Questions M3 - A-Level Maths

Find the distance of the centre of mass of $S$ from $O$.
(6) The lower plane face of $S$ rests in equilibrium on a desk lid which is inclined at an angle $\theta$ to the horizontal. Assuming that the lid is sufficiently rough to prevent $S$ from slipping, and that $S$ is on the point of toppling when $\theta = \alpha$,
find the value of $\alpha$.
(3) Given instead that the coefficient of friction between $S$ and the lid is 0.8 , and that $S$ is on the point of sliding down the lid when $\theta = \beta$,
find the value of $\beta$.
(3)

Edexcel M3 2004 January Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{c4b453e7-8a32-458b-8041-58c9e4ef9533-6_710_729_172_672}

\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 a point $O$. The particle is held with the string taut and $O P$ horizontal. It is then projected vertically downwards with speed $u$, where $u ^ { 2 } = \frac { 3 } { 2 } g a$. When $O P$ has turned through an angle $\theta$ and the string is still taut, the speed of $P$ is $v$ and the tension in the string is $T$, as shown in Fig. 3.

Find an expression for $v ^ { 2 }$ in terms of $a , g$ and $\theta$.
Find an expression for $T$ in terms of $m , g$ and $\theta$.
Prove that the string becomes slack when $\theta = 210 ^ { \circ }$.
State, with a reason, whether $P$ would complete a vertical circle if the string were replaced by a light rod. After the string becomes slack, $P$ moves freely under gravity and is at the same level as $O$ when it is at the point $A$.
Explain briefly why the speed of $P$ at $A$ is $\sqrt { } \left( \frac { 3 } { 2 } g a \right)$. The direction of motion of $P$ at $A$ makes an angle $\varphi$ with the horizontal.
Find $\varphi$.

Edexcel M3 2005 January Q1

A particle $P$ of mass 0.5 kg is attached to one end of a light inextensible string of length 1.5 m . The other end of the string is attached to a fixed point $A$. The particle is moving, with the string taut, in a horizontal circle with centre $O$ vertically below $A$. The particle is moving with constant angular speed $2.7 \mathrm { rad } \mathrm { s } ^ { - 1 }$. Find
1. the tension in the string,
2. the angle, to the nearest degree, that $A P$ makes with the downward vertical.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-2_966_706_841_641}
\end{figure} A child's toy consists of a uniform solid hemisphere, of mass $M$ and base radius $r$, joined to a uniform solid right circular cone of mass $m$, where $2 m < M$. The cone has vertex $O$, base radius $r$ and height $3 r$. Its plane face, with diameter $A B$, coincides with the plane face of the hemisphere, as shown in Figure 1.
Show that the distance of the centre of mass of the trom $A B$ is $$\frac { 3 ( M - 2 m ) } { 8 ( M + m ) } r$$ The toy is placed with $O A$ on a horizontal surface. The toy is released from rest and does not remain in equilibrium.
Show that $M > 26 m$.

Edexcel M3 2005 January Q3

3. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-3_531_899_299_497}

\end{figure} A uniform lamina occupies the region $R$ bounded by the $x$-axis and the curve $$y = \sin x , \quad 0 \leq x \leq \pi$$ as shown in Figure 2.

Show, by integration, that the $y$-coordinate of the centre of mass of the lamina is $\frac { \pi } { 8 }$. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-3_652_792_1439_568}
\end{figure} A uniform prism $S$ has cross-section $R$. The prism is placed with its rectangular face on a table which is inclined at an angle $\theta ^ { \circ }$ to the horizontal. The cross-section $R$ lies in a vertical plane as shown in Figure 3. The table is sufficiently rough to prevent $S$ sliding. Given that $S$ does not topple,
find the largest possible value of $\theta$.
(3)

Edexcel M3 2005 January Q4

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-4_284_1077_294_429}

\end{figure} In a game at a fair, a small target $C$ moves horizontally with simple harmonic motion between the points $A$ and $B$, where $A B = 4 L$. The target moves inside a box and takes 3 s to travel from $A$ to $B$. A player has to shoot at $C$, but $C$ is only visible to the player when it passes a window $P Q$, where $P Q = b$. The window is initially placed with $Q$ at the point as shown in Figure 4. The target $C$ takes 0.75 s to pass from $Q$ to $P$.

Show that $b = ( 2 - \sqrt { 2 } ) L$.
Find the speed of $C$ as it passes $P$. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-4_286_1082_1327_424}
\end{figure} For advanced players, the window $P Q$ is moved to the centre of $A B$ so that $A P = Q B$, as shown in Figure 5.
Find the time, in seconds to 2 decimal places, taken for $C$ to pass from $Q$ to $P$ in this new position.

Edexcel M3 2005 January Q5

5. At time $t = 0$, a particle $P$ is at the origin $O$, moving with speed $18 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along the $x$-axis, in the positive $x$-direction. At time $t$ seconds $( t > 0 )$ the acceleration of $P$ has magnitude $\frac { 3 } { \sqrt { } ( t + 4 ) } \mathrm { m } \mathrm { s } ^ { - 2 }$ and is directed towards $O$.

Show that, at time $t$ seconds, the velocity of $P$ is $[ 30 - 6 \sqrt { } ( t + 4 ) ] \mathrm { m } \mathrm { s } ^ { - 1 }$.
Find the distance of $P$ from $O$ when $P$ comes to instantaneous rest.

Edexcel M3 2005 January Q6

6. A light spring of natural length $L$ has one end attached to a fixed point $A$. A particle $P$ of mass $m$ is attached to the other end of the spring. The particle is moving vertically. As it passes through the point $B$ below $A$, where $A B = L$, its speed is $\sqrt { } ( 2 g L )$. The particle comes to instantaneous rest at a point $C , 4 L$ below $A$.

Show that the modulus of elasticity of the spring is $\frac { 8 m g } { 9 }$. At the point $D$ the tension in the spring is $m g$.
Show that $P$ performs simple harmonic motion with centre $D$.
Find, in terms of $L$ and $g$,
1. the period of the simple harmonic motion,
2. the maximum speed of $P$.
  (5)

Edexcel M3 2005 January Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 6} \includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-6_451_1360_296_356}

\end{figure} A trapeze artiste of mass 60 kg is attached to the end $A$ of a light inextensible rope $O A$ of length 5 m . The artiste must swing in an arc of a vertical circle, centre $O$, from a platform $P$ to another platform $Q$, where $P Q$ is horizontal. The other end of the rope is attached to the fixed point $O$ which lies in the vertical plane containing $P Q$, with $\angle P O Q = 120 ^ { \circ }$ and $O P = O Q = 5 \mathrm {~m}$, as shown in Figure 6. As part of her act, the artiste projects herself from $P$ with speed $\sqrt { } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a direction perpendicular to the rope $O A$ and in the plane $P O Q$. She moves in a circular arc towards $Q$. At the lowest point of her path she catches a ball of mass $m \mathrm {~kg}$ which is travelling towards her with speed $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and parallel to $Q P$. After catching the ball, she comes to rest at the point $Q$. By modelling the artiste and the ball as particles and ignoring her air resistance, find

the speed of the artiste immediately before she catches the ball,
the value of $m$,
the tension in the rope immediately after she catches the ball.

Edexcel M3 2006 January Q1

1. \section*{Figure 1}

\includegraphics[max width=\textwidth, alt={}]{67a9cf74-833f-4b4a-9fde-3c62dcc08e8c-2_515_1157_276_516}

A particle $P$ of mass 0.8 kg is attached to one end of a light inelastic string, of natural length 1.2 m and modulus of elasticity 24 N . The other end of the string is attached to a fixed point $A$. A horizontal force of magnitude $F$ newtons is applied to $P$. The particle $P$ in in equilibrium with the string making an angle $60 ^ { \circ }$ with the downward vertical, as shown in Figure 1. Calculate

the value of $F$,
the extension of the string,
the elasticity stored in the string.

Edexcel M3 2006 January Q2

2. A particle $P$ moves along the $x$-axis. At time $t$ seconds the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and its acceleration is $2 \sin \frac { 1 } { 2 } t \mathrm {~m} \mathrm {~s} ^ { - 2 }$, both measured in the direction of $O x$. Given that $v = 4$ when $t = 0$,

find $v$ in terms of $t$,
calculate the distance travelled by $P$ between the times $t = 0$ and $t = \frac { \pi } { 2 }$.

Edexcel M3 2006 January Q3

3. A rocket is fired vertically upwards with speed $U$ from a point on the Earth's surface. The rocket is modelled as a particle $P$ of constant mass $m$, and the Earth as a fixed sphere of radius $R$. At a distance $x$ from the centre of the Earth, the speed of $P$ is $v$. The only force acting on $P$ is directed towards the centre of the Earth and has magnitude $\frac { c m } { x ^ { 2 } }$, where $c$ is a constant.

Show that $v ^ { 2 } = U ^ { 2 } + 2 c \left( \frac { 1 } { x } - \frac { 1 } { R } \right)$. The kinetic energy of $P$ at $x = 2 R$ is half of its kinetic energy at $x = R$.
Find $c$ in terms of $U$ and $R$.
(3)

Edexcel M3 2006 January Q4

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{67a9cf74-833f-4b4a-9fde-3c62dcc08e8c-3_531_387_1226_845}

\end{figure} A body consists of a uniform solid circular cylinder $C$, together with a uniform solid hemisphere $H$ which is attached to $C$. The plane face of $H$ coincides with the upper plane face of $C$, as shown in Figure 2. The cylinder $C$ has base radius $r$, height $h$ and mass 3M. The mass of $H$ is $2 M$. The point $O$ is the centre of the base of $C$.

Show that the distance of the centre of mass of the body from $O$ is $$\frac { 14 h + 3 r } { 20 } .$$ The body is placed with its plane face on a rough plane which is inclined at an angle $\alpha$ to the horizontal, where tan $\alpha = \frac { 4 } { 3 }$. The plane is sufficiently rough to prevent slipping. Given that the body is on the point of toppling,
find $h$ in terms of $r$.

Edexcel M3 2006 January Q5

5. A light elastic string of natural length $l$ has one end attached to a fixed point $A$. A particle $P$ of mass $m$ is attached tot he other end of the string and hangs in equilibrium at the point $O$, where $A O = \frac { 5 } { 4 } l$.

Find the modulus of the elasticity of the string. The particle $P$ is then pulled down and released from rest. At time $t$ the length of the string is $\frac { 5 l } { 4 } + x$.
Prove that, while the string is taut, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g x } { l }$$ When $P$ is released, $A P = \frac { 7 } { 4 } l$. The point $B$ is a distance $l$ vertically below $A$.
Find the speed of $P$ at $B$.
Describe briefly the motion of $P$ after it has passed through $B$ for the first time until it next passes through $O$.

Edexcel M3 2006 January Q6

6. 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 hanging freely at rest at point $B$. The particle $P$ is projected horizontally from $B$ with speed $\sqrt { } ( 3 g l )$. When $A P$ makes an angle $\theta$ with the downward vertical and the string remains taut, the tension in the string is $T$.

Show that $T = m g ( 1 + 3 \cos \theta )$.
Find the speed of $P$ at the instant when the string becomes slack.
Find the maximum height above the level of $B$ reached by $P$.

Edexcel M3 2006 January Q7

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{67a9cf74-833f-4b4a-9fde-3c62dcc08e8c-5_625_1141_319_424}

\end{figure} A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $2 a$. The other end of the string is fixed to a point $A$ which is vertically above the point $O$ on a smooth horizontal table. The particle $P$ remains in contact with the surface of the table and moves in a circle with centre $O$ and with angular speed $\sqrt { \left( \frac { k g } { 3 a } \right) }$, where $k$ is a constant. Throughout the motion the string remains taut and $\angle A P O = 30 ^ { \circ }$, as shown in Figure 3.

Show that the tension in the string is $\frac { 2 k m g } { 3 }$.
Find, in terms of $m , g$ and $k$, the normal reaction between $P$ and the table.
Deduce the range of possible values of $k$. The angular speed of $P$ is changed to $\sqrt { \left( \frac { 2 g } { a } \right) }$. The particle $P$ now moves in a horizontal circle above the table. The centre of this circle is $X$.
Show that $X$ is the mid-point of $O A$.

Edexcel M3 2007 January Q1

A particle $P$ moves along the $x$-axis. At time $t = 0 , P$ passes through the origin $O$, moving in the positive $x$-direction. At time $t$ seconds, the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $O P = x$ metres. The acceleration of $P$ is $\frac { 1 } { 12 } ( 30 - x ) \mathrm { m } \mathrm { s } ^ { - 2 }$, measured in the positive $x$-direction.
1. Give a reason why the maximum speed of $P$ occurs when $x = 30$.
Given that the maximum speed of $P$ is $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$,
find an expression for $v ^ { 2 }$ in terms of $x$.

Edexcel M3 2007 January Q2

2. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-03_513_399_303_785}

\end{figure} A uniform solid right circular cone has base radius $a$ and semi-vertical angle $\alpha$, where $\tan \alpha = \frac { 1 } { 3 }$. The cone is freely suspended by a string attached at a point $A$ on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of $\theta ^ { \circ }$ with the upward vertical, as shown in Figure 1. Find, to one decimal place, the value of $\theta$.

Edexcel M3 2007 January Q3

3. A particle $P$ of mass $m$ is attached to one end of a light elastic string, of natural length $a$ and modulus of elasticity 3.6 mg . The other end of the string is fixed at a point $O$ on a rough horizontal table. The particle is projected along the surface of the table from $O$ with speed $\sqrt { } ( 2 a g )$. At its furthest point from $O$, the particle is at the point $A$, where $O A = \frac { 4 } { 3 } a$.

Find, in terms of $m , g$ and $a$, the elastic energy stored in the string when $P$ is at $A$.
Using the work-energy principle, or otherwise, find the coefficient of friction between $P$ and the table.

Edexcel M3 2007 January Q4

4. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-05_574_510_324_726}

\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 attached to a point $O$. The point $A$ is vertically below $O$, and $O A = a$. The particle is projected horizontally from $A$ with speed $\sqrt { } ( 3 a g )$. When $O P$ makes an angle $\theta$ with the upward vertical through $O$ and the string is still taut, the tension in the string is $T$ and the speed of $P$ is $v$, as shown in Figure 2.

Find, in terms of $a , g$ and $\theta$, an expression for $v ^ { 2 }$.
Show that $T = ( 1 - 3 \cos \theta ) m g$. The string becomes slack when $P$ is at the point $B$.
Find, in terms of $a$, the vertical height of $B$ above $A$. After the string becomes slack, the highest point reached by $P$ is $C$.
Find, in terms of $a$, the vertical height of $C$ above $B$.

Edexcel M3 2007 January Q5

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-07_531_691_299_657}

\end{figure} One end of a light inextensible string is attached to a fixed point $A$. The other end of the string is attached to a fixed point $B$, vertically below $A$, where $A B = h$. A small smooth ring $R$ of mass $m$ is threaded on the string. The ring $R$ moves in a horizontal circle with centre $B$, as shown in Figure 3. The upper section of the string makes a constant angle $\theta$ with the downward vertical and $R$ moves with constant angular speed $\omega$. The ring is modelled as a particle.

Show that $\omega ^ { 2 } = \frac { g } { h } \left( \frac { 1 + \sin \theta } { \sin \theta } \right)$.
Deduce that $\omega > \sqrt { \frac { 2 g } { h } }$. Given that $\omega = \sqrt { \frac { 3 g } { h } }$,
find, in terms of $m$ and $g$, the tension in the string.

Edexcel M3 2007 January Q6

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-09_515_1015_319_477}

\end{figure} The shaded region $R$ is bounded by the curve with equation $y = \frac { 1 } { 2 x ^ { 2 } }$, the $x$-axis and the lines $x = 1$ and $x = 2$, as shown in Figure 4. The unit of length on each axis is 1 m . A uniform solid $S$ has the shape made by rotating $R$ through $360 ^ { \circ }$ about the $x$-axis.

Show that the centre of mass of $S$ is $\frac { 2 } { 7 } \mathrm {~m}$ from its larger plane face. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{25b3ece7-69ed-4ec4-a6c7-4cd83ec2cc5e-09_616_431_1420_778}
\end{figure} A sporting trophy $T$ is a uniform solid hemisphere $H$ joined to the solid $S$. The hemisphere has radius $\frac { 1 } { 2 } \mathrm {~m}$ and its plane face coincides with the larger plane face of $S$, as shown in Figure 5. Both $H$ and $S$ are made of the same material.
Find the distance of the centre of mass of $T$ from its plane face.

Edexcel M3 2007 January Q7

A particle $P$ of mass 0.25 kg is attached to one end of a light elastic string. The string has natural length 0.8 m and modulus of elasticity $\lambda \mathrm { N }$. The other end of the string is attached to a fixed point $A$. In its equilibrium position, $P$ is 0.85 m vertically below $A$.
1. Show that $\lambda = 39.2$.
The particle is now displaced to a point $B , 0.95 \mathrm {~m}$ vertically below $A$, and released from rest.
Prove that, while the string remains stretched, $P$ moves with simple harmonic motion of period $\frac { \pi } { 7 } \mathrm {~s}$.
Calculate the speed of $P$ at the instant when the string first becomes slack. The particle first comes to instantaneous rest at the point $C$.
Find, to 3 significant figures, the time taken for $P$ to move from $B$ to $C$.

Edexcel M3 2008 January Q1

A light elastic string of natural length 0.4 m has one end $A$ attached to a fixed point. The other end of the string is attached to a particle $P$ of mass 2 kg . When $P$ hangs in equilibrium vertically below $A$, the length of the string is 0.56 m .
1. Find the modulus of elasticity of the string.
A horizontal force is applied to $P$ so that it is held in equilibrium with the string making an angle $\theta$ with the downward vertical. The length of the string is now 0.72 m .
Find the angle $\theta$.

Edexcel M3 2008 January Q2

2. A particle $P$ of mass 0.1 kg moves in a straight line on a smooth horizontal table. When $P$ is a distance $x$ metres from a fixed point $O$ on the line, it experiences a force of magnitude $\frac { 16 } { 5 x ^ { 2 } } \mathrm {~N}$ away from $O$ in the direction $O P$. Initially $P$ is at a point 2 m from $O$ and is moving towards $O$ with speed $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the distance of $P$ from $O$ when $P$ first comes to rest.

Edexcel M3 2008 January Q3

3. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{39c2d25a-a39b-4eb9-a17b-6e741ab5ae98-04_519_709_315_603}

\end{figure} A uniform solid $S$ is formed by taking a uniform solid right circular cone, of base radius $2 r$ and height $2 h$, and removing the cone, with base radius $r$ and height $h$, which has the same vertex as the original cone, as shown in Figure 1.

Show that the distance of the centre of mass of $S$ from its larger plane face is $\frac { 11 } { 28 } h$. The solid $S$ lies with its larger plane face on a rough table which is inclined at an angle $\theta ^ { \circ }$ to the horizontal. The table is sufficiently rough to prevent $S$ from slipping. Given that $h = 2 r$,
find the greatest value of $\theta$ for which $S$ does not topple.

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Edexcel M3 2004 January Q6

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