Questions M3 - A-Level Maths

the magnitude and direction of the acceleration of $P$ when it passes through $A$,
the speed of $P$ when it passes through $A$,
the time $P$ takes to move directly from $C$ to $A$.

Edexcel M3 2012 January Q3

3. A particle $P$ is moving in a straight line. At time $t$ seconds, $P$ is at a distance $x$ metres from a fixed point $O$ on the line and is moving away from $O$ with speed $\frac { 10 } { x + 6 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the acceleration of $P$ when $x = 14$ Given that $x = 2$ when $t = 1$,
find the value of $t$ when $x = 14$

Edexcel M3 2012 January Q4

4. A light elastic string $A B$ has natural length 0.8 m and modulus of elasticity 19.6 N . The end $A$ is attached to a fixed point. A particle of mass 0.5 kg is attached to the end $B$. The particle is moving with constant angular speed $\omega$ rad s $^ { - 1 }$ in a horizontal circle whose centre is vertically below $A$. The string is inclined at $60 ^ { \circ }$ to the vertical.

Show that the extension of the string is 0.4 m .
Find the value of $\omega$.

Edexcel M3 2012 January Q5

5. Above the Earth's surface, the magnitude of the gravitational force on a particle due to the Earth is inversely proportional to the square of the distance of the particle from the centre of the Earth. The Earth is modelled as a sphere of radius $R$ and the acceleration due to gravity at the Earth's surface is $g$. A particle $P$ of mass $m$ is at a height $x$ above the surface of the Earth.

Show that the magnitude of the gravitational force acting on $P$ is $$\frac { m g R ^ { 2 } } { ( R + x ) ^ { 2 } }$$ A rocket is fired vertically upwards from the surface of the Earth. When the rocket is at height $2 R$ above the surface of the Earth its speed is $\sqrt { } \left( \frac { g R } { 2 } \right)$. You may assume that air resistance can be ignored and that the engine of the rocket is switched off before the rocket reaches height $R$. Modelling the rocket as a particle,
find the speed of the rocket when it was at height $R$ above the surface of the Earth.

Edexcel M3 2012 January Q6

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 hanging in equilibrium at the point $A$, vertically below $O$, when it is set in motion with a horizontal speed $\frac { 1 } { 2 } \sqrt { } ( 11 g l )$. When the string has turned through an angle $\theta$ and the string is still taut, the tension in the string is $T$.
1. Show that $T = 3 m g \left( \cos \theta + \frac { 1 } { 4 } \right)$.
At the instant when $P$ reaches the point $B$, the string becomes slack. Find
the speed of $P$ at $B$,
the maximum height above $B$ reached by $P$ before it starts to fall.

Edexcel M3 2012 January Q7

7. Diagram NOT accurately drawn \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{bbd531ab-05f8-48ff-8a68-ec6f33ac0a2f-12_444_768_253_603} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The shaded region $R$ is bounded by the curve with equation $y = \frac { 1 } { 2 } x ( 6 - x )$, the $x$-axis and the line $x = 2$, as shown in Figure 1. The unit of length on both axes is 1 cm . A uniform solid $P$ is formed by rotating $R$ through $360 ^ { \circ }$ about the $x$-axis.

Show that the centre of mass of $P$ is, to 3 significant figures, 1.42 cm from its plane face. The uniform solid $P$ is placed with its plane face on an inclined plane which makes an angle $\theta$ with the horizontal. Given that the plane is sufficiently rough to prevent $P$ from sliding and that $P$ is on the point of toppling when $\theta = \alpha$,
find the angle $\alpha$. Given instead that $P$ is on the point of sliding down the plane when $\theta = \beta$ and that the coefficient of friction between $P$ and the plane is 0.3 ,
find the angle $\beta$.

Edexcel M3 2013 January Q1

A particle $P$ is moving along the positive $x$-axis. When the displacement of $P$ from the origin is $x$ metres, the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the acceleration of $P$ is $9 x \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

When $x = 2 , v = 6$
Show that $v ^ { 2 } = 9 x ^ { 2 }$.
(4)

Edexcel M3 2013 January Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d19c7390-0332-4cab-82e5-72976bd499a2-03_636_529_322_662} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform solid consists of a right circular cone of radius $r$ and height $k r$, where $k > \sqrt { } 3$, 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 1.

Show that the distance of the centre of mass of the solid from $O$ is $$\frac { \left( k ^ { 2 } - 3 \right) r } { 4 ( k + 2 ) }$$ 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. The angle between $A O$ and the vertical is $\theta$, where $\tan \theta = \frac { 11 } { 14 }$
Find the value of $k$.

Edexcel M3 2013 January Q3

A particle $P$ of mass 0.6 kg is moving along the $x$-axis in the positive direction. At time $t = 0 , P$ passes through the origin $O$ with speed $15 \mathrm {~ms} ^ { - 1 }$. At time $t$ seconds the distance $O P$ is $x$ metres, the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the resultant force acting on $P$ has magnitude $\frac { 12 } { ( t + 2 ) ^ { 2 } }$ newtons. The resultant force is directed towards $O$.
1. Show that $v = 5 \left( \frac { 4 } { t + 2 } + 1 \right)$.
2. Find the value of $x$ when $t = 5$

Edexcel M3 2013 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d19c7390-0332-4cab-82e5-72976bd499a2-07_503_618_242_646} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle $P$ of mass $m$ is attached to one end of a light elastic string, of natural length $2 a$ and modulus of elasticity $6 m g$. The other end of the string is attached to a fixed point $A$. The particle moves with constant speed $v$ in a horizontal circle with centre $O$, where $O$ is vertically below $A$ and $O A = 2 a$, as shown in Figure 2 .

Show that the extension in the string is $\frac { 2 } { 5 } a$.
Find $v ^ { 2 }$ in terms of $a$ and $g$.

Edexcel M3 2013 January Q5

5. A particle $P$ is moving in a straight line with simple harmonic motion on a smooth horizontal floor. The particle comes to instantaneous rest at points $A$ and $B$ where $A B$ is 0.5 m . The mid-point of $A B$ is $O$. The mid-point of $O A$ is $C$. The mid-point of $O B$ is $D$. The particle takes 0.2 s to travel directly from $C$ to $D$. At time $t = 0 , P$ is moving through $O$ towards $A$.

Show that the period of the motion is $\frac { 6 } { 5 } \mathrm {~s}$.
Find the distance of $P$ from $B$ when $t = 2 \mathrm {~s}$.
Find the maximum magnitude of the acceleration of $P$.
Find the maximum speed of $P$.

Edexcel M3 2013 January Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d19c7390-0332-4cab-82e5-72976bd499a2-11_412_533_258_685} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A smooth hollow cylinder of internal radius $a$ is fixed with its axis horizontal. A particle $P$ moves on the inner surface of the cylinder in a vertical circle with radius $a$ and centre $O$, where $O$ lies on the axis of the cylinder. The particle is projected vertically downwards with speed $u$ from point $A$ on the circle, where $O A$ is horizontal. The particle first loses contact with the cylinder at the point $B$, where $\angle A O B = 150 ^ { \circ }$, as shown in Figure 3. Given that air resistance can be ignored,

show that the speed of $P$ at $B$ is $\sqrt { } \left( \frac { a g } { 2 } \right)$,
find $u$ in terms of $a$ and $g$. After losing contact with the cylinder, $P$ crosses the diameter through $A$ at the point $D$. At $D$ the velocity of $P$ makes an angle $\theta ^ { \circ }$ with the horizontal.
Find the value of $\theta$.

Edexcel M3 2013 January Q7

7. A particle $P$ of mass 1.5 kg is attached to the mid-point of a light elastic string of natural length 0.30 m and modulus of elasticity $\lambda$ newtons. The ends of the string are attached to two fixed points $A$ and $B$, where $A B$ is horizontal and $A B = 0.48 \mathrm {~m}$. Initially $P$ is held at rest at the mid-point, $M$, of the line $A B$ and the tension in the string is 240 N .

Show that $\lambda = 400$ The particle is now held at rest at the point $C$, where $C$ is 0.07 m vertically below $M$. The particle is released from rest at $C$.
Find the magnitude of the initial acceleration of $P$.
Find the speed of $P$ as it passes through $M$.

Edexcel M3 2001 June Q1

A particle $P$ moves along the $x$-axis in the positive direction. At time $t$ seconds, the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and its acceleration is $\frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 6 } t } \mathrm {~m} \mathrm {~s} ^ { - 2 }$. When $t = 0$ the speed of $P$ is $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
1. Express $v$ in terms of $t$.
2. Find, to 3 significant figures, the speed of $P$ when $t = 3$.
3. Find the limiting value of $v$.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c3026c4b-d499-4756-9e01-9b9929f2e04e-2_526_1186_1142_466}
\end{figure} A smooth solid hemisphere, of radius 0.8 m and centre $O$, is fixed with its plane face on a horizontal table. A particle of mass 0.5 kg is projected horizontally with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from the highest point $A$ of the hemisphere. The particle leaves the hemisphere at the point $B$, which is a vertical distance of 0.2 m below the level of $A$. The speed of the particle at $B$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the angle between $O A$ and $O B$ is $\theta$, as shown in Fig. 1.
Find the value of $\cos \theta$.
Show that $v ^ { 2 } = 5.88$.
Find the value of $u$.

Edexcel M3 2001 June Q3

3. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c3026c4b-d499-4756-9e01-9b9929f2e04e-3_481_1262_541_390}

\end{figure} A light horizontal spring, of natural length 0.25 m and modulus of elasticity 52 N , is fastened at one end to a point $A$. The other end of the spring is fastened to a small wooden block $B$ of mass 1.5 kg which is on a horizontal table, as shown in Fig. 2. The block is modelled as a particle. The table is initially assumed to be smooth. The block is released from rest when it is a distance 0.3 m from $A$. By using the principle of the conservation of energy,

find, to 3 significant figures, the speed of $B$ when it is a distance 0.25 m from $A$. It is now assumed that the table is rough and the coefficient of friction between $B$ and the table is 0.6 .
Find, to 3 significant figures, the minimum distance from $A$ at which $B$ can rest in equilibrium.
(5)

Edexcel M3 2001 June Q4

4. A projectile $P$ is fired vertically upwards from a point on the earth's surface. When $P$ is at a distance $x$ from the centre of the earth its speed is $v$. Its acceleration is directed towards the centre of the earth and has magnitude $\frac { k } { x ^ { 2 } }$, where $k$ is a constant. The earth may be assumed to be a sphere of radius $R$.

Show that the motion of $P$ may be modelled by the differential equation $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \frac { g R ^ { 2 } } { x ^ { 2 } }$$ The initial speed of $P$ is $U$, where $U ^ { 2 } < 2 g R$. The greatest distance of $P$ from the centre of the earth is $X$.
Find $X$ in terms of $U , R$ and $g$.
(6) \section*{5.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{c3026c4b-d499-4756-9e01-9b9929f2e04e-5_792_732_513_593}
\end{figure} An ornament $S$ is formed by removing a solid right circular cone, of radius $r$ and height $\frac { 1 } { 2 } h$, from a solid uniform cylinder, of radius $r$ and height $h$, as shown in Fig. 3.
Show that the distance of the centre of mass $S$ from its plane face is $\frac { 17 } { 40 } h$. The ornament is suspended from a point on the circular rim of its open end. It hangs in equilibrium with its axis of symmetry inclined at an angle $\alpha$ to the horizontal. Given that $h = 4 r$,
find, in degrees to one decimal place, the value of $\alpha$.
(4) \section*{6.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{c3026c4b-d499-4756-9e01-9b9929f2e04e-6_841_942_507_605}
\end{figure} A particle $P$ of mass $m$ is attached to two light inextensible strings. The other ends of the string are attached to fixed points $A$ and $B$. The point $A$ is a distance $h$ vertically above $B$. The system rotates about the line $A B$ with constant angular speed $\omega$. Both strings are taut and inclined at $60 ^ { \circ }$ to $A B$, as shown in Fig. 4. The particle moves in a circle of radius $r$.
Show that $r = \frac { \sqrt { 3 } } { 2 } h$.
Find, in terms of $m , g , h$ and $\omega$, the tension in $A P$ and the tension in $B P$. The time taken for $P$ to complete one circle is $T$.
Show that $T < \pi \sqrt { \left( \frac { 2 h } { g } \right) }$.
(4)

Edexcel M3 2001 June Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{c3026c4b-d499-4756-9e01-9b9929f2e04e-7_492_929_504_543}

\end{figure} A small ring $R$ of mass in is free to slide on a smooth straight wire which is fixed at an angle of $30 ^ { \circ }$ to the horizontal. The ring is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $\lambda$. The other end of the string is attached to a fixed point $A$ of the wire, as shown in Fig. 5. The ring rests in equilibrium at the point $B$, where $A B = \frac { 9 } { 8 } a$.

Show that $\lambda = 4 m g$. The ring is pulled down to the point $C$, where $B C = \frac { 1 } { 4 } a$, and released from rest. At time $t$ after $R$ is released the extension of the string is ( $\frac { 1 } { 8 } a + x$ ).
Obtain a differential equation for the motion of $R$ while the string remains taut, and show that it represents simple harmonic motion with period $\pi \sqrt { \left( \frac { a } { g } \right) }$.
(6)
Find, in terms of $g$, the greatest magnitude of the acceleration of $R$ while the string remains taut.
(2)
Find, in terms of $a$ and $g$, the time taken for $R$ to move from the point at which it first reaches maximum speed to the point where the string becomes slack for the first time. END

Edexcel M3 2002 June Q1

A particle $P$ moves in a straight line with simple harmonic motion about a fixed centre $O$ with period 2 s . At time $t$ seconds the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When $t = 0$, $v = 0$ and $P$ is at a point $A$ where $O A = 0.25 \mathrm {~m}$.

Find the smallest positive value of $t$ for which $A P = 0.375 \mathrm {~m}$. \section*{2.} \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6d6b6de9-ec06-4b55-8dc6-4923a3554ffa-2_882_985_648_486}

\end{figure} A metal ball $B$ of mass $m$ is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point $A$. The ball $B$ moves in a horizontal circle with centre $O$ vertically below $A$, as shown in Fig. 1. The string makes a constant angle $\alpha ^ { \circ }$ with the downward vertical and $B$ moves with constant angular speed $\sqrt { } ( 2 g k )$, where $k$ is a constant. The tension in the string is $3 m g$. By modelling $B$ as a particle. find

the value of $\alpha$,
the length of the string.

Edexcel M3 2002 June Q3

3. A particle $P$ of mass 2.5 kg moves along the positive $x$-axis. It moves away from a fixed origin $O$, under the action of a force directed away from $O$. When $O P = x$ metres the magnitude of the force is $2 \mathrm { e } ^ { - 0.1 x }$ newtons and the speed of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When $x = 0 , v = 2$. Find

$v ^ { 2 }$ in terms of $x$,
the value of $x$ when $v = 4$.
Give a reason why the speed of $P$ does not exceed $\sqrt { } 20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Edexcel M3 2002 June Q4

4. A light elastic string $A B$ of natural length 1.5 m has modulus of elasticity 20 N . The end $A$ is fixed to a point on a smooth horizontal table. A small ball $S$ of mass 0.2 kg is attached to the end $B$. Initially $S$ is at rest on the table with $A B = 1.5 \mathrm {~m}$. The ball $S$ is then projected horizontally directly away from $A$ with a speed of $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. By modelling $S$ as a particle,

find the speed of $S$ when $A S = 2 \mathrm {~m}$. When the speed of $S$ is $1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the string breaks.
Find the tension in the string immediately before the string breaks.
(5)

Edexcel M3 2002 June Q5

5.
\includegraphics[max width=\textwidth, alt={}, center]{6d6b6de9-ec06-4b55-8dc6-4923a3554ffa-4_1185_1059_239_416} A model tree is made by joining a uniform solid cylinder to a uniform solid cone made of the same material. The centre $O$ of the base of the cone is also the centre of one end of the cylinder, as shown in Fig. 2. The radius of the cylinder is $r$ and the radius of the base of the cone is $2 r$. The height of the cone and the height of the cylinder are each $h$. The centre of mass of the model is at the point $G$.

Show that $O G = \frac { 1 } { 14 } h$.
(8) The model stands on a desk top with its plane face in contact with the desk top. The desk top is tilted until it makes an angle $\alpha$ with the horizontal, where tan $\alpha = \frac { 7 } { 26 }$. The desk top is rough enough to prevent slipping and the model is about to topple.
Find $r$ in terms of $h$.
(4)

Edexcel M3 2002 June Q6

6. A light elastic string, of natural length $4 a$ and modulus of elasticity $8 m g$, has one end attached to a fixed point $A$. A particle $P$ of mass $m$ is attached to the other end of the string and hangs in equilibrium at the point $O$.

Find the distance $A O$.
(2) The particle is now pulled down to a point $C$ vertically below $O$, where $O C = d$. It is released from rest. In the subsequent motion the string does not become slack.
Show that $P$ moves with simple harmonic motion of period $\pi \sqrt { \left( \frac { 2 a } { g } \right) }$. The greatest speed of $P$ during this motion is $\frac { 1 } { 2 } \sqrt { } ( g a )$.
Find $d$ in terms of $a$.
(3) Instead of being pulled down a distance $d$, the particle is pulled down a distance $a$. Without further calculation,
describe briefly the subsequent motion of $P$.
(2)

Edexcel M3 2002 June Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{6d6b6de9-ec06-4b55-8dc6-4923a3554ffa-6_682_553_264_828}

\end{figure} 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 $O$. The particle is hanging at the point $A$, which is vertically below $O$. It is projected horizontally with speed $u$. When the particle is at the point $P , \angle A O P = \theta$, as shown in Fig. 3. The string oscillates through an angle $\alpha$ on either side of $O A$ where $\cos \alpha = \frac { 2 } { 3 }$.

Find $u$ in terms of g and $l$. When $\angle A O P = \theta$, the tension in the string is $T$.
Show that $T = \frac { m g } { 3 } ( 9 \cos \theta - 4 )$.
Find the range of values of $T$. END

Edexcel M3 2003 June Q1

A particle $P$ of mass $m$ is held at a point $A$ on a rough horizontal plane. The coefficient of friction between $P$ and the plane is $\frac { 2 } { 3 }$. The particle is attached to one end of a light elastic string, of natural length $a$ and modulus of elasticity $4 m g$. The other end of the string is attached to a fixed point $O$ on the plane, where $O A = \frac { 3 } { 2 } a$. The particle $P$ is released from rest and comes to rest at a point $B$, where $O B < a$.

Using the work-energy principle, or otherwise, calculate the distance $A B$.
(6)

Edexcel M3 2003 June Q2

2. A car moves round a bend which is banked at a constant angle of $10 ^ { \circ }$ to the horizontal. When the car is travelling at a constant speed of $18 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, there is no sideways frictional force on the car. The car is modelled as a particle moving in a horizontal circle of radius $r$ metres. Calculate the value of $r$.
(6)

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