Questions - Edexcel FM2 - A-Level Maths

Edexcel FM2 2019 June Q2

A particle, $P$, of mass 0.4 kg is moving along the positive $x$-axis, in the positive $x$ direction under the action of a single force. At time $t$ seconds, $t > 0 , P$ is $x$ metres from the origin $O$ and the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The force is acting in the direction of $x$ increasing and has magnitude $\frac { k } { v }$ newtons, where $k$ is a constant.

At $x = 3 , v = 2$ and at $x = 6 , v = 2.5$

Show that $v ^ { 3 } = \frac { 61 x + 9 } { 24 }$ The time taken for the speed of $P$ to increase from $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is $T$ seconds.
Use algebraic integration to show that $T = \frac { 81 } { 61 }$

Edexcel FM2 2019 June Q3

Numerical (calculator) integration is not acceptable in this question.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-08_547_550_303_753} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The shaded region $O A B$ in Figure 2 is bounded by the $x$-axis, the line with equation $x = 4$ and the curve with equation $y = \frac { 1 } { 4 } ( x - 2 ) ^ { 3 } + 2$. The point $A$ has coordinates (4, 4) and the point $B$ has coordinates $( 4,0 )$. A uniform lamina $L$ has the shape of $O A B$. The unit of length on both axes is one centimetre. The centre of mass of $L$ is at the point with coordinates $( \bar { x } , \bar { y } )$. Given that the area of $L$ is $8 \mathrm {~cm} ^ { 2 }$,

show that $\bar { y } = \frac { 8 } { 7 }$ The lamina is freely suspended from $A$ and hangs in equilibrium with $A B$ at an angle $\theta ^ { \circ }$ to the downward vertical.
Find the value of $\theta$.

Edexcel FM2 2019 June Q4

A flagpole, $A B$, is 4 m long. The flagpole is modelled as a non-uniform rod so that, at a distance $x$ metres from $A$, the mass per unit length of the flagpole, $m \mathrm {~kg} \mathrm {~m} ^ { - 1 }$, is given by $m = 18 - 3 x$.
1. Show that the mass of the flagpole is 48 kg .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-12_515_439_502_806} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The end $A$ of the flagpole is fixed to a point on a vertical wall. A cable has one end attached to the midpoint of the flagpole and the other end attached to a point on the wall that is vertically above $A$. The cable is perpendicular to the flagpole. The flagpole and the cable lie in the same vertical plane that is perpendicular to the wall. A small ball of mass 4 kg is attached to the flagpole at $B$. The cable holds the flagpole and ball in equilibrium, with the flagpole at $45 ^ { \circ }$ to the wall, as shown in Figure 3. The tension in the cable is $T$ newtons.
The cable is modelled as a light inextensible string and the ball is modelled as a particle.
Using the model, find the value of $T$.
Give a reason why the answer to part (b) is not likely to be the true value of $T$.

Edexcel FM2 2019 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-16_560_560_283_749} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} The region $R$, shown shaded in Figure 4, is bounded by part of the curve with equation $y ^ { 2 } = 2 x$, the line with equation $y = 2$ and the $y$-axis. The unit of length on both axes is one centimetre. A uniform solid, $S$, is formed by rotating $R$ through $360 ^ { \circ }$ about the $y$-axis.
Given that the volume of $S$ is $\frac { 8 } { 5 } \pi \mathrm {~cm} ^ { 3 }$,

show that the centre of mass of $S$ is $\frac { 1 } { 3 } \mathrm {~cm}$ from its plane face. A uniform solid cylinder, $C$, has base radius 2 cm and height 4 cm . The cylinder $C$ is attached to $S$ so that the plane face of $S$ coincides with a plane face of $C$, to form the paperweight $P$, shown in Figure 5. The density of the material used to make $S$ is three times the density of the material used to make $C$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-16_572_456_1617_758} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} The plane face of $P$ rests in equilibrium on a desk lid that is inclined at an angle $\theta ^ { \circ }$ to the horizontal. The lid is sufficiently rough to prevent $P$ from slipping. Given that $P$ is on the point of toppling,
find the value of $\theta$.

Edexcel FM2 2019 June Q6

The points $A$ and $B$ lie on a smooth horizontal surface with $A B = 4.5 \mathrm {~m}$.

A light elastic string has natural length 1.5 m and modulus of elasticity 15 N . One end of the string is attached to $A$ and the other end of the string is attached to $B$. A particle, $P$, of mass 0.2 kg , is attached to the stretched string so that $A P B$ is a straight line and $A P = 1.5 \mathrm {~m}$. The particle rests in equilibrium on the surface. The particle is now moved directly towards $A$ and is held on the surface so $A P B$ is a straight line with $A P = 1 \mathrm {~m}$. The particle is released from rest.

Prove that $P$ moves with simple harmonic motion.
Find
1. the maximum speed of $P$ during the motion,
2. the maximum acceleration of $P$ during the motion.
Find the total time, in each complete oscillation of $P$, for which the speed of $P$ is greater than $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Edexcel FM2 2019 June Q7

A particle, $P$, of mass $m$ is attached to one end of a light rod of length $L$. The other end of the rod is attached to a fixed point $O$ so that the rod is free to rotate in a vertical plane about $O$. The particle is held with the rod horizontal and is then projected vertically downwards with speed $u$. The particle first comes to instantaneous rest at the point $A$.
1. Explain why the acceleration of $P$ at $A$ is perpendicular to $O A$.
At the instant when $P$ is at the point $A$ the acceleration of $P$ is in a direction making an angle $\theta$ with the horizontal. Given that $u ^ { 2 } = \frac { 2 g L } { 3 }$,
find
1. the magnitude of the acceleration of $P$ at the point $A$,
2. the size of $\theta$.
Find, in terms of $m$ and $g$, the magnitude of the tension in the rod at the instant when $P$ is at its lowest point.

Edexcel FM2 2020 June Q1

Three particles of masses $3 m$, $4 m$ and $2 m$ are placed at the points $( - 2,2 ) , ( 3,1 )$ and ( $p , p$ ) respectively.

The value of $p$ is such that the distance of the centre of mass of the three particles from the point ( 0,0 ) is as small as possible. Find the value of $p$.

Edexcel FM2 2020 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{962c2b40-3c45-4eed-a0af-a59068bda0e1-04_506_590_255_429} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{962c2b40-3c45-4eed-a0af-a59068bda0e1-04_296_327_456_1311} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A uniform plane figure $R$, shown shaded in Figure 1, is bounded by the $x$-axis, the line with equation $x = \ln 5$, the curve with equation $y = 8 \mathrm { e } ^ { - x }$ and the line with equation $x = \ln 2$. The unit of length on each axis is one metre. The area of $R$ is $2.4 \mathrm {~m} ^ { 2 }$
The centre of mass of $R$ is at the point with coordinates $( \bar { x } , \bar { y } )$.

Use algebraic integration to show that $\bar { y } = 1.4$ Figure 2 shows a uniform lamina $A B C D$, which is the same size and shape as $R$. The lamina is freely suspended from $C$ and hangs in equilibrium with $C B$ at an angle $\theta ^ { \circ }$ to the downward vertical.
Find the value of $\theta$

Edexcel FM2 2020 June Q3

A particle $P$ of mass 0.5 kg is moving along the positive $x$-axis in the direction of $x$ increasing. At time $t$ seconds $( t \geqslant 0 ) , P$ is $x$ metres from the origin $O$ and the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resultant force acting on $P$ is directed towards $O$ and has magnitude $k v ^ { 2 } \mathrm {~N}$, where $k$ is a positive constant.

When $x = 1 , v = 4$ and when $x = 2 , v = 2$

Show that $v = a b ^ { x }$, where $a$ and $b$ are constants to be found. The time taken for the speed of $P$ to decrease from $4 \mathrm {~ms} ^ { - 1 }$ to $2 \mathrm {~ms} ^ { - 1 }$ is $T$ seconds.
Show that $T = \frac { 1 } { 4 \ln 2 }$

Edexcel FM2 2020 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{962c2b40-3c45-4eed-a0af-a59068bda0e1-12_492_412_246_824} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform solid cylinder of base radius $r$ and height $\frac { 4 } { 3 } r$ has the same density as a uniform solid hemisphere of radius $r$. The plane face of the hemisphere is joined to a plane face of the cylinder to form the composite solid $S$ shown in Figure 3. The point $O$ is the centre of the plane face of $S$.

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

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

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

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

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

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

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

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

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

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

\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

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

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

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

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 $g$

Questions — Edexcel FM2 (51 questions)

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Edexcel FM2 2019 June Q1

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Edexcel FM2 2022 June Q1

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