle leaves the surface of the hemisphere.

Edexcel FM2 2023 June Q5

7 marks Standard +0.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-16_730_442_223_877} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform lamina $O A B$ is modelled by the finite region bounded by the $x$-axis, the $y$-axis and the curve with equation $y = 9 - x ^ { 2 }$, for $x \geqslant 0$, as shown shaded in Figure 3. The unit of length on both axes is 1 m . The area of the lamina is $18 \mathrm {~m} ^ { 2 }$

Show that the centre of mass of the lamina is 3.6 m from $\boldsymbol { O B }$.
[0pt] [ Solutions relying on calculator technology are not acceptable.] A light string has one end attached to the lamina at $O$ and the other end attached to the ceiling. A second light string has one end attached to the lamina at $A$ and the other end attached to the ceiling.
The lamina hangs in equilibrium with the strings vertical and $O A$ horizontal.
The weight of the lamina is $W$
The tension in the string attached to the lamina at $A$ is $\lambda W$
Find the value of $\lambda$

Edexcel FM2 2023 June Q6

9 marks Standard +0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-20_611_782_210_660} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A hollow right circular cone, of internal base radius 0.6 m and height 0.8 m , is fixed with its axis vertical and its vertex $V$ pointing downwards, as shown in Figure 4. A particle $P$ of mass $m \mathrm {~kg}$ moves in a horizontal circle of radius 0.5 m on the rough inner surface of the cone. The particle $P$ moves with constant angular speed $\omega$ rads $^ { - 1 }$
The coefficient of friction between the particle $P$ and the inner surface of the cone is 0.25 Find the greatest possible value of $\omega$

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