Questions - Edexcel M3 - A-Level Maths

use the principle of conservation of mechanical energy to show that $$v ^ { 2 } = u ^ { 2 } - \frac { 36 } { 13 } g r$$
show that $u ^ { 2 } \geqslant \frac { 41 } { 13 } g r$ The point $C$ is such that $B C$ is a diameter of the rim of the bowl.
Given that $u ^ { 2 } = 4 g r$
use the model to show that, after leaving the inner surface of the bowl at $B$, the ball falls back into the bowl before reaching $C$.

Edexcel M3 2014 June Q1

A particle $P$ moves in a straight line with simple harmonic motion. The period of the motion is $\frac { \pi } { 4 }$ seconds. At time $t = 0 , P$ is at rest at the point $A$ and the acceleration of $P$ has magnitude $20 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Find

the amplitude of the motion,
the greatest speed of $P$ during the motion,
the time $P$ takes to travel a total distance of 1.5 m after it has first left $A$.

Edexcel M3 2014 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-03_522_654_223_646} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform lamina $L$ is in the shape of an equilateral triangle of side $2 a$. The lamina is placed in the $x y$-plane with one vertex at the origin $O$ and an axis of symmetry along the $x$-axis, as shown in Figure 1. Use algebraic integration to find the $x$ coordinate of the centre of mass of $L$.

Edexcel M3 2014 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-05_951_750_121_635} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass 3 kg is attached by two light inextensible strings to two fixed points $A$ and $B$ on a fixed vertical pole. Both strings are taut and $P$ is moving in a horizontal circle with constant angular speed $6 \mathrm { rad } \mathrm { s } ^ { - 1 }$. String $A P$ is inclined at $30 ^ { \circ }$ to the vertical. String $B P$ has length 0.4 m and $A$ is 0.4 m vertically above $B$, as shown in Figure 2 . Find the tension in

$A P$,
$B P$.

Edexcel M3 2014 June Q4

At time $t = 0$, a particle $P$ of mass 0.4 kg is at the origin $O$ moving with speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the $x$-axis in the positive $x$ direction. At time $t$ seconds, $t \geqslant 0$, the resultant force acting on $P$ has magnitude $\frac { 4 } { ( t + 5 ) ^ { 2 } } \mathrm {~N}$ and is directed away from $O$.
1. Show that the speed of $P$ cannot exceed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
The particle passes through the point $A$ when $t = 2$ and passes through the point $B$ when $t = 7$
Find the distance $A B$.
Find the gain in kinetic energy of $P$ as it moves from $A$ to $B$.

Edexcel M3 2014 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-09_485_442_221_758} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A particle $P$ of mass $2 m$ 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 $O$. Initially the particle is at the point $A$ where $O A = a$ and $O A$ makes an angle $60 ^ { \circ }$ with the downward vertical. The particle is projected downwards from $A$ with speed $u$ in a direction perpendicular to the string, as shown in Figure 3. The point $B$ is vertically below $O$ and $O B = a$. As $P$ passes through $B$ it strikes and adheres to another particle $Q$ of mass $m$ which is at rest at $B$.

Show that the speed of the combined particle immediately after the impact is $$\frac { 2 } { 3 } \sqrt { u ^ { 2 } + a g } .$$
Find, in terms of $a , g , m$ and $u$, the tension in the string immediately after the impact. The combined particle moves in a complete circle.
Show that $u ^ { 2 } \geqslant \frac { 41 a g } { 4 }$.

Edexcel M3 2014 June Q6

6. A particle of mass $m$ is attached to one end of a light elastic string, of natural length $6 a$ and modulus of elasticity 9 mg . The other end of the string is attached to a fixed point $A$ on a ceiling. The particle hangs in equilibrium at the point $B$, where $B$ is vertically below $A$ and $A B = ( 6 + p ) a$.

Show that $p = \frac { 2 } { 3 }$ The particle is now released from rest at a point $C$ vertically below $B$, where $A C < \frac { 22 } { 3 } a$.
Show that the particle moves with simple harmonic motion.
Find the period of this motion.
Explain briefly the significance of the condition $A C < \frac { 22 } { 3 } a$. The point $D$ is vertically below $A$ and $A D = 8 a$. The particle is now released from rest at $D$. The particle first comes to instantaneous rest at the point $E$.
Find, in terms of $a$, the distance $A E$.

Edexcel M3 2014 June Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-13_449_668_221_641} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Diagram not drawn to scale A uniform right circular solid cylinder has radius $3 a$ and height $2 a$. A right circular cone of height $\frac { 3 a } { 2 }$ and base radius $2 a$ is removed from the cylinder to form a solid $S$, as shown in Figure 4. The plane face of the cone coincides with the upper plane face of the cylinder and the centre $O$ of the plane face of the cone is also the centre of the upper plane face of the cylinder.

Show that the distance of the centre of mass of $S$ from $O$ is $\frac { 69 a } { 64 }$. The point $A$ is on the open face of $S$ such that $O A = 3 a$, as shown in Figure 4. The solid is now suspended from $A$ and hangs freely in equilibrium.
Find the angle between $O A$ and the horizontal.
(3) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-13_543_826_1653_557} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} The solid is now placed on a rough inclined plane with the face through $A$ in contact with the inclined plane, as shown in Figure 5. The solid rests in equilibrium on this plane. The coefficient of friction between the plane and $S$ is 0.6 and the plane is inclined at an angle $\phi ^ { \circ }$ to the horizontal. Given that $S$ is on the point of sliding down the plane,
show that $\phi = 31$ to 2 significant figures.

Edexcel M3 2015 June Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-02_406_537_264_715} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A hemispherical bowl, of internal radius $r$, is fixed with its circular rim upwards and horizontal. A particle $P$ of mass $m$ moves on the smooth inner surface of the bowl. The particle moves with constant angular speed in a horizontal circle. The centre of the circle is at a distance $\frac { 1 } { 2 } r$ vertically below the centre of the bowl, as shown in Figure 1.
The time taken by $P$ to complete one revolution of its circular path is $T$.
Show that $T = \pi \sqrt { \frac { 2 r } { g } }$.

Edexcel M3 2015 June Q2

2. A spacecraft $S$ of mass $m$ moves in a straight line towards the centre of the Earth. The Earth is modelled as a sphere of radius $R$ and $S$ is modelled as a particle. When $S$ is at a distance $x , x \geqslant R$, from the centre of the Earth, the force exerted by the Earth on $S$ is directed towards the centre of the Earth. The force has magnitude $\frac { K } { x ^ { 2 } }$, where $K$ is a constant.

Show that $K = m g R ^ { 2 }$
(2) When $S$ is at a distance $3 R$ from the centre of the Earth, the speed of $S$ is $V$. Assuming that air resistance can be ignored,
find, in terms of $g , R$ and $V$, the speed of $S$ as it hits the surface of the Earth.

Edexcel M3 2015 June Q3

At time $t = 0$, a particle $P$ is at the origin $O$, moving with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the positive $x$ direction. At time $t$ seconds, $t \geqslant 0$, the acceleration of $P$ has magnitude $2 ( t + 4 ) ^ { - \frac { 1 } { 2 } } \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and is directed towards $O$.
1. Show that, at time $t$ seconds, the velocity of $P$ is $16 - 4 ( t + 4 ) ^ { \frac { 1 } { 2 } } \mathrm {~ms} ^ { - 1 }$
2. Find the distance of $P$ from $O$ when $P$ comes to instantaneous rest.

Edexcel M3 2015 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-07_408_509_246_705} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle of mass $3 m$ 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 $O A$ is horizontal and $O A = a$. The particle is projected vertically downwards from $A$ with speed $u$, as shown in Figure 2. The particle moves in complete vertical circles.

Show that $u ^ { 2 } \geqslant 3 a g$. Given that the greatest tension in the string is three times the least tension in the string, (b) show that $u ^ { 2 } = 6 a g$.

Edexcel M3 2015 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-09_205_941_262_513} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Two fixed points $A$ and $B$ are 5 m apart on a smooth horizontal floor. A particle $P$ of mass 0.5 kg is attached to one end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to $A$. A second light elastic string, of natural length 1.2 m and modulus of elasticity 15 N , has one end attached to $P$ and the other end attached to $B$. Initially $P$ rests in equilibrium at the point $O$, as shown in Figure 3.

Show that $A O = 3 \mathrm {~m}$. The particle is now pulled towards $A$ and released from rest at the point $C$, where $A C B$ is a straight line and $O C = 1 \mathrm {~m}$.
Show that, while both strings are taut, $P$ moves with simple harmonic motion.
Find the speed of $P$ at the instant when the string $P B$ becomes slack. The particle first comes to instantaneous rest at the point $D$.
Find the distance $D B$.

Edexcel M3 2015 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-11_442_727_237_603} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} The shaded region $R$ is bounded by part of the curve with equation $y = x ^ { 2 } + 3$, the $x$-axis, the $y$-axis and the line with equation $x = 2$, as shown in Figure 4. The unit of length on each axis is one centimetre. The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a uniform solid $S$.
Using algebraic integration,

show that the volume of $S$ is $\frac { 202 } { 5 } \pi \mathrm {~cm} ^ { 3 }$,
show that, to 2 decimal places, the centre of mass of $S$ is 1.30 cm from $O$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-11_478_472_1407_762} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A uniform right circular solid cone, of base radius 7 cm and height 6 cm , is joined to $S$ to form a solid $T$. The base of the cone coincides with the larger plane face of $S$, as shown in Figure 5. The vertex of the cone is $V$.
The mass per unit volume of $S$ is twice the mass per unit volume of the cone.
Find the distance from $V$ to the centre of mass of $T$. The point $A$ lies on the circumference of the base of the cone. The solid $T$ is suspended from $A$ and hangs freely in equilibrium.
Find the size of the angle between $V A$ and the vertical.

Edexcel M3 2016 June Q1

A particle is attached to one end of a light inextensible string of length $l$. The other end of
A light elastic spring, of natural length $5 a$ and modulus of elasticity 10 mg , has one end attached to a fixed point $A$ on a ceiling. A particle $P$ of mass $2 m$ is attached to the other end of the spring and $P$ hangs freely in equilibrium at the point $O$.
1. Find the distance $A O$.
  (3)
The particle is now pulled vertically downwards a distance $\frac { 1 } { 2 } a$ from $O$ and released from rest.
Show that $P$ moves with simple harmonic motion.
Find the period of the motion.

Edexcel M3 2016 June Q3

3. A particle $P$ of mass $m$ is attached to one end of a light elastic string, of natural length $l$ and modulus of elasticity $4 m g$. 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 a point $A$ on the plane, where $O A = \frac { 5 } { 4 } l$, and released from rest. The particle comes to rest at the point $B$.

Show that $O B < l$
Find the distance $O B$.

Edexcel M3 2016 June Q4

4. 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 a magnitude which is inversely proportional to $x ^ { 2 }$. 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 acting on $P$ is $\frac { m g R ^ { 2 } } { x ^ { 2 } }$ The particle was fired with initial speed $U$ and the greatest height above the surface of the Earth reached by $P$ is $\frac { R } { 20 }$ Given that air resistance can be ignored,
find $U$ in terms of $g$ and $R$.

Edexcel M3 2016 June Q5

5. A vertical ladder is fixed to a wall in a harbour. On a particular day the minimum depth of water in the harbour occurs at 0900 hours. The next time the water is at its minimum depth is 2115 hours on the same day. The bottom step of the ladder is 1 m above the lowest level of the water and 9 m below the highest level of the water. The rise and fall of the water level can be modelled as simple harmonic motion and the thickness of the step can be assumed to be negligible. Find

the speed, in metres per hour, at which the water level is moving when it reaches the bottom step of the ladder,
the length of time, on this day, between the water reaching the bottom step of the ladder and the ladder being totally out of the water once more.

Edexcel M3 2016 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8338c3a4-cb37-4979-a424-e7cf4901207a-09_410_1025_255_520} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A smooth solid hemisphere of radius 0.5 m is fixed with its plane face on a horizontal floor. The plane face has centre $O$ and the highest point of the surface of the hemisphere is $A$. A particle $P$ has mass 0.2 kg . The particle is projected horizontally with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from $A$ and leaves the hemisphere at the point $B$, where $O B$ makes an angle $\theta$ with $O A$, as shown in Figure 1. The point $B$ is at a vertical distance of 0.1 m below the level of $A$. The speed of $P$ at $B$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$

Show that $v ^ { 2 } = u ^ { 2 } + 1.96$
Find the value of $u$. The particle first strikes the floor at the point $C$.
Find the length of $O C$.

Edexcel M3 2016 June Q7

7. (a) Use algebraic integration to show that the centre of mass of a uniform solid right circular cone of height $h$ is at a distance $\frac { 3 } { 4 } h$ from the vertex of the cone.
[0pt] [You may assume that the volume of a cone of height $h$ and base radius $r$ is $\frac { 1 } { 3 } \pi r ^ { 2 } h$ ] \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8338c3a4-cb37-4979-a424-e7cf4901207a-11_389_634_502_657} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A uniform solid $S$ consists of a right circular cone, of radius $r$ and height $5 r$, fixed to a hemisphere of radius $r$. The centre of the plane face of the hemisphere is $O$ and this plane face coincides with the base of the cone, as shown in Figure 2.
(b) Find the distance of the centre of mass of $S$ from $O$. The point $A$ lies on the circumference of the base of the cone. The solid is suspended by a string attached at $A$ and hangs freely in equilibrium.
(c) Find the size of the angle between $O A$ and the vertical. The mass of the hemisphere is $M$. A particle of mass $k M$ is fixed to the surface of the hemisphere on the axis of symmetry of $S$. The solid is again suspended by the string attached at $A$ and hangs freely in equilibrium. The axis of symmetry of $S$ is now horizontal.
(d) Find the value of $k$.

Edexcel M3 2017 June Q1

The region enclosed by the curve with equation $y = \frac { 1 } { 2 } \sqrt { x }$, the $x$-axis and the lines $x = 2$ and $x = 4$, is rotated through $2 \pi$ radians about the $x$-axis to form a uniform solid $S$. Use algebraic integration to find the exact value of the $x$ coordinate of the centre of mass of $S$.
(6)

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Edexcel M3 2017 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a67e3644-13fa-4196-a2ef-ea1e26f5726c-04_264_438_269_753} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform solid right circular cone $R$, with vertex $V$, has base radius $4 r$ and height $4 h$. A right circular cone $S$, also with vertex $V$ and the same axis of symmetry as $R$, has base radius $3 r$ and height $3 h$. The cone $S$ is cut away from the cone $R$ leaving a solid $T$. The centre of the larger plane face of $T$ is $O$. Figure 1 shows the solid $T$.

Find the distance from $O$ to the centre of mass of $T$. The point $A$ lies on the circumference of the smaller plane face of $T$. The solid is freely suspended from $A$ and hangs in equilibrium. Given that $h = r$
find the size of the angle between $O A$ and the downward vertical.

Edexcel M3 2017 June Q3

3. A particle $P$ of mass 0.5 kg moves in a straight line with simple harmonic motion, completing 4 oscillations per second. The particle comes to instantaneous rest at the fixed points $A$ and $B$, where $A B = 0.5 \mathrm {~m}$.

Find the maximum magnitude of the acceleration of $P$. When $P$ is moving at its maximum speed it receives an impulse. The direction of this impulse is opposite to the direction in which $P$ is moving when it receives the impulse. The impulse causes $P$ to reverse its direction of motion but $P$ continues to move with simple harmonic motion. The centre and period of this new simple harmonic motion are the same as the centre and period of the original simple harmonic motion. The amplitude is now half the original amplitude.
Find the magnitude of the impulse.
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Edexcel M3 2017 June Q4

4. A light elastic string has natural length 0.4 m and modulus of elasticity 49 N . A particle $P$ of mass 0.3 kg is attached to one end of the string. The other end of the string is attached to a fixed point $A$ on a ceiling. The particle is released from rest at $A$ and falls vertically. The particle first comes to instantaneous rest at the point $B$.

Find the distance $A B$. The particle is now held at the point 0.6 m vertically below $A$ and released from rest.
Find the speed of $P$ immediately before it hits the ceiling.

Edexcel M3 2017 June Q5

5. A particle $P$ of mass 0.4 kg moves on the positive $x$-axis under the action of a single force. The force is directed towards the origin $O$ and has magnitude $\frac { k } { x ^ { 2 } }$ newtons, where $O P = x$ metres and $k$ is a constant. Initially $P$ is moving away from $O$. At $x = 2$ the speed of $P$ is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and at $x = 5$ the speed of $P$ is $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the value of $k$. The particle first comes to instantaneous rest at the point $A$.
Find the value of $x$ at $A$.

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