Questions - Edexcel M3 - A-Level Maths

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

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

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 $\mu \geqslant \frac { 1 } { 3 } \tan \theta$.

Edexcel M3 2014 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{aafbccd2-7ba9-426b-9023-73b556ac3bed-07_574_472_219_744} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A smooth sphere of radius $a$ is fixed with a point $A$ of its surface in contact with a fixed vertical wall. A particle is placed on the highest point of the sphere and is projected towards the wall and perpendicular to the wall with horizontal speed $\sqrt { \frac { 2 a g } { 5 } }$, as shown in Figure 2. The particle leaves the surface of the sphere with speed $V$.

Show that $V = \sqrt { \frac { 4 a g } { 5 } }$ The particle strikes the wall at the point $X$.
Find the distance $A X$.

Edexcel M3 2014 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{aafbccd2-7ba9-426b-9023-73b556ac3bed-09_867_1289_214_397} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform solid right circular cylinder has height $h$ and radius $r$. The centre of one plane face is $O$ and the centre of the other plane face is $Y$. A cylindrical hole is made by removing a solid cylinder of radius $\frac { 1 } { 4 } r$ and height $\frac { 1 } { 4 } h$ from the end with centre $O$. The axis of the cylinder removed is parallel to $O Y$ and meets the end with centre $O$ at $X$, where $O X = \frac { 1 } { 4 } r$. One plane face of the cylinder removed coincides with the plane face through $O$ of the original cylinder. The resulting solid $S$ is shown in Figure 3.

Show that the centre of mass of $S$ is at a distance $\frac { 85 h } { 168 }$ from the plane face
containing $O$. containing $O$. The solid $S$ is freely suspended from $O$. In equilibrium the line $O Y$ is inclined at an angle $\arctan ( 17 )$ to the horizontal.
Find $r$ in terms of $h$.

Edexcel M3 2014 June Q6

6. A light elastic string, of natural length $l$ and modulus of elasticity $4 m g$, has one end attached to a fixed point $A$. The other end is attached to a particle $P$ of mass $m$. The particle hangs freely at rest in equilibrium at the point $E$. The distance of $E$ below $A$ is $( l + e )$.

Find $e$ in terms of $l$. At time $t = 0$, the particle is projected vertically downwards from $E$ with speed $\sqrt { g l }$.
Prove that, while the string is taut, $P$ moves with simple harmonic motion.
Find the amplitude of the simple harmonic motion.
Find the time at which the string first goes slack.

Edexcel M3 2014 June Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e500e20b-9060-4c69-af13-fb97b9a86dfd-02_389_524_221_712} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A hemispherical bowl of internal radius $4 r$ is fixed with its circular rim horizontal. The centre of the circular rim is $O$ and the point $A$ on the surface of the bowl is vertically below $O$. A particle $P$ moves in a horizontal circle, with centre $C$, on the smooth inner surface of the bowl. The particle moves with constant angular speed $\sqrt { \frac { 3 g } { 8 r } }$
The point $C$ lies on $O A$, as shown in Figure 1.
Find, in terms of $r$, the distance $O C$.

Edexcel M3 2014 June Q2

2. A particle $P$ of mass $m$ is fired vertically upwards from a point on the surface of the Earth and initially moves in a straight line directly away from the centre of the Earth. When $P$ is at a distance $x$ from the centre of the Earth, the gravitational force exerted by the Earth on $P$ is directed towards the centre of the Earth and has magnitude $\frac { k } { x ^ { 2 } }$, where $k$ is a constant. 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 $k = m g R ^ { 2 }$. When $P$ is at a height $\frac { R } { 4 }$ above the surface of the Earth, the speed of $P$ is $\sqrt { \frac { g R } { 2 } }$ Given that air resistance can be ignored,
find, in terms of $R$, the greatest distance from the centre of the Earth reached by $P$.

Edexcel M3 2014 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e500e20b-9060-4c69-af13-fb97b9a86dfd-05_639_422_223_769} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a container in the shape of a uniform right circular conical shell of height 6r. The radius of the open circular face is $r$. The container is suspended by two vertical strings attached to two points at opposite ends of a diameter of the open circular face. It hangs with the open circular face uppermost and axis vertical. Molten wax is poured into the container. The wax solidifies and adheres to the container, forming a uniform solid right circular cone. The depth of the wax in the container is $2 r$. The container together with the wax forms a solid $S$. The mass of the container when empty is $m$ and the mass of the wax in the container is $3 m$.

Find the distance of the centre of mass of the solid $S$ from the vertex of the container. One of the strings is now removed and the solid $S$ hangs freely in equilibrium suspended by the remaining vertical string.
Find the size of the angle between the axis of the container and the downward vertical.

Edexcel M3 2014 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e500e20b-9060-4c69-af13-fb97b9a86dfd-07_486_874_223_495} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} One end of a light elastic string, of natural length $l$ and modulus of elasticity $3 m g$, is fixed to a point $A$ on a fixed plane inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 3 } { 5 }$ A small ball of mass $2 m$ is attached to the free end of the string. The ball is held at a point $C$ on the plane, where $C$ is below $A$ and $A C = l$ as shown in Figure 3. The string is parallel to a line of greatest slope of the plane. The ball is released from rest. In an initial model the plane is assumed to be smooth.

Find the distance that the ball moves before first coming to instantaneous rest. In a refined model the plane is assumed to be rough. The coefficient of friction between the ball and the plane is $\mu$. The ball first comes to instantaneous rest after moving a distance $\frac { 2 } { 5 } l$.
Find the value of $\mu$.

Edexcel M3 2014 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e500e20b-9060-4c69-af13-fb97b9a86dfd-09_529_713_223_612} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows the region $R$ bounded by part of the curve with equation $y = \cos x$, the $x$-axis and the $y$-axis. A uniform solid $S$ is formed by rotating $R$ through $2 \pi$ radians about the $x$-axis.

Show that the volume of $S$ is $\frac { \pi ^ { 2 } } { 4 }$
Find, using algebraic integration, the $x$ coordinate of the centre of mass of $S$.

Edexcel M3 2014 June Q6

6. 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. The particle is hanging freely at rest, with the string vertical, when it is projected horizontally with speed $U$. The particle moves in a complete vertical circle.

Show that $U \geqslant \sqrt { 5 a g }$ As $P$ moves in the circle the least tension in the string is $T$ and the greatest tension is $k T$. Given that $U = 3 \sqrt { a g }$
find the value of $k$.

Edexcel M3 2014 June Q7

7. A particle $P$ of mass $m$ is attached to one end of a light elastic spring of natural length $l$. The other end of the spring is attached to a fixed point $A$. The particle is hanging freely in equilibrium at the point $B$, where $A B = 1.5 l$

Show that the modulus of elasticity of the spring is $2 m g$. The particle is pulled vertically downwards from $B$ to the point $C$, where $A C = 1.8 \mathrm { l }$, and released from rest.
Show that $P$ moves in simple harmonic motion with centre $B$.
Find the greatest magnitude of the acceleration of $P$. The midpoint of $B C$ is $D$. The point $E$ lies vertically below $A$ and $A E = 1.2 l$
Find the time taken by $P$ to move directly from $D$ to $E$.

Edexcel M3 2015 June Q1

A particle $P$ of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 1.2 m and modulus of elasticity $\lambda$ newtons. The other end of the spring is attached to a fixed point $A$ on a ceiling. The particle is hanging freely in equilibrium at a distance 1.5 m vertically below $A$.
1. Find the value of $\lambda$.
The particle is now raised to the point $B$, where $B$ is vertically below $A$ and $A B = 0.8 \mathrm {~m}$. The spring remains straight. The particle is released from rest and first comes to instantaneous rest at the point $C$.
Find the distance $A C$.

Edexcel M3 2015 June Q2

2. The finite region bounded by the $x$-axis, the curve with equation $y = 2 \mathrm { e } ^ { x }$, the $y$-axis and the line $x = 1$ is rotated through one complete revolution about the $x$-axis to form a uniform solid. Use algebraic integration to

show that the volume of the solid is $2 \pi \left( \mathrm { e } ^ { 2 } - 1 \right)$,
find, in terms of e, the $x$ coordinate of the centre of mass of the solid.

Edexcel M3 2015 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{00388805-5d60-4327-a10e-c0df74a0cb75-05_776_791_223_573} \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 $4 l$. The ends of the string are attached to fixed points $A$ and $B$, where $A$ is vertically above $B$. Both strings are taut and $A P$ makes an angle of $30 ^ { \circ }$ with $A B$, as shown in Figure 1. The ball is moving in a horizontal circle with constant angular speed $\omega$.

Find, in terms of $m , g , l$ and $\omega$,
1. the tension in $A P$,
2. the tension in $B P$.
Show that $\omega ^ { 2 } \geqslant \frac { g \sqrt { 3 } } { 3 l }$.

Edexcel M3 2015 June Q4

A vehicle of mass 900 kg moves along a straight horizontal road. At time $t$ seconds the resultant force acting on the vehicle has magnitude $\frac { 63000 } { k t ^ { 2 } } \mathrm {~N}$, where $k$ is a positive constant. The force acts in the direction of motion of the vehicle. At time $t$ seconds, $t \geqslant 1$, the speed of the vehicle is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the vehicle is a distance $x$ metres from a fixed point $O$ on the road. When $t = 1$ the vehicle is at rest at $O$ and when $t = 4$ the speed of the vehicle is $10.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
1. Show that $v = 14 - \frac { 14 } { t }$
2. Hence deduce that the speed of the vehicle never reaches $14 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
3. Use the trapezium rule, with 4 equal intervals, to estimate the value of $x$ when $v = 7$

Edexcel M3 2015 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{00388805-5d60-4327-a10e-c0df74a0cb75-09_403_790_210_577} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a uniform solid spindle which is made by joining together the circular faces of two right circular cones. The common circular face has radius $r$ and centre $O$. The smaller cone has height $h$ and the larger cone has height $k h$. The point $A$ lies on the circumference of the common circular face. The spindle is suspended from $A$ and hangs freely in equilibrium with $A O$ at an angle of $30 ^ { \circ }$ to the vertical. Show that $k = \frac { 4 r } { h \sqrt { 3 } } + 1$

Edexcel M3 2015 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{00388805-5d60-4327-a10e-c0df74a0cb75-11_186_1042_223_452} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two points $A$ and $B$ are 6 m apart on a smooth horizontal floor. A particle $P$ of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 2.5 m and modulus of elasticity 20 N . The other end of the spring is attached to $A$. A second light elastic spring, of natural length 1.5 m and modulus of elasticity 18 N , has one end attached to $P$ and the other end attached to $B$, as shown in Figure 3. Initially $P$ rests in equilibrium at the point $O$, where $A O B$ is a straight line.

Find the length of $A O$. The particle $P$ now receives an impulse of magnitude 6 N s acting in the direction $O B$ and $P$ starts to move towards $B$.
Show that $P$ moves with simple harmonic motion about $O$.
Find the amplitude of the motion.
Find the time taken by $P$ to travel 1.2 m from $O$.

Edexcel M3 2015 June Q7

A solid smooth sphere, with centre $O$ and radius $r$, is fixed to a point $A$ on a horizontal floor. A particle $P$ is placed on the surface of the sphere at the point $B$, where $B$ is vertically above $A$. The particle is projected horizontally from $B$ with speed $\frac { \sqrt { g r } } { 2 }$ and starts to move on the surface of the sphere. When $O P$ makes an angle $\theta$ with the upward vertical and $P$ remains in contact with the sphere, the speed of $P$ is $v$.
1. Show that $v ^ { 2 } = \frac { g r } { 4 } ( 9 - 8 \cos \theta )$.
The particle leaves the surface of the sphere when $\theta = \alpha$.
Find the value of $\cos \alpha$. After leaving the surface of the sphere, $P$ moves freely under gravity and hits the floor at the point $C$. Given that $r = 0.5 \mathrm {~m}$,
find, to 2 significant figures, the distance $A C$.

Edexcel M3 2016 June Q1

A particle $P$ of mass 0.5 kg is moving along the positive $x$-axis under the action of a resultant force. The force acts along the $x$-axis. At time $t$ seconds, $P$ is $x$ metres from the origin $O$ and is moving away from $O$ in the positive $x$ direction with speed $\frac { 12 } { x + 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$
1. Find the magnitude of the force acting on $P$ when $x = 3$
Given that $x = 4$ when $t = 2$
find the value of $t$ when $x = 10$

Edexcel M3 2016 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4c1c51ff-6ae8-402d-b303-b656d26e4230-03_430_739_324_607} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a uniform triangular lamina $A B C$ in which $A B = 6 \mathrm {~cm} , B C = 9 \mathrm {~cm}$ and angle $A B C = 90 ^ { \circ }$. The centre of mass of the lamina is $G$. Use algebraic integration to find the distance of $G$ from $A B$.
(6)

Edexcel M3 2016 June Q3

3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N ,
3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N , is attached to a fixed point $O$ on a ceiling. A particle $P$ of mass 0.6 kg is attached to the free end of the string. The particle is held at $O$ and released from rest. The particle comes to instantaneous rest for the first time at the point $A$. Find

the distance $O A$,
the magnitude of the instantaneous acceleration of $P$ at $A$.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c1c51ff-6ae8-402d-b303-b656d26e4230-05_620_956_118_500} \captionsetup{labelformat=empty} \caption{igure 2}
\end{figure} A uniform solid $S$ consists of two right circular cones of base radius $r$. The smaller cone has height $2 h$ and the centre of the plane face of this cone is $O$. The larger cone has height $k h$ where $k > 2$. The two cones are joined so that their plane faces coincide, as shown in Figure 2.
Show that the distance of the centre of mass of $S$ from $O$ is $$\frac { h } { 4 } ( k - 2 )$$ The point $A$ lies on the circumference of the base of one of the cones. The solid is suspended by a string attached at $A$ and hangs freely in equilibrium. Given that $r = 3 h$ and $k = 6$
find the size of the angle between $A O$ and the vertical.

Edexcel M3 2016 June Q4

4. A uniform solid $S$ consists of two right circular cones of base rate
has height $2 h$ and the centre of the plane face of this cone is $O$. T
$k h$ where $k > 2$. The two cones are joined so that their plane fac
Figure 2 .

Show that the distance of the centre of mass of $S$ from $O$ is
$\frac { h } { 4 } ( k - 2 )$

Edexcel M3 2016 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4c1c51ff-6ae8-402d-b303-b656d26e4230-07_842_449_248_826} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A particle $P$ of mass $m$ is attached to the ends of two light inextensible strings. The other ends of the strings are attached to fixed points $A$ and $B$, where $B$ is vertically below $A$ and $A B = l$. The particle is moving with constant angular speed $\omega$ in a horizontal circle. Both strings are taut and inclined at $30 ^ { \circ }$ to $A B$, as shown in Figure 3.

1. Show that the tension in $A P$ is $\frac { m \sqrt { 3 } } { 6 } \left( 2 g + l \omega ^ { 2 } \right)$
2. Find the tension in $B P$.
Show that the time taken by $P$ to complete one revolution is less than $\pi \sqrt { \frac { 2 l } { g } }$

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