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Edexcel M3 2011 June Q5

12 marks Standard +0.8

A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $l$ and modulus of elasticity $3 m g$. 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 $O A = \frac { 7 } { 6 } l$. The coefficient of friction between $P$ and the table is $\mu$.
1. Show that $\mu \geqslant \frac { 1 } { 2 }$.
The particle is now moved along the table to the point $B$, where $O B = \frac { 3 } { 2 } l$, and released from rest. Given that $\mu = \frac { 1 } { 2 }$, find
the speed of $P$ at the instant when the string becomes slack,
the total distance moved by $P$ before it comes to rest again.

Edexcel M3 2011 June Q6

12 marks Standard +0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-11_574_540_226_701} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} A particle $P$ 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 point $B$ is vertically above $O$ and the point $C$ is vertically below $O$, with $O B = O C = a$, as shown in Figure 5. The particle is projected vertically upwards with speed $3 \sqrt { } ( a g )$.

Show that $P$ will pass through $B$.
Find the speed of $P$ as it reaches $C$. As $P$ passes through $C$ it receives an impulse. Immediately after this, the speed of $P$ is $\frac { 5 } { 12 } \sqrt { } ( 11 a g )$ and the direction of motion of $P$ is unchanged.
Find the angle between the string and the downward vertical when $P$ comes to instantaneous rest.

Edexcel M3 2011 June Q7

15 marks Challenging +1.2

A particle $P$ of mass 0.5 kg is attached to the mid-point of a light elastic string of natural length 1.4 m and modulus of elasticity 2 N . The ends of the string are attached to the points $A$ and $B$ on a smooth horizontal table, where $A B = 2 \mathrm {~m}$. The mid-point of $A B$ is $O$ and the point $C$ is on the table between $O$ and $B$ where $O C = 0.2 \mathrm {~m}$. At time $t = 0$ the particle is released from rest at $C$. At time $t$ seconds the length of the string $A P$ is $( 1 + x ) \mathrm { m }$.
1. Show that the tension in $B P$ is $\frac { 2 } { 7 } ( 3 - 10 x ) \mathrm { N }$.
2. Find, in terms of $x$, the tension in $A P$.
3. Show that $P$ performs simple harmonic motion with period $2 \pi \sqrt { } \left( \frac { 7 } { 80 } \right)$ s.
4. Find the greatest speed of $P$ during the motion.
The point $D$ lies between $O$ and $A$, where $O D = 0.1 \mathrm {~m}$.
Find the time taken by $P$ to move directly from $C$ to $D$.

Edexcel M3 2012 June Q1

9 marks Standard +0.3

A particle $P$ is moving along the positive $x$-axis. At time $t = 0 , P$ is at the origin $O$. At time $t$ seconds, $P$ is $x$ metres from $O$ and has velocity $v = 2 \mathrm { e } ^ { - x } \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the direction of $x$ increasing.
1. Find the acceleration of $P$ in terms of $x$.
2. Find $x$ in terms of $t$.
3. A particle $P$ moves in a straight line with simple harmonic motion about a fixed centre $O$. The period of the motion is $\frac { \pi } { 2 }$ seconds. At time $t$ seconds the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When $t = 0 , P$ is at $O$ and $v = 6$. Find
4. the greatest distance of $P$ from $O$ during the motion,
5. the greatest magnitude of the acceleration of $P$ during the motion,
6. the smallest positive value of $t$ for which $P$ is 1 m from $O$.

Edexcel M3 2012 June Q3

10 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2c7ac0e1-14fd-4e50-935a-d82e7127c2f8-05_638_1320_233_312} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A particle $Q$ of mass 5 kg is attached by two light inextensible strings to two fixed points $A$ and $B$ on a vertical pole. Each string has length 0.6 m and $A$ is 0.4 m vertically above $B$, as shown in Figure 1. Both strings are taut and $Q$ is moving in a horizontal circle with constant angular speed $10 \mathrm { rad } \mathrm { s } ^ { - 1 }$. Find the tension in

$A Q$,
$B Q$.

Edexcel M3 2012 June Q4

10 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{2c7ac0e1-14fd-4e50-935a-d82e7127c2f8-07_707_481_228_733} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows the cross-section $A V B C$ of the solid $S$ formed when a uniform right circular cone of base radius $a$ and height $a$, is removed from a uniform right circular cone of base radius $a$ and height $2 a$. Both cones have the same axis VCO, where $O$ is the centre of the base of each cone.

Show that the distance of the centre of mass of $S$ from the vertex $V$ is $\frac { 5 } { 4 } a$. The mass of $S$ is $M$. A particle of mass $k M$ is attached to $S$ at $B$. The system is suspended by a string attached to the vertex $V$, and hangs freely in equilibrium. Given that $V A$ is at an angle $45 ^ { \circ }$ to the vertical through $V$,
find the value of $k$.

Edexcel M3 2012 June Q5

12 marks Standard +0.3

A fixed smooth sphere has centre $O$ and radius $a$. A particle $P$ is placed on the surface of the sphere at the point $A$, where $O A$ makes an angle $\alpha$ with the upward vertical through $O$. The particle is released from rest at $A$. When $O P$ makes an angle $\theta$ to the upward vertical through $O , P$ is on the surface of the sphere and the speed of $P$ is $v$.

Given that $\cos \alpha = \frac { 3 } { 5 }$

show that $$v ^ { 2 } = \frac { 2 g a } { 5 } ( 3 - 5 \cos \theta )$$
find the speed of $P$ at the instant when it loses contact with the sphere.

Edexcel M3 2012 June Q6

12 marks Challenging +1.2

6. Figure 3 Figure 3 shows a uniform equilateral triangular lamina $P R T$ with sides of length $2 a$.

Using calculus, prove that the centre of mass of $P R T$ is at a distance $\frac { 2 \sqrt { } 3 } { 3 } a$ from $R$. (6) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2c7ac0e1-14fd-4e50-935a-d82e7127c2f8-11_545_588_1121_678} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The circular sector $P Q U$, of radius $a$ and centre $P$, and the circular sector TUS, of radius $a$ and centre $T$, are removed from $P R T$ to form the uniform lamina $Q R S U$ shown in Figure 4.
Show that the distance of the centre of mass of QRSU from $U$ is $\frac { 2 a } { 3 \sqrt { 3 } - \pi }$

Edexcel M3 2012 June Q7

14 marks Challenging +1.2

7. A particle $B$ of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.75 m and modulus of elasticity 24.5 N . The other end of the string is attached to a fixed point $A$. The particle is hanging in equilibrium at the point $E$, vertically below $A$.

Show that $A E = 0.9 \mathrm {~m}$. The particle is held at $A$ and released from rest. The particle first comes to instantaneous rest at the point $C$.
Find the distance $A C$.
Show that while the string is taut, $B$ is moving with simple harmonic motion with centre $E$.
Calculate the maximum speed of $B$.

Find the acceleration of $P$ immediately after it is released from rest. The particle comes to instantaneous rest for the first time at the point $C$.
Find the distance $B C$.

Edexcel M3 2013 June Q4

10 marks Standard +0.8

A particle $P$ is moving along the positive $x$-axis. At time $t$ seconds, $t \geqslant 0 , P$ is $x$ metres from the origin $O$ and is moving away from $O$ with velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $v = \frac { 4 } { ( x + 2 ) }$. When $t = 0 , P$ is at $O$. Find
1. the distance of $P$ from $O$ when $t = 2$
2. the magnitude and direction of the acceleration of $P$ when $t = 2$

Edexcel M3 2013 June Q5

12 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{daa795f0-2c5e-4617-a295-fbe74c22be4a-08_504_1429_212_264} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Part of a hollow spherical shell, centre $O$ and radius $r$, forms a bowl with a plane circular rim. The bowl is fixed to a horizontal surface at $A$ with the rim uppermost and horizontal. The point $A$ is the lowest point of the bowl. The point $B$, where $\angle A O B = \alpha$ and $\tan \alpha = \frac { 3 } { 4 }$, is on the rim of the bowl, as shown in Figure 2. A small smooth marble $M$ is placed inside the bowl at $A$, and given an initial horizontal speed $\sqrt { } ( g r )$. The motion of $M$ takes place in the vertical plane $O A B$.

Show that the speed of $M$ as it reaches $B$ is $\sqrt { } \left( \frac { 3 } { 5 } g r \right)$. After leaving the surface of the bowl at $B , M$ moves freely under gravity and first strikes the horizontal surface at the point $C$. Given that $r = 0.4 \mathrm {~m}$,
find the distance $A C$.

Edexcel M3 2013 June Q6

15 marks Challenging +1.2

6. (a) A uniform lamina is in the shape of a quadrant of a circle of radius $a$. Show, by integration, that the centre of mass of the lamina is at a distance of $\frac { 4 a } { 3 \pi }$ from each of its straight edges. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{daa795f0-2c5e-4617-a295-fbe74c22be4a-10_809_802_484_571} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A second uniform lamina $A B C D E F A$ is shown shaded in Figure 3. The straight sides $A C$ and $A E$ are perpendicular and $A C = A E = 2 a$. In the figure, the midpoint of $A C$ is $B$, the midpoint of $A E$ is $F$, and $A B D F$ and $D G E F$ are squares of side $a$. $B C D$ is a quadrant of a circle with centre $B$. $D G E$ is a quadrant of a circle with centre $G$.
(b) Find the distance of the centre of mass of the lamina from the side $A E$. The lamina is smoothly hinged to a horizontal axis which passes through $E$ and is perpendicular to the plane of the lamina. The lamina has weight $W$ newtons. The lamina is held in equilibrium in a vertical plane, with $A$ vertically above $E$, by a horizontal force of magnitude $X$ newtons applied at $C$.
(c) Find $X$ in terms of $W$.

Edexcel M3 2013 June Q7

14 marks Challenging +1.2

Two points $A$ and $B$ are 4 m apart on a smooth horizontal surface. A light elastic string, of natural length 0.8 m and modulus of elasticity 15 N , has one end attached to the point A. A light elastic string, of natural length 0.8 m and modulus of elasticity 10 N , has one end attached to the point $B$. A particle $P$ of mass 0.2 kg is attached to the free end of each string. The particle rests in equilibrium on the surface at the point $C$ on the straight line between $A$ and $B$.
1. Show that the length of $A C$ is 1.76 m .
The particle $P$ is now held at the point $D$ on the line $A B$ such that $A D = 2.16 \mathrm {~m}$. The particle is then released from rest and in the subsequent motion both strings remain taut.
Show that $P$ moves with simple harmonic motion.
Find the speed of $P$ as it passes through the point $C$.
Find the time from the instant when $P$ is released from $D$ until the instant when $P$ is first moving with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Edexcel M3 2013 June Q1

6 marks Moderate -0.3

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-02_515_976_285_484} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A rough disc is rotating in a horizontal plane with constant angular speed 20 revolutions per minute about a fixed vertical axis through its centre $O$. A particle $P$ rests on the disc at a distance 0.4 m from $O$, as shown in Figure 1. The coefficient of friction between $P$ and the disc is $\mu$. The particle $P$ is on the point of slipping. Find the value of $\mu$.

Edexcel M3 2013 June Q2

9 marks Standard +0.3

2. A particle $P$ of mass 0.5 kg is moving along the positive $x$-axis in the positive $x$-direction. The only force on $P$ is a force of magnitude $\left( 2 t + \frac { 1 } { 2 } \right) \mathrm { N }$ acting in the direction of $x$ increasing, where $t$ seconds is the time after $P$ leaves the origin $O$. When $t = 0$, $P$ is at rest at $O$.

Find an expression, in terms of $t$, for the velocity of $P$ at time $t$ seconds. The particle passes through the point $A$ with speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the distance $O A$.

Edexcel M3 2013 June Q3

8 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-04_707_1006_258_427} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Two particles $P$ and $Q$, of mass $m$ and $2 m$ respectively, are attached to the ends of a light inextensible string of length 6l. The string passes through a small smooth fixed ring at the point $A$. The particle $Q$ is hanging freely at a distance $l$ vertically below $A$. The particle $P$ is moving in a horizontal circle with constant angular speed $\omega$. The centre $O$ of the circle is vertically below $A$. The particle $Q$ does not move and $A P$ makes a constant angle $\theta$ with the downward vertical, as shown in Figure 2. Show that

$\theta = 60 ^ { \circ }$
$\omega = \sqrt { } \left( \frac { 2 g } { 5 l } \right)$

Edexcel M3 2013 June Q4

9 marks Standard +0.8

A particle $P$ of mass 2 kg is attached to one end of a light elastic string of natural length 1.2 m . The other end of the string is attached to a fixed point $O$ on a rough horizontal plane. The coefficient of friction between $P$ and the plane is $\frac { 2 } { 5 }$. The particle is held at rest at a point $B$ on the plane, where $O B = 1.5 \mathrm {~m}$. When $P$ is at $B$, the tension in the string is 20 N . The particle is released from rest.
1. Find the speed of $P$ when $O P = 1.2 \mathrm {~m}$.
The particle comes to rest at the point $C$.
Find the distance $B C$.

Edexcel M3 2013 June Q5

13 marks Standard +0.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-08_622_1186_251_443} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} The shaded region $R$ is bounded by the curve with equation $y = ( x + 1 ) ^ { 2 }$, the $x$-axis, the $y$-axis and the line with equation $x = 2$, as shown in Figure 3. The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a uniform solid $S$.

Use algebraic integration to find the $x$ coordinate of the centre of mass of $S$.
(8) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-08_558_492_1263_703} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A uniform solid hemisphere is fixed to $S$ to form a solid $T$. The hemisphere has the same radius as the smaller plane face of $S$ and its plane face coincides with the smaller plane face of $S$, as shown in Figure 4. The mass per unit volume of the hemisphere is 10 times the mass per unit volume of $S$. The centre of the circular plane face of $T$ is $A$. All lengths are measured in centimetres.
Find the distance of the centre of mass of $T$ from $A$.

Edexcel M3 2013 June Q6

14 marks Standard +0.8

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-10_191_972_276_484} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} The points $A$ and $B$ are 3.75 m apart on a smooth horizontal floor. A particle $P$ has mass 0.8 kg . One end of a light elastic spring, of natural length 1.5 m and modulus of elasticity 24 N , is attached to $P$ and the other end is attached to $A$. The ends of another light elastic spring, of natural length 0.75 m and modulus of elasticity 18 N , are attached to $P$ and $B$. 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 $A O = 2.4 \mathrm {~m}$. The point $C$ lies on the straight line $A O B$ between $O$ and $B$. The particle $P$ is held at $C$ and released from rest.
Show that $P$ moves with simple harmonic motion. The maximum speed of $P$ is $\sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Find the time taken by $P$ to travel 0.3 m from $C$.

Edexcel M3 2013 June Q7

16 marks Challenging +1.2

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-12_499_833_262_664} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} A particle $P$ of mass $5 m$ 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, as shown in Figure 6. The particle is projected vertically downwards with speed $\sqrt { } \left( \frac { 9 a g } { 5 } \right)$. 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 $T = 3 m g ( 5 \cos \theta + 3 )$. At the instant when the particle reaches the point $B$ the string becomes slack.
Find the speed of $P$ at $B$. At time $t = 0 , P$ is at $B$. At time $t$, before the string becomes taut once more, the coordinates of $P$ are $( x , y )$ referred to horizontal and vertical axes with origin $O$. The $x$-axis is directed along $O A$ produced and the $y$-axis is vertically upward.
Find
1. $x$ in terms of $t , a$ and $g$,
2. $y$ in terms of $t , a$ and $g$.

Edexcel M3 2014 June Q1

8 marks Moderate -0.3

A particle $P$ of mass 0.25 kg is moving along the positive $x$-axis under the action of a single force. At time $t$ seconds $P$ is $x$ metres from the origin $O$ and is moving away from $O$ with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ where $\frac { \mathrm { d } v } { \mathrm {~d} x } = 3$. It is given that $x = 2$ and $v = 3$ when $t = 0$
1. Find the magnitude of the force acting on $P$ when $x = 5$
2. Find the value of $t$ when $x = 5$
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aafbccd2-7ba9-426b-9023-73b556ac3bed-03_676_822_280_546} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A cone of semi-vertical angle $60 ^ { \circ }$ is fixed with its axis vertical and vertex upwards. A particle 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 vertically above the vertex of the cone. The particle moves in a horizontal circle on the smooth outer surface of the cone with constant angular speed $\omega$, with the string making a constant angle $60 ^ { \circ }$ with the horizontal, as shown in Figure 1.
Find the tension in the string, in terms of $m , l , \omega$ and $g$. The particle remains on the surface of the cone.
Show that the time for the particle to make one complete revolution is greater than $$2 \pi \sqrt { \frac { l \sqrt { 3 } } { 2 g } }$$

Edexcel M3 2014 June Q3

11 marks Standard +0.8

One end $A$ of a light elastic string $A B$, of modulus of elasticity $m g$ and natural length $a$, is fixed to a point on a rough plane inclined at an angle $\theta$ to the horizontal. The other end $B$ of the string is attached to a particle of mass $m$ which is held at rest on the plane. The string $A B$ lies along a line of greatest slope of the plane, with $B$ lower than $A$ and $A B = a$. The coefficient of friction between the particle and the plane is $\mu$, where $\mu < \tan \theta$. The particle is released from rest.
1. Show that when the particle comes to rest it has moved a distance $2 a ( \sin \theta - \mu \cos \theta )$ down the plane.
2. Given that there is no further motion, show that $\mu \geqslant \frac { 1 } { 3 } \tan \theta$.

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