Questions - Edexcel FM2 AS

Show that the centre of mass of the frame is $\frac { 3 } { 2 } a$ from $A B$. The frame is freely suspended from $A$. A horizontal force of magnitude $k M g$, where $k$ is a constant, is applied to the frame at $B$. The line of action of the force lies in the vertical plane containing the frame. The frame hangs in equilibrium with $A B$ vertical.
Find the value of $k$.

Edexcel FM2 AS 2018 June Q2

A car moves round a bend which is banked at a constant angle of $\theta ^ { \circ }$ to the horizontal.

When the car is travelling at a constant speed of $80 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ there is no sideways frictional force on the car. The car is modelled as a particle moving in a horizontal circle of radius 500 m .

Find the value of $\theta$.
Identify one limitation of this model. The speed of the car is increased so that it is now travelling at a constant speed of $90 \mathrm { kmh } ^ { - 1 }$ The car is still modelled as a particle moving in a horizontal circle of radius 500 m .
Describe the extra force that will now be acting on the car, stating the direction of this force.
VILU SIHI NI IIIUM ION OC VGHV SIHILNI IMAM ION OO VJYV SIHI NI JIIYM ION OC

Edexcel FM2 AS 2018 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{66c0f4c8-938e-4c05-93a7-99ea26ea0348-08_694_710_382_780} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The lamina $L$, shown in Figure 2, consists of a uniform square lamina $A B D F$ and two uniform triangular laminas $B D C$ and $F D E$. The square has sides of length $2 a$. The two triangles are identical. The straight lines $B D E$ and $F D C$ are perpendicular with $B D = D F = 2 a$ and $D C = D E = a$.
The mass per unit of area of the square is $M$.
The mass per unit area of each triangle is $3 M$.
The centre of mass of $L$ is at the point $G$.

Without doing any calculations, explain why $G$ lies on $A D$.
Show that the distance of $G$ from $D$ is $\frac { \sqrt { 2 } } { 2 } a$ The lamina $L$ is freely suspended from $B$ and hangs in equilibrium.
Find the size of the angle between $B E$ and the downward vertical.
V349 SIHI NI IMIMM ION OC VJYV SIHIL NI LIIIM ION OO VJYV SIHIL NI JIIYM ION OC

Edexcel FM2 AS 2018 June Q4

A particle, $P$, moves on the $x$-axis. At time $t$ seconds, $t \geqslant 0$, the velocity of $P$ is $v \mathrm {~ms} ^ { - 1 }$ in the direction of $x$ increasing and the acceleration of $P$ is $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$ in the direction of $x$ increasing.

When $t = 0$ the particle is at rest at the origin $O$.
Given that $a = \frac { 5 } { 2 } ( 5 - v )$

show that $v = 5 \left( 1 - \mathrm { e } ^ { - 2.5 t } \right)$
state the limiting value of $v$ as $t$ increases. At the instant when $v = 2.5$, the particle is $d$ metres from $O$.
Show that $d = 2 \ln 2 - 1$

Edexcel FM2 AS 2019 June Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{269e7aef-d7b7-4c3b-8d55-5a00696c97cc-02_369_625_301_721} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Five identical uniform rods are joined together to form the rigid framework $A B C D$ shown in Figure 1. Each rod has weight $W$ and length 4a. The points $A , B , C$ and $D$ all lie in the same plane. The centre of mass of the framework is at the point $G$.

Explain why $G$ is the midpoint of $A C$. The framework is suspended from the ceiling by two vertical light inextensible strings. One string is attached to the framework at $A$ and the other string is attached to the framework at $B$. The framework hangs freely in equilibrium with $A B$ horizontal.
Find
1. the tension in the string attached at $A$,
2. the tension in the string attached at $B$. A particle of weight $k W$ is now attached to the framework at $D$ and a particle of weight $2 k W$ is now attached to the framework at $C$. The framework remains in equilibrium with $A B$ horizontal and the strings vertical. Either string will break if the tension in it exceeds $6 W$.
Find the greatest possible value of $k$.

Edexcel FM2 AS 2019 June Q2

A car moves in a straight line along a horizontal road. The car is modelled as a particle. At time $t$ seconds, where $t \geqslant 0$, the speed of the car is $v \mathrm {~ms} ^ { - 1 }$

At the instant when $t = 0$, the car passes through the point $A$ with speed $2 \mathrm {~ms} ^ { - 1 }$
The acceleration, $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$, of the car is modelled by $$a = \frac { 4 } { 2 + v }$$ in the direction of motion of the car.

Use algebraic integration to show that $v = \sqrt { 8 t + 16 } - 2$ At the instant when the car passes through the point $B$, the speed of the car is $4 \mathrm {~ms} ^ { - 1 }$
Use algebraic integration to find the distance $A B$.

Edexcel FM2 AS 2019 June Q3

A light inextensible string has length $8 a$. One end of the string is attached to a fixed point $A$ and the other end of the string is attached to a fixed point $B$, with $A$ vertically above $B$ and $A B = 4 a$. A small ball of mass $m$ is attached to a point $P$ on the string, where $A P = 5 a$.

The ball moves in a horizontal circle with constant speed $v$, with both $A P$ and $B P$ taut.
The string will break if the tension in it exceeds $\frac { 3 m g } { 2 }$
By modelling the ball as a particle and assuming the string does not break,

show that $\frac { 9 a g } { 4 } < v ^ { 2 } \leqslant \frac { 27 a g } { 4 }$
find the least possible time needed for the ball to make one complete revolution.

Edexcel FM2 AS 2019 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{269e7aef-d7b7-4c3b-8d55-5a00696c97cc-14_888_1322_294_374} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The uniform triangular lamina $A B C D E$ is such that angle $C E A = 90 ^ { \circ } , C E = 9 a$ and $E A = 6 a$. The point $D$ lies on $C E$, with $D E = 3 a$. The point $B$ on $C A$ is such that $D B$ is parallel to $E A$ and $D B = 4 a$. The triangular lamina is folded along the line $D B$ to form the folded lamina $A B D E C F$, as shown in Figure 2. The distance of the centre of mass of the triangular lamina from $D C$ is $d _ { 1 }$
The distance of the centre of mass of the folded lamina from $D C$ is $d _ { 2 }$

Explain why $d _ { 1 } = d _ { 2 }$ The folded lamina is freely suspended from $B$ and hangs in equilibrium with $B A$ inclined at an angle $\alpha$ to the downward vertical through $B$.
Find, to the nearest degree, the size of angle $\alpha$.

Edexcel FM2 AS 2020 June Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0567d068-e23c-446e-9e11-f0c292972093-02_490_824_253_588} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a uniform rectangular lamina $A B C D$ with $A B = 2 a$ and $A D = a$ The mass of the lamina is $6 m$. A particle of mass $2 m$ is attached to the lamina at $A$, a particle of mass $m$ is attached to the lamina at $B$ and a particle of mass $3 m$ is attached to the lamina at $D$, to form a loaded lamina $L$ of total mass $12 m$.

Write down the distance of the centre of mass of $L$ from $A B$. You must give a reason for your answer.
Show that the distance of the centre of mass of $L$ from $A D$ is $\frac { 2 a } { 3 }$ A particle of mass $k m$ is now also attached to $L$ at $D$ to form a new loaded lamina $N$.
Show that the distance of the centre of mass of $N$ from $A B$ is $\frac { ( k + 6 ) a } { ( k + 12 ) }$ When $N$ is freely suspended from $A$ and is hanging in equilibrium, the side $A B$ makes an angle $\alpha$ with the vertical, where $\tan \alpha = \frac { 3 } { 2 }$
Find the value of $k$.

Edexcel FM2 AS 2020 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0567d068-e23c-446e-9e11-f0c292972093-06_531_837_258_632} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} One end of a string of length $3 a$ is attached to a point $A$ and the other end is attached to a point $B$ on a smooth horizontal table. The point $B$ is vertically below $A$ with $A B = a \sqrt { 3 }$ A small smooth bead, $P$, of mass $m$ is threaded on to the string. The bead $P$ moves on the table in a horizontal circle, with centre $B$, with constant speed $U$. Both portions, $A P$ and $B P$, of the string are taut, as shown in Figure 2. The string is modelled as being light and inextensible and the bead is modelled as a particle.

Show that $A P = 2 a$
Find, in terms of $m , U$ and $a$, the tension in the string.
Show that $U ^ { 2 } < a g \sqrt { 3 }$
Describe what would happen if $U ^ { 2 } > a g \sqrt { 3 }$
State briefly how the tension in the string would be affected if the string were not modelled as being light.

Edexcel FM2 AS 2020 June Q3

At time $t = 0$, a toy electric car is at rest at a fixed point $O$. The car then moves in a horizontal straight line so that at time $t$ seconds $( t > 0 )$ after leaving $O$, the velocity of the car is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the acceleration of the car is modelled as $( p + q v ) \mathrm { ms } ^ { - 2 }$, where $p$ and $q$ are constants.

When $t = 0$, the acceleration of the car is $3 \mathrm {~ms} ^ { - 2 }$
When $t = T$, the acceleration of the car is $\frac { 1 } { 2 } \mathrm {~ms} ^ { - 2 }$ and $v = 4$

Show that $$8 \frac { \mathrm {~d} v } { \mathrm {~d} t } = ( 24 - 5 v )$$
Find the exact value of $T$, simplifying your answer.

Edexcel FM2 AS 2021 June Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a7901165-1679-4d30-9444-0c27020e32ea-02_744_805_246_632} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform rod of length $72 a$ is cut into pieces. The pieces are used to make two rigid squares, $A B C D$ and $P Q R S$, with sides of length $10 a$ and $8 a$ respectively. The two squares are joined to form the rigid framework shown in Figure 1. The squares both lie in the same plane with the rod $A B$ parallel to the rod $P Q$.
Given that

$A D$ cuts $P Q$ in the ratio $3 : 5$
$D C$ cuts $Q R$ in the ratio 5:3
1. explain why the centre of mass of square $A B C D$ is at $Q$.
2. Find the distance of the centre of mass of the framework from $B$.

Edexcel FM2 AS 2021 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a7901165-1679-4d30-9444-0c27020e32ea-04_572_889_246_589} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A small smooth ring $P$, of mass $m$, is threaded onto a light inextensible string of length 4a. One end of the string is attached to a fixed point $A$ on a smooth horizontal table. The other end of the string is attached to a fixed point $B$ which is vertically above $A$. The ring moves in a horizontal circle with centre $A$ and radius $a$, as shown in Figure 2. The ring moves with constant angular speed $\sqrt { \frac { 2 g } { 3 a } }$ about $A B$.
The string remains taut throughout the motion.

Find, in terms of $m$ and $g$, the magnitude of the normal reaction between $P$ and the table. The angular speed of $P$ is now gradually increased.
Find, in terms of $a$ and $g$, the angular speed of $P$ at the instant when it loses contact with the table.
Explain how you have used the fact that $P$ is smooth.

Edexcel FM2 AS 2021 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a7901165-1679-4d30-9444-0c27020e32ea-08_547_410_246_829} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The uniform lamina $A B C D E F G H I J$ is shown in Figure 3.
The lamina has $A J = 8 a , A B = 5 a$ and $B C = D E = E F = F G = G H = H I = I J = 2 a$.
All the corners are right angles.

Show that the distance of the centre of mass of the lamina from $A J$ is $\frac { 49 } { 26 } a$ A light inextensible rope is attached to the lamina at $A$ and another light inextensible rope is attached to the lamina at $B$. The lamina hangs in equilibrium with both ropes vertical and $A B$ horizontal. The weight of the lamina is $W$.
Find, in terms of $W$, the tension in the rope attached to the lamina at $B$. The rope attached to $B$ breaks and subsequently the lamina hangs freely in equilibrium, suspended from $A$.
Find the size of the angle between $A J$ and the downward vertical.

Edexcel FM2 AS 2021 June Q4

A particle $P$ moves on the $x$-axis. At time $t$ seconds, $t \geqslant 0 , P$ is $x$ metres from the origin $O$ and moving with velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the direction of $x$ increasing, where

$$v = 5 \sin 2 t$$ When $t = 0 , x = 1$ and $P$ is at rest.

Find the magnitude and direction of the acceleration of $P$ at the instant when $P$ is next at rest.
Show that $1 \leqslant x \leqslant 6$
Find the total time, in the first $4 \pi$ seconds of the motion, for which $P$ is more than 3 metres from $O$
\includegraphics[max width=\textwidth, alt={}]{a7901165-1679-4d30-9444-0c27020e32ea-16_2260_52_309_1982}

Edexcel FM2 AS 2022 June Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{99e1d643-7408-4793-9ebc-b33c91bc5fab-02_474_716_246_676} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform plane lamina is in the shape of an isosceles trapezium $A B C D E F$, as shown shaded in Figure 1.

$B C E F$ is a square
$A B = C D = a$
$B C = 3 a$
1. Show that the distance of the centre of mass of the lamina from $A D$ is $\frac { 11 a } { 8 }$

The mass of the lamina is $M$
The lamina is suspended by two light vertical strings, one attached to the lamina at $A$ and the other attached to the lamina at $F$ The lamina hangs freely in equilibrium, with $B F$ horizontal.

Find, in terms of $M$ and $g$, the tension in the string attached at $A$

Edexcel FM2 AS 2022 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{99e1d643-7408-4793-9ebc-b33c91bc5fab-06_554_547_246_758} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Uniform wire is used to form the framework shown in Figure 2.
In the framework

$A B C D$ is a rectangle with $A D = 2 a$ and $D C = a$
$B E C$ is a semicircular arc of radius $a$ and centre $O$, where $O$ lies on $B C$

The diameter of the semicircle is $B C$ and the point $E$ is such that $O E$ is perpendicular to $B C$. The points $A , B , C , D$ and $E$ all lie in the same plane.

Show that the distance of the centre of mass of the framework from $B C$ is $$\frac { a } { 6 + \pi }$$ The framework is freely suspended from $A$ and hangs in equilibrium with $A E$ at an angle $\theta ^ { \circ }$ to the downward vertical.
Find the value of $\theta$. The mass of the framework is $M$.
A particle of mass $k M$ is attached to the framework at $B$.
The centre of mass of the loaded framework lies on $O A$.
Find the value of $k$.

Edexcel FM2 AS 2022 June Q3

A cyclist is travelling around a circular track which is banked at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac { 3 } { 4 }$

The cyclist moves with constant speed in a horizontal circle of radius $r$.
In an initial model,

the cyclist and her cycle are modelled as a particle
the track is modelled as being rough so that there is sideways friction between the tyres of the cycle and the track, with coefficient of friction $\mu$, where $\mu < \frac { 4 } { 3 }$
Using this model, the maximum speed that the cyclist can travel around the track in a horizontal circle of radius $r$, without slipping sideways, is $V$.
1. Show that $V = \sqrt { \frac { ( 3 + 4 \mu ) r g } { 4 - 3 \mu } }$

In a new simplified model,

the cyclist and her cycle are modelled as a particle
the motion is now modelled so that there is no sideways friction between the tyres of the cycle and the track

Using this new model, the speed that the cyclist can travel around the track in a horizontal circle of radius $r$, without slipping sideways, is $U$.

Find $U$ in terms of $r$ and $g$.

Show that $U < V$.

Edexcel FM2 AS 2022 June Q4

A particle $P$ moves on the $x$-axis. At time $t$ seconds the velocity of $P$ is $v \mathrm {~ms} ^ { - 1 }$ in the direction of $x$ increasing, where

$$v = \frac { 1 } { 2 } \left( 3 \mathrm { e } ^ { 2 t } - 1 \right) \quad t \geqslant 0$$ The acceleration of $P$ at time $t$ seconds is $a \mathrm {~ms} ^ { - 2 }$

Show that $a = 2 v + 1$
Find the acceleration of $P$ when $t = 0$
Find the exact distance travelled by $P$ in accelerating from a speed of $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to a speed of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

Edexcel FM2 AS 2023 June Q1

Three particles of masses $4 m , 2 m$ and $k m$ are placed at the points with coordinates $( - 3 , - 1 ) , ( 6,1 )$ and $( - 1,5 )$ respectively.

Given that the centre of mass of the three particles is at the point with coordinates $( \bar { x } , \bar { y } )$

show that $\bar { x } = \frac { - k } { k + 6 }$
find $\bar { y }$ in terms of $k$. Given that the centre of mass of the three particles lies on the line with equation $y = 2 x + 3$
find the value of $k$. A fourth particle is placed at the point with coordinates $( \lambda , 4 )$.
Given that the centre of mass of the four particles also lies on the line with equation $y = 2 x + 3$
find the value of $\lambda$.

Edexcel FM2 AS 2023 June Q2

A particle $P$ is moving along the $x$-axis.

At time $t$ seconds, $t \geqslant 0 , P$ has acceleration $a \mathrm {~ms} ^ { - 2 }$ and velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the direction of $x$ increasing, where $$v = \mathrm { e } ^ { 2 t } + 6 \mathrm { e } ^ { t } - k t$$ and $k$ is a positive constant.
When $t = \ln 2$, $a = 0$

Find the value of $k$. When $t = 0$, the particle passes through the fixed point $A$.
When $t = \ln 2$, the particle is $d$ metres from $A$.
Showing all stages of your working, find the value of $d$ correct to 2 significant figures.
[0pt] [Solutions relying entirely on calculator technology are not acceptable.]

Edexcel FM2 AS 2023 June Q3

A girl is cycling round a circular track.

The girl and her bicycle have a combined mass of 55 kg .
The coefficient of friction between the track surface and the tyres of the bicycle is $\mu$.
The track is banked at an angle of $15 ^ { \circ }$ to the horizontal.
The girl and her bicycle are modelled as a particle moving in a horizontal circle of radius 50 m
The minimum speed at which the girl can cycle round this circle without slipping is $4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Using the model, find the value of $\mu$.

Edexcel FM2 AS 2023 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8fcae18f-6588-4b71-8b7f-c8408de591f4-12_819_853_255_607} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform triangular lamina $A B C$ is isosceles, with $A C = B C$. The midpoint of $A B$ is $M$. The length of $A B$ is $18 a$ and the length of $C M$ is $18 a$. The triangular lamina $C D E$, with $D E = 6 a$ and $C D = 12 a$, has $E D$ parallel to $A B$ and $M D C$ is a straight line. Triangle $C D E$ is removed from triangle $A B C$ to form the lamina $L$, shown shaded in Figure 1. The distance of the centre of mass of $L$ from $M C$ is $d$.

Show that $d = \frac { 4 } { 7 } a$ The lamina $L$ is suspended by two light inextensible strings. One string is attached to $L$ at $A$ and the other string is attached to $L$ at $B$.
The lamina hangs in equilibrium in a vertical plane with the strings vertical and $A B$ horizontal.
The weight of $L$ is $W$
Find, in terms of $W$, the tension in the string attached to $L$ at $B$ The string attached to $L$ at $B$ breaks, so that $L$ is now suspended from $A$. When $L$ is hanging in equilibrium in a vertical plane, the angle between $A B$ and the downward vertical through $A$ is $\theta ^ { \circ }$
Find the value of $\theta$

Edexcel FM2 AS 2024 June Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-02_586_824_244_623} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform rod of length $24 a$ is cut into seven pieces which are used to form the framework $A B C D E F$ shown in Figure 1. It is given that

$A F = B E = C D = A B = F E = 4 a$
$B C = E D = 2 a$
the rods $A F , B E$ and $C D$ are parallel
the rods $A B , B C , F E$ and $E D$ are parallel
$A F$ is perpendicular to $A B$
the rods all lie in the same plane

The distance of the centre of mass of the framework from $A F$ is $d$.

Show that $d = \frac { 19 } { 6 } a$
Find the distance of the centre of mass of the framework from $A$.

Edexcel FM2 AS 2024 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-04_529_794_246_639} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A thin hollow hemisphere, with centre $O$ and radius $a$, is fixed with its axis vertical, as shown in Figure 2. A small ball $B$ of mass $m$ moves in a horizontal circle on the inner surface of the hemisphere. The circle has centre $C$ and radius $r$. The point $C$ is vertically below $O$ such that $O C = h$. The ball moves with constant angular speed $\omega$
The inner surface of the hemisphere is modelled as being smooth and $B$ is modelled as a particle. Air resistance is modelled as being negligible.

Show that $\omega ^ { 2 } = \frac { g } { h }$ Given that the magnitude of the normal reaction between $B$ and the surface of the hemisphere is $3 m g$
find $\omega$ in terms of $g$ and $a$.
State how, apart from ignoring air resistance, you have used the fact that $B$ is modelled as a particle.

Questions — Edexcel FM2 AS (30 questions)

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