Questions M3 - A-Level Maths

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Edexcel M3 2017 January Q5

A particle $P$ of mass $4 m$ is attached to one end of a light elastic string of natural length $l$ and modulus of elasticity 3 mg . The other end of the string is attached to a fixed point $O$ on a rough horizontal table. The particle lies at rest at the point $A$ on the table, where $O A = \frac { 4 } { 3 } l$. The coefficient of friction between $P$ and the table is $\mu$.
1. Show that $\mu \geqslant \frac { 1 } { 4 }$
The particle is now moved along the table to the point $B$, where $O B = 2 l$, and released from rest. Given that $\mu = \frac { 2 } { 5 }$
show that $P$ comes to rest before the string becomes slack.
(5)

Edexcel M3 2017 January Q6

6. One end of a light elastic string, of natural length 5 l and modulus of elasticity 20 mg , is attached to a fixed point $A$. A particle $P$ of mass $2 m$ is attached to the free end of the string and $P$ hangs freely in equilibrium at the point $B$.

Find the distance $A B$.
(3) The particle is now pulled vertically downwards from $B$ to the point $C$ and released from rest. In the subsequent motion the string does not become slack.
Show that $P$ moves with simple harmonic motion with centre $B$.
Find the period of this motion. The greatest speed of $P$ during this motion is $\frac { 1 } { 5 } \sqrt { g l }$
Find the amplitude of this motion. The point $D$ is the midpoint of $B C$ and the point $E$ is the highest point reached by $P$.
Find the time taken by $P$ to move directly from $D$ to $E$.

Edexcel M3 2017 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{85d8fc7d-8863-419e-8eef-8751a6fb6315-11_412_1054_260_447} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A hollow sphere has internal radius $r$ and centre $O$. A bowl with a plane circular rim is formed by removing part of the sphere. The bowl is fixed to a horizontal floor with the rim uppermost and horizontal. The point $B$ is the lowest point of the inner surface of the bowl. The point $A$, where angle $A O B = 120 ^ { \circ }$, lies on the rim of the bowl, as shown in Figure 4. A particle $P$ of mass $m$ is projected from $A$, with speed $U$ at $90 ^ { \circ }$ to $O A$, and moves on the smooth inner surface of the bowl. The motion of $P$ takes place in the vertical plane $O A B$.

Find, in terms of $m , g , U$ and $r$, the magnitude of the force exerted on $P$ by the bowl at the instant when $P$ passes through $B$.
Find, in terms of $g , U$ and $r$, the greatest height above the floor reached by $P$. Given that $U > \sqrt { 2 g r }$
show that, after leaving the surface of the bowl, $P$ does not fall back into the bowl.

Edexcel M3 2018 January Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-02_333_890_264_529} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform solid $S$ consists of a right solid circular cone of base radius $r$ and a right solid cylinder, also of radius $r$. The cone has height $4 h$ and the centre of the plane face of the cone is $O$. The cylinder has height $3 h$. The cone and cylinder are joined so that the plane face of the cone coincides with one of the plane faces of the cylinder, as shown in Figure 1. Find the distance from $O$ to the centre of mass of $S$.

Edexcel M3 2018 January Q2

A particle of mass 0.9 kg is attached to one end of a light elastic string, of natural length 1.2 m and modulus of elasticity 29.4 N . The other end of the string is attached to a fixed point $A$ on a ceiling.

The particle is held at $A$ and then released from rest. The particle first comes to instantaneous rest at the point $B$. Find the distance $A B$.
(5)

Edexcel M3 2018 January Q3

A particle $P$ of mass 0.4 kg moves along the $x$-axis in the positive direction. At time $t = 0 , P$ passes through the origin $O$ with speed $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At time $t$ seconds $P$ is $x$ metres from $O$ and the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resultant force acting on $P$ has magnitude $\frac { 8 } { ( t + 4 ) ^ { 2 } } \mathrm {~N}$ and is directed towards $O$.
1. Show that $v = \frac { 20 } { t + 4 } + 5$
When $v = 6 , x = a + b \ln 5$, where $a$ and $b$ are integers.
Using algebraic integration, find the value of $a$ and the value of $b$.

Edexcel M3 2018 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-10_547_841_244_555} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A small smooth bead $P$ is threaded on a light inextensible string of length $8 a$. 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 the fixed point $B$, where $B$ is vertically above $A$ and $A B = 4 a$, as shown in Figure 2. The bead moves with constant angular speed, in a horizontal circle, centre $A$, with $A P$ horizontal. The bead remains in contact with the table and both parts of the string, $A P$ and $B P$, are taut. The time for $P$ to complete one revolution is $S$. Show that $\quad S \geqslant \pi \sqrt { \frac { 6 a } { g } }$

Edexcel M3 2018 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-14_510_723_269_607} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows the finite region $R$ which is bounded by part of the curve with equation $y = \sin x$, the $x$-axis and the line with equation $x = \frac { \pi } { 2 }$. A uniform solid $S$ is formed by rotating $R$ through $2 \pi$ radians about the $x$-axis. Using algebraic integration,

show that the volume of $S$ is $\frac { \pi ^ { 2 } } { 4 }$
find, in terms of $\pi$, the $x$ coordinate of the centre of mass of $S$.

Edexcel M3 2018 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-18_483_730_242_609} \captionsetup{labelformat=empty} \caption{Figure 4}

\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 at the point $A$, where $O A = l$ and $O A$ is horizontal. The particle is then projected vertically downwards from $A$ with speed $\sqrt { 2 g l }$, as shown in Figure 4 . When the string makes an angle $\theta$ with the downward vertical through $O$ and the string is still taut, the tension in the string is $T$.

Show that $T = m g ( 3 \cos \theta + 2 )$ At the instant when the particle reaches the point $B$, the string becomes slack.
Find the speed of $P$ at $B$.
Find the greatest height above $O$ reached by $P$ in the subsequent motion.

Edexcel M3 2018 January Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-22_197_945_251_497} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} The fixed points $A$ and $B$ are 4.2 m apart on a smooth horizontal floor. One end of a light elastic spring, of natural length 1.8 m and modulus of elasticity 20 N , is attached to a particle $P$ and the other end is attached to $A$. One end of another light elastic spring, of natural length 0.9 m and modulus of elasticity 15 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 5.

Show that $A O = 2.7 \mathrm {~m}$. The particle $P$ now receives an impulse acting in the direction $O B$ and moves away from $O$ towards $B$. In the subsequent motion $P$ does not reach $B$.
Show that $P$ moves with simple harmonic motion about centre $O$. The mass of $P$ is 10 kg and the magnitude of the impulse is $J \mathrm { Ns }$. Given that $P$ first comes to instantaneous rest at the point $C$ where $A C = 2.9 \mathrm {~m}$,
1. find the value of $J$,
2. find the time taken by $P$ to travel a total distance of 0.5 m from when it first leaves $O$.

Edexcel M3 2019 January Q1

A particle $P$ moves on the $x$-axis. At time $t$ seconds, $t \geqslant 0$, the displacement of $P$ from the origin $O$ is $x$ metres and the acceleration of $P$ is $\left( \frac { 7 } { 2 } - 2 x \right) \mathrm { m } \mathrm { s } ^ { - 2 }$, measured in the positive $x$ direction. At time $t = 0 , P$ passes through $O$ moving with speed $3 \mathrm {~ms} ^ { - 1 }$ in the positive $x$ direction. Find the distance of $P$ from $O$ when $P$ first comes to instantaneous rest.
(6)

Edexcel M3 2019 January Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-04_573_456_264_712} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A small ball $P$ of mass $m$ is attached to the midpoint of a light inextensible string of length $2 a$. The ends of the string are attached to fixed points $A$ and $B$, where $A$ is vertically above $B$ and $A B = a$, as shown in Figure 1. The system rotates about the line $A B$ with constant angular speed $\omega$. The ball moves in a horizontal circle with both parts of the string taut. The tension in the string must be less than $3 m g$ otherwise the string will break. Given that the time taken by the ball to complete one revolution is $S$, show that $$\pi \sqrt { \frac { a } { g } } < S < \pi \sqrt { \frac { k a } { g } }$$ stating the value of the constant $k$.

Edexcel M3 2019 January Q3

3. A particle $P$ is moving in a straight line with simple harmonic motion between two points $A$ and $B$, where $A B$ is $2 a$ metres. The point $C$ lies on the line $A B$ and $A C = \frac { 1 } { 2 } a$ metres. The particle passes through $C$ with speed $\frac { 3 a \sqrt { 3 } } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the period of the motion. The maximum magnitude of the acceleration of $P$ is $45 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Find
the value of $a$,
the maximum speed of $P$. The point $D$ lies on $A B$ and $P$ takes a quarter of one period to travel directly from $C$ to $D$.
Find the distance CD.

Edexcel M3 2019 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-12_364_718_278_612} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The ends of a light elastic string, of natural length $4 l$ and modulus of elasticity $\lambda$, are attached to two fixed points $A$ and $B$, where $A B$ is horizontal and $A B = 4 l$. A particle $P$ of mass $2 m$ is attached to the midpoint of the string. The particle hangs freely in equilibrium at a distance $\frac { 3 } { 2 } l$ vertically below the midpoint of $A B$, as shown in Figure 2.

Show that $\lambda = \frac { 20 } { 3 } m g$. The particle is pulled vertically downwards from its equilibrium position until the total length of the string is 6l. The particle is then released from rest.
Show that $P$ comes to instantaneous rest before reaching the line $A B$.

Edexcel M3 2019 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-16_492_442_237_744} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The region $R$, shown shaded in Figure 3, is bounded by the circle with centre $O$ and radius $r$, the line with equation $x = \frac { 3 } { 5 } r$ and the $x$-axis. The region is rotated through one complete revolution about the $x$-axis to form a uniform solid $S$.

Use algebraic integration to show that the $x$ coordinate of the centre of mass of $S$ is $\frac { 48 } { 65 } r$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-16_394_643_1311_653} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A bowl is made from a uniform solid hemisphere of radius 6 cm by removing a hemisphere of radius 5 cm . Both hemispheres have the same centre $A$ and the same axis of symmetry. The bowl is fixed with its open plane face uppermost and horizontal. Liquid is poured into the bowl. The depth of the liquid is 2 cm , as shown in Figure 4. The mass of the empty bowl is $5 M \mathrm {~kg}$ and the mass of the liquid is $2 M \mathrm {~kg}$.
Find, to 3 significant figures, the distance from $A$ to the centre of mass of the bowl with its liquid.

Edexcel M3 2019 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-20_497_643_237_653} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a hollow sphere, with centre $O$ and internal radius $a$, which is fixed to a horizontal surface. A particle $P$ of mass $m$ is projected horizontally with speed $\sqrt { \frac { 7 a g } { 2 } }$ from the lowest point $A$ of the inner surface of the sphere. The particle moves in a vertical circle with centre $O$ on the smooth inner surface of the sphere. The particle passes through the point $B$, on the inner surface of the sphere, where $O B$ is horizontal.

Find, in terms of $m$ and $g$, the normal reaction exerted on $P$ by the surface of the sphere when $P$ is at $B$. The particle leaves the inner surface of the sphere at the point $C$, where $O C$ makes an angle $\theta , \theta > 0$, with the upward vertical.
Show that, after leaving the surface of the sphere at $C$, the particle is next in contact with the surface at $A$.
END

Edexcel M3 2021 January Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-02_469_758_251_593} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The region $R$, shown shaded in Figure 1, is bounded by the curve with equation $y = \frac { 1 } { x }$, the line with equation $x = 1$, the positive $x$-axis and the line with equation $x = a$ where $a > 1$ A uniform solid $S$ is formed by rotating $R$ through $2 \pi$ radians about the $x$-axis.

Show that the volume of $S$ is $$\pi \left( 1 - \frac { 1 } { a } \right)$$
Find the $x$ coordinate of the centre of mass of $S$.

Edexcel M3 2021 January Q2

2. A particle $P$ of mass $m$ is at a distance $x$ above the surface of the Earth. The Earth exerts a gravitational force on $P$. This force is directed towards the centre of the Earth. The magnitude of this force is inversely proportional to the square of the distance of $P$ from the centre of the Earth. At the surface of the Earth the acceleration due to gravity is $g$. The Earth is modelled as a fixed sphere of radius $R$.

Show that the magnitude of the gravitational force on $P$ is $\frac { m g R ^ { 2 } } { ( x + R ) ^ { 2 } }$ A particle is released from rest from a point above the surface of the Earth. When the particle is at a distance $R$ above the surface of the Earth, the particle has speed $U$. Air resistance is modelled as being negligible.
Find, in terms of $U , g$ and $R$, the speed of the particle when it strikes the surface of the Earth.

VIAV SIHI NI III IM I ON OC VIAV SIMI NI III M M O N OO VIUV SIMI NI JIIYM ION OC

Edexcel M3 2021 January Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-08_506_527_251_712} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A fairground ride consists of a cabin $C$ that travels in a horizontal circle with a constant angular speed about a fixed vertical central axis. The cabin is attached to one end of each of two rigid arms, each of length 5 m . The other end of the top arm is attached to the fixed point $A$ at the top of the central axis of the ride. The other end of the lower arm is attached to the fixed point $B$ on the central axis, where $A B$ is 8 m , as shown in Figure 2. Both arms are free to rotate about the central axis. The arms are modelled as light inextensible rods. The cabin, together with the people inside, is modelled as a particle. The cabin completes one revolution every 2 seconds. Given that the combined mass of the cabin and the people is 600 kg ,

find
1. the tension in the upper arm of the ride,
2. the tension in the lower arm of the ride. In a refined model, it is assumed that both arms stretch to a length of 5.1 m .
State how this would affect the sum of the tensions in the two arms, justifying your answer.

Edexcel M3 2021 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-12_442_506_251_721} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform right solid cone $C$ has diameter $6 a$ and height $8 a$, as shown in Figure 3.
The solid $S$ is formed by removing a cone of height $4 a$ from the top of $C$ and then removing an identical, inverted cone. The vertex of the removed cone is at the point $O$ in the centre of the base of $C$, as shown in Figure 4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-12_236_502_1126_721} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure}

Find the distance of the centre of mass of $S$ from $O$.
(5) The point $A$ lies on the circumference of the base of $S$ and the point $B$ lies on the circumference of the top of $S$. The points $O$, $A$ and $B$ all lie in the same vertical plane, as shown in Figure 5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-12_248_449_1845_749} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} The solid $S$ is freely suspended from the point $B$ and hangs in equilibrium.
Find the size of the angle that $A B$ makes with the downward vertical.

Edexcel M3 2021 January Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-16_720_232_251_858} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} The fixed points, $A$ and $B$, are a distance $10 a$ apart, with $B$ vertically above $A$. One end of a light elastic string, of natural length $2 a$ and modulus of elasticity $2 m g$, is attached to a particle $P$ of mass $m$ and the other end is attached to $A$. One end of another light elastic string, of natural length $4 a$ and modulus of elasticity $6 m g$, is attached to $P$ and the other end is attached to $B$. The particle $P$ rests in equilibrium at the point $C$, as shown in Figure 6.

Show that each string has an extension of $2 a$.
(5) The particle $P$ is now pulled down vertically, so that it is a distance $a$ below $C$ and then released from rest.
Show that in the subsequent motion, $P$ performs simple harmonic motion.
Find, in terms of $a$ and $g$, the speed of $P$ when it is a distance $\frac { 7 } { 2 } a$ above $A$.

Edexcel M3 2021 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-20_789_858_121_536} \captionsetup{labelformat=empty} \caption{Figure 7}

\end{figure} A particle of mass $m$ is attached to one end of a light inextensible string of length $2 a$. The other end of the string is attached to a fixed point $O$. The particle is initially held at the point $A$ with the string taut and $O A$ making an angle of $60 ^ { \circ }$ with the downward vertical. The particle is then projected upwards with a speed of $3 \sqrt { a g }$, perpendicular to $O A$, in the vertical plane containing $O A$, as shown in Figure 7. In an initial model of the motion of the particle, it is assumed that the string does not break. Using this model,

show that the particle performs complete vertical circles. In a refined model it is assumed that the string will break if the tension in it exceeds 7 mg . Using this refined model,
show that the particle still performs complete vertical circles.
\includegraphics[max width=\textwidth, alt={}, center]{8a687d17-ec7e-463f-84dd-605f5c230db1-20_2249_50_314_1982}