Questions M3 - A-Level Maths

Show that $k = m g R ^ { 2 }$. Given that $S$ starts from rest when its distance from the centre of the earth is $2 R$, and that air resistance can be ignored,
find the speed of $S$ as it crashes into the surface of the earth.

Edexcel M3 2007 June Q4

4. A light inextensible string of length $l$ has one end attached to a fixed point $A$. The other end is attached to a particle $P$ of mass $m$. The particle moves with constant speed $v$ in a horizontal circle with the string taut. The centre of the circle is vertically below $A$ and the radius of the circle is $r$. Show that $$g r ^ { 2 } = v ^ { 2 } \sqrt { } \left( l ^ { 2 } - r ^ { 2 } \right) .$$

Edexcel M3 2007 June Q5

A particle $P$ moves on the $x$-axis with simple harmonic motion about the origin $O$ as centre. When $P$ is a distance 0.04 m from $O$, its speed is $0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the magnitude of its acceleration is $1 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
1. Find the period of the motion.
The amplitude of the motion is $a$ metres. Find
the value of $a$,
the total time, within one complete oscillation, for which the distance $O P$ is greater than $\frac { 1 } { 2 } a$ metres.

Edexcel M3 2007 June Q6

6. A particle $P$ is free to move on the smooth inner surface of a fixed thin hollow sphere of internal radius $a$ and centre $O$. The particle passes through the lowest point of the spherical surface with speed $U$. The particle loses contact with the surface when $O P$ is inclined at an angle $\alpha$ to the upward vertical.

Show that $\quad U ^ { 2 } = a g ( 2 + 3 \cos \alpha )$. The particle has speed $W$ as it passes through the level of $O$. Given that $\cos \alpha = \frac { 1 } { \sqrt { } 3 }$, (b) show that $\quad W ^ { 2 } = a g \sqrt { } 3$.

Edexcel M3 2007 June Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a20270d9-4a30-45d8-ac33-2e4fc9c7fb06-10_487_696_316_632}

\end{figure} A light elastic string, of natural length $3 l$ and modulus of elasticity $\lambda$, has its ends attached to two points $A$ and $B$, where $A B = 3 l$ and $A B$ is horizontal. A particle $P$ of mass $m$ is attached to the mid-point of the string. Given that $P$ rests in equilibrium at a distance $2 l$ below $A B$, as shown in Figure 1,

show that $\lambda = \frac { 15 m g } { 16 }$. The particle is pulled vertically downwards from its equilibrium position until the total length of the elastic string is $7.8 l$. The particle is released from rest.
Show that $P$ comes to instantaneous rest on the line $A B$.

Edexcel M3 2008 June Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-02_259_659_283_642} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A light elastic spring, of natural length $L$ and modulus of elasticity $\lambda$, has a particle $P$ of mass $m$ attached to one end. The other end of the spring is fixed to a point $O$ on the closed end of a fixed smooth hollow tube of length $L$. The tube is placed horizontally and $P$ is held inside the tube with $O P = \frac { 1 } { 2 } L$, as shown
in Figure 1. The particle $P$ is released and passes through the open end of the tube with speed $\sqrt { } ( 2 g L )$.

Show that $\lambda = 8 \mathrm { mg }$. The tube is now fixed vertically and $P$ is held inside the tube with $O P = \frac { 1 } { 2 } L$ and $P$ above $O$. The particle $P$ is released and passes through the open top of the tube with speed $u$.
Find $u$.

Edexcel M3 2008 June Q2

2. A particle $P$ moves with simple harmonic motion and comes to rest at two points $A$ and $B$ which are 0.24 m apart on a horizontal line. The time for $P$ to travel from $A$ to $B$ is 1.5 s . The midpoint of $A B$ is $O$. At time $t = 0 , P$ is moving through $O$, towards $A$, with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the value of $u$.
Find the distance of $P$ from $B$ when $t = 2 \mathrm {~s}$.
Find the speed of $P$ when $t = 2 \mathrm {~s}$.

Edexcel M3 2008 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-05_495_972_239_484} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a particle $B$, of mass $m$, attached to one end of a light elastic string. The other end of the string is attached to a fixed point $A$, at a distance $h$ vertically above a smooth horizontal table. The particle moves on the table in a horizontal circle with centre $O$, where $O$ is vertically below $A$. The string makes a constant angle $\theta$ with the downward vertical and $B$ moves with constant angular speed $\omega$ about $O A$.

Show that $\omega ^ { 2 } \leqslant \frac { g } { h }$. The elastic string has natural length $h$ and modulus of elasticity $2 m g$.
Given that $\tan \theta = \frac { 3 } { 4 }$,
find $\omega$ in terms of $g$ and $h$.

Edexcel M3 2008 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-07_454_614_239_662} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform solid hemisphere, of radius $6 a$ and centre $O$, has a solid hemisphere of radius $2 a$, and centre $O$, removed to form a bowl $B$ as shown in Figure 3.

Show that the centre of mass of $B$ is $\frac { 30 } { 13 } a$ from $O$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-07_735_614_1126_662} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The bowl $B$ is fixed to a plane face of a uniform solid cylinder made from the same material as $B$. The cylinder has radius $2 a$ and height $6 a$ and the combined solid $S$ has an axis of symmetry which passes through $O$, as shown in Figure 4.
Show that the centre of mass of $S$ is $\frac { 201 } { 61 } a$ from $O$. The plane surface of the cylindrical base of $S$ is placed on a rough plane inclined at $12 ^ { \circ }$ to the horizontal. The plane is sufficiently rough to prevent slipping.
Determine whether or not $S$ will topple. \section*{
\includegraphics[max width=\textwidth, alt={}]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-08_56_366_251_178}
}

Edexcel M3 2008 June Q5

A particle $P$ of mass $m$ 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 released from rest with the string taut and $O P$ horizontal.
1. Find the tension in the string when $O P$ makes an angle of $60 ^ { \circ }$ with the downward vertical.
A particle $Q$ of mass $3 m$ is at rest at a distance $a$ vertically below $O$. When $P$ strikes $Q$ the particles join together and the combined particle of mass $4 m$ starts to move in a vertical circle with initial speed $u$.
Show that $u = \sqrt { } \left( \frac { g a } { 8 } \right)$. The combined particle comes to instantaneous rest at $A$.
Find
1. the angle that the string makes with the downward vertical when the combined particle is at $A$,
2. the tension in the string when the combined particle is at $A$.
  \section*{LU
  $\_\_\_\_$}

Edexcel M3 2008 June Q6

A particle $P$ of mass 0.5 kg moves along the positive $x$-axis. It moves away from the origin $O$ under the action of a single force directed away from $O$. When $O P = x$ metres, the magnitude of the force is $\frac { 3 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}$ and the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Initially $P$ is at rest at $O$.
1. Show that $v ^ { 2 } = 6 \left( 1 - \frac { 1 } { ( x + 1 ) ^ { 2 } } \right)$.
2. Show that the speed of $P$ never reaches $\sqrt { } 6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
3. Find $x$ when $P$ has been moving for 2 seconds.
\section*{LL
$\_\_\_\_$}

Edexcel M3 2009 June Q1

A light elastic string has natural length 8 m and modulus of elasticity 80 N .

The ends of the string are attached to fixed points $P$ and $Q$ which are on the same horizontal level and 12 m apart. A particle is attached to the mid-point of the string and hangs in equilibrium at a point 4.5 m below $P Q$.

Calculate the weight of the particle.
Calculate the elastic energy in the string when the particle is in this position.

Edexcel M3 2009 June Q2

2. [The centre of mass of a uniform hollow cone of height $h$ is $\frac { 1 } { 3 } h$ above the base on the line from the centre of the base to the vertex.] \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5c9d14ac-0757-4cdd-9534-337e6b3acee0-03_641_614_388_662} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A marker for the route of a charity walk consists of a uniform hollow cone fixed on to a uniform solid cylindrical ring, as shown in Figure 1. The hollow cone has base radius $r$, height $9 h$ and mass $m$. The solid cylindrical ring has outer radius $r$, height $2 h$ and mass $3 m$. The marker stands with its base on a horizontal surface.

Find, in terms of $h$, the distance of the centre of mass of the marker from the horizontal surface. When the marker stands on a plane inclined at $\arctan \frac { 1 } { 12 }$ to the horizontal it is on the point of toppling over. The coefficient of friction between the marker and the plane is large enough to be certain that the marker will not slip.
Find $h$ in terms of $r$.

Edexcel M3 2009 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5c9d14ac-0757-4cdd-9534-337e6b3acee0-05_454_835_239_548} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass $m$ moves on the smooth inner surface of a hemispherical bowl of radius $r$. The bowl is fixed with its rim horizontal as shown in Figure 2. The particle moves with constant angular speed $\sqrt { } \left( \frac { 3 g } { 2 r } \right)$ in a horizontal circle at depth $d$ below the centre of the bowl.

Find, in terms of $m$ and $g$, the magnitude of the normal reaction of the bowl on $P$.
Find $d$ in terms of $r$.

Edexcel M3 2009 June Q4

The finite region bounded by the $x$-axis, the curve $y = \frac { 1 } { x ^ { 2 } }$, the line $x = \frac { 1 } { 4 }$ and the line $x = 1$, is rotated through one complete revolution about the $x$-axis to form a uniform solid of revolution.
1. Show that the volume of the solid is $21 \pi$.
2. Find the coordinates of the centre of mass of the solid.
3. One end of a light inextensible string of length $l$ is attached to a fixed point $A$. The other end is attached to a particle $P$ of mass $m$, which is held at a point $B$ with the string taut and $A P$ making an angle $\arccos \frac { 1 } { 4 }$ with the downward vertical. The particle is released from rest. When $A P$ makes an angle $\theta$ with the downward vertical, the string is taut and the tension in the string is $T$.
4. Show that
$$T = 3 m g \cos \theta - \frac { m g } { 2 }$$ (6) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c9d14ac-0757-4cdd-9534-337e6b3acee0-08_678_629_815_653} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} At an instant when $A P$ makes an angle of $60 ^ { \circ }$ to the downward vertical, $P$ is moving upwards, as shown in Figure 3. At this instant the string breaks. At the highest point reached in the subsequent motion, $P$ is at a distance $d$ below the horizontal through $A$.
Find $d$ in terms of $l$.

Edexcel M3 2009 June Q6

A cyclist and her bicycle have a combined mass of 100 kg . She is working at a constant rate of 80 W and is moving in a straight line on a horizontal road. The resistance to motion is proportional to the square of her speed. Her initial speed is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and her maximum possible speed under these conditions is $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When she is at a distance $x \mathrm {~m}$ from a fixed point $O$ on the road, she is moving with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ away from $O$.
1. Show that
$$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = \frac { 8000 - v ^ { 3 } } { 10000 v }$$
Find the distance she travels as her speed increases from $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
Use the trapezium rule, with 2 intervals, to estimate how long it takes for her speed to increase from $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ to $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Edexcel M3 2009 June Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5c9d14ac-0757-4cdd-9534-337e6b3acee0-12_195_922_237_511} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} $A$ and $B$ are two points on a smooth horizontal floor, where $A B = 5 \mathrm {~m}$. A particle $P$ has mass 0.5 kg . One end of a light elastic spring, of natural length 2 m and modulus of elasticity 16 N , is attached to $P$ and the other end is attached to $A$. The ends of another light elastic spring, of natural length 1 m and modulus of elasticity 12 N , are attached to $P$ and $B$, as shown in Figure 4.

Find the extensions in the two springs when the particle is at rest in equilibrium. Initially $P$ is at rest in equilibrium. It is then set in motion and starts to move towards $B$. In the subsequent motion $P$ does not reach $A$ or $B$.
Show that $P$ oscillates with simple harmonic motion about the equilibrium position.
Given that the initial speed of $P$ is $\sqrt { } 10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, find the proportion of time in each complete oscillation for which $P$ stays within 0.25 m of the equilibrium position.

Edexcel M3 2010 June Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-02_458_516_281_712} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A garden game is played with a small ball $B$ of mass $m$ attached to one end of a light inextensible string of length 13l. The other end of the string is fixed to a point $A$ on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius $5 l$ and centre $C$, where $C$ is vertically below $A$. Modelling the ball as a particle, find

the tension in the string,
the speed of the ball.

Edexcel M3 2010 June Q2

2. A particle $P$ of mass $m$ is above the surface of the Earth at distance $x$ from the centre of the Earth. The Earth exerts a gravitational force on $P$. The magnitude of this force 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 sphere of radius $R$.

Prove that the magnitude of the gravitational force on $P$ is $\frac { m g R ^ { 2 } } { x ^ { 2 } }$. A particle is fired vertically upwards from the surface of the Earth with initial speed $3 U$. At a height $R$ above the surface of the Earth the speed of the particle is $U$.
Find $U$ in terms of $g$ and $R$.

Edexcel M3 2010 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-05_342_718_255_610} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.9 m and modulus of elasticity $\lambda$ newtons. The other end of the spring is attached to a fixed point $O$ on a rough plane which is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac { 3 } { 5 }$. The coefficient of friction between the particle and the plane is 0.15 . The particle is held on the plane at a point which is 1.5 m down the line of greatest slope from $O$, as shown in Figure 2. The particle is released from rest and first comes to rest again after moving 0.7 m up the plane. Find the value of $\lambda$.

Edexcel M3 2010 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-07_431_604_260_667} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A container is formed by removing a right circular solid cone of height $4 l$ from a uniform solid right circular cylinder of height $6 l$. The centre $O$ of the plane face of the cone coincides with the centre of a plane face of the cylinder and the axis of the cone coincides with the axis of the cylinder, as shown in Figure 3. The cylinder has radius $2 l$ and the base of the cone has radius $l$.

Find the distance of the centre of mass of the container from $O$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-07_460_588_1254_676} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The container is placed on a plane which is inclined at an angle $\theta ^ { \circ }$ to the horizontal. The open face is uppermost, as shown in Figure 4. The plane is sufficiently rough to prevent the container from sliding. The container is on the point of toppling.
Find the value of $\theta$.

Edexcel M3 2010 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-10_474_465_269_735} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is fixed at the point $O$. The particle is initially held with $O P$ horizontal and the string taut. It is then projected vertically upwards with speed $u$, where $u ^ { 2 } = 5 a g$. When $O P$ has turned through an angle $\theta$ the speed of $P$ is $v$ and the tension in the string is $T$, as shown in Figure 5.

Find, in terms of $a , g$ and $\theta$, an expression for $v ^ { 2 }$.
Find, in terms of $m , g$ and $\theta$, an expression for $T$.
Prove that $P$ moves in a complete circle.
Find the maximum speed of $P$.

Edexcel M3 2010 June Q6

At time $t = 0$, a particle $P$ is at the origin $O$ moving with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the $x$-axis in the positive $x$-direction. At time $t$ seconds $( t > 0 )$, the acceleration of $P$ has magnitude $\frac { 3 } { ( t + 1 ) ^ { 2 } } \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and is directed towards $O$.
1. Show that at time $t$ seconds the velocity of $P$ is $\left( \frac { 3 } { t + 1 } - 1 \right) \mathrm { m } \mathrm { s } ^ { - 1 }$.
2. Find, to 3 significant figures, the distance of $P$ from $O$ when $P$ is instantaneously at rest.

Edexcel M3 2010 June Q7

A light elastic string, of natural length $3 a$ and modulus of elasticity $6 m g$, has one end attached to a fixed point $A$. A particle $P$ of mass $2 m$ is attached to the other end of the string and hangs in equilibrium at the point $O$, vertically below $A$.
1. Find the distance $A O$.
The particle is now raised to point $C$ vertically below $A$, where $A C > 3 a$, and is released from rest.
Show that $P$ moves with simple harmonic motion of period $2 \pi \sqrt { } \left( \frac { a } { g } \right)$. It is given that $O C = \frac { 1 } { 4 } a$.
Find the greatest speed of $P$ during the motion. The point $D$ is vertically above $O$ and $O D = \frac { 1 } { 8 } a$. The string is cut as $P$ passes through $D$, moving upwards.
Find the greatest height of $P$ above $O$ in the subsequent motion.

Edexcel M3 2011 June Q1

A particle $P$ of mass 0.5 kg moves on the positive $x$-axis under the action of a single force directed towards the origin $O$. At time $t$ seconds the distance of $P$ from $O$ is $x$ metres, the magnitude of the force is $0.375 x ^ { 2 } \mathrm {~N}$ and the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

When $t = 0 , O P = 8 \mathrm {~m}$ and $P$ is moving towards $O$ with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $v ^ { 2 } = 260 - \frac { 1 } { 2 } \chi ^ { 3 }$.
Find the distance of $P$ from $O$ at the instant when $v = 5$.

Questions M3 (745 questions)

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Edexcel M3 2007 June Q3

Edexcel M3 2007 June Q4

Edexcel M3 2007 June Q5

Edexcel M3 2007 June Q6

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Edexcel M3 2008 June Q1

Edexcel M3 2008 June Q2

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Edexcel M3 2008 June Q5

Edexcel M3 2008 June Q6

Edexcel M3 2009 June Q1

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Edexcel M3 2009 June Q7

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Edexcel M3 2010 June Q5

Edexcel M3 2010 June Q6

Edexcel M3 2010 June Q7

Edexcel M3 2011 June Q1