Questions M5 - A-Level Maths

Show that the moment of inertia of $S$ about an axis through $O$ and perpendicular to its plane is $\frac { 5 } { 2 } M a ^ { 2 }$.
(3) The lamina is free to rotate about a fixed smooth horizontal axis $L$. The axis $L$ lies in the plane of $S$ and is a tangent to its outer circumference, as shown in Figure 2.
Show that the moment of inertia of $S$ about $L$ is $\frac { 21 } { 4 } M a ^ { 2 }$.
(4)
$S$ is displaced through a small angle from its position of stable equilibrium and, at time $t = 0$, it is released from rest. Using the equation of motion of $S$, with a suitable approximation,
find the time when $S$ first passes through its position of stable equilibrium.
(6)

Edexcel M5 2007 June Q7

7. A motor boat of mass $M$ is moving in a straight line, with its engine switched off, across a stretch of still water. The boat is moving with speed $U$ when, at time $t = 0$, it develops a leak. The water comes in at a constant rate so that at time $t$, the mass of water in the boat is $\lambda t$. At time $t$ the speed of the boat is $v$ and it experiences a total resistance to motion of magnitude $2 \lambda v$.

Show that $( M + \lambda t ) \frac { \mathrm { d } v } { \mathrm {~d} t } + 3 \lambda v = 0$.
(6)
Show that the time taken for the speed of the boat to reduce to $\frac { 1 } { 2 } U$ is $\frac { M } { \lambda } \left( 2 ^ { \frac { 1 } { 3 } } - 1 \right)$.
(6) The boat sinks when the mass of water inside the boat is $M$.
Show that the boat does not sink before the speed of the boat is $\frac { 1 } { 2 } U$.

Edexcel M5 2007 June Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5180a4e0-dafe-4595-a517-e3501f7aed40-5_533_584_292_703} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A uniform rod $A B$ has mass $3 m$ and length $2 a$. It is free to rotate in a vertical plane about a smooth fixed horizontal axis through the point $X$ on the rod, where $A X = \frac { 1 } { 2 } a$. A particle of mass $m$ is attached to the rod at $B$. At time $t = 0$, the rod is vertical, with $B$ above $A$, and is given an initial angular speed $\sqrt { \frac { g } { a } }$. When the rod makes an angle $\theta$ with the upward vertical, the angular speed of the rod is $\omega$, as shown in Figure 3.

By using the principle of the conservation of energy, show that $$\omega ^ { 2 } = \frac { g } { 2 a } ( 5 - 3 \cos \theta )$$
Find the angular acceleration of the rod when it makes an angle $\theta$ with the upward vertical. When $\theta = \phi$, the resultant force of the axis on the rod is in a direction perpendicular to the rod.
Find $\cos \phi$.

Edexcel M5 2008 June Q1

\hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors.]

A small bead of mass 0.5 kg is threaded on a smooth horizontal wire. The bead is initially at rest at the point with position vector $( \mathbf { i } - 6 \mathbf { j } ) \mathrm { m }$. A constant horizontal force $\mathbf { P } \mathrm { N }$ then acts on the bead causing it to move along the wire. The bead passes through the point with position vector ( $7 \mathbf { i } - 14 \mathbf { j }$ ) m with speed $2 \sqrt { 7 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Given that $\mathbf { P }$ is parallel to ( $6 \mathbf { i } + \mathbf { j }$ ), find $\mathbf { P }$.
(6)

Edexcel M5 2008 June Q2

2. The velocity $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$ of a particle $P$ at time $t$ seconds satisfies the vector differential equation $$\frac { \mathrm { d } \mathbf { v } } { \mathrm {~d} t } + 4 \mathbf { v } = \mathbf { 0 }$$ The position vector of $P$ at time $t$ seconds is $\mathbf { r }$ metres.
Given that at $t = 0 , \mathbf { r } = ( \mathbf { i } - \mathbf { j } )$ and $\mathbf { v } = ( - 8 \mathbf { i } + 4 \mathbf { j } )$, find $\mathbf { r }$ at time $t$ seconds.

Edexcel M5 2008 June Q3

3. A system of forces consists of two forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ acting on a rigid body.
$\mathbf { F } _ { 1 } = ( - 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { N }$ and acts at the point with position vector $\mathbf { r } _ { 1 } = ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }$.
$\mathbf { F } _ { 2 } = ( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }$ and acts at the point with position vector $\mathbf { r } _ { 2 } = ( 4 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) \mathrm { m }$.
Given that the system is equivalent to a single force $\mathbf { R } \mathrm { N }$, acting at the point with position vector $( 5 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { m }$, together with a couple $\mathbf { G N m }$, find

$\mathbf { R }$,
the magnitude of $\mathbf { G }$.
(9)

Edexcel M5 2008 June Q4

4. At time $t = 0$ a rocket is launched from rest vertically upwards. The rocket propels itself upwards by expelling burnt fuel vertically downwards with constant speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$ relative to the rocket. The initial mass of the rocket is $M _ { 0 } \mathrm {~kg}$. At time $t$ seconds, where $t < 2$, its mass is $M _ { 0 } \left( 1 - \frac { 1 } { 2 } t \right) \mathrm { kg }$, and it is moving upwards with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { U } { ( 2 - t ) } - 9.8 .$$
Hence show that $U > 19.6$.
Find, in terms of $U$, the speed of the rocket one second after its launch.

Edexcel M5 2008 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{dadd5bac-b547-42dd-838e-60a786555472-3_303_1301_1089_390} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A pendulum $P$ is modelled as a uniform rod $A B$, of length $9 a$ and mass $m$, rigidly fixed to a uniform circular disc of radius $a$ and mass $2 m$. The end $B$ of the rod is attached to the centre of the disc, and the rod lies in the plane of the disc, as shown in Figure 1. The pendulum is free to rotate in a vertical plane about a fixed smooth horizontal axis $L$ which passes through the end $A$ and is perpendicular to the plane of the disc.

Show that the moment of inertia of $P$ about $L$ is $190 m a ^ { 2 }$. The pendulum makes small oscillations about $L$.
By writing down an equation of motion for $P$, find the approximate period of these small oscillations.

Edexcel M5 2008 June Q6

6. A uniform solid right circular cylinder has mass $M$, height $h$ and radius $a$. Find, using integration, its moment of inertia about a diameter of one of its circular ends.
[0pt] [You may assume without proof that the moment of inertia of a uniform circular disc, of mass $m$ and radius $a$, about a diameter is $\frac { 1 } { 4 } m a ^ { 2 }$.]

Edexcel M5 2008 June Q7

7. A uniform square lamina $A B C D$, of mass $2 m$ and side $3 a \sqrt { 2 }$, is free to rotate in a vertical plane about a fixed smooth horizontal axis $L$ which passes through $A$ and is perpendicular to the plane of the lamina. The moment of inertia of the lamina about $L$ is $24 m a ^ { 2 }$. The lamina is at rest with $C$ vertically above $A$. At time $t = 0$ the lamina is slightly displaced. At time $t$ the lamina has rotated through an angle $\theta$.

Show that $$2 a \left( \frac { d \theta } { d t } \right) ^ { 2 } = g ( 1 - \cos \theta )$$
Show that, at time $t$, the magnitude of the component of the force acting on the lamina at $A$, in a direction perpendicular to $A C$, is $\frac { 1 } { 2 } m g \sin \theta$. When the lamina reaches the position with $C$ vertically below $A$, it receives an impulse which acts at $C$, in the plane of the lamina and in a direction which is perpendicular to the line $A C$. As a result of this impulse the lamina is brought immediately to rest.
Find the magnitude of the impulse.

Edexcel M5 2009 June Q1

At time $t = 0$, a particle $P$ of mass 3 kg is at rest at the point $A$ with position vector $( \mathbf { j } - 3 \mathbf { k } ) \mathrm { m }$. Two constant forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ then act on the particle $P$ and it passes through the point $B$ with position vector $( 8 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k } ) \mathrm { m }$.

Given that $\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } ) \mathrm { N }$ and $\mathbf { F } _ { 2 } = ( 8 \mathbf { i } - 4 \mathbf { j } + 7 \mathbf { k } ) \mathrm { N }$ and that $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ are the only two forces acting on $P$, find the velocity of $P$ as it passes through $B$, giving your answer as a vector.

Edexcel M5 2009 June Q2

2. At time $t$ seconds, the position vector of a particle $P$ is $\mathbf { r }$ metres, where $\mathbf { r }$ satisfies the vector differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 4 \mathbf { r } = \mathrm { e } ^ { 2 t } \mathbf { j }$$ When $t = 0 , P$ has position vector $( \mathbf { i } + \mathbf { j } ) \mathrm { m }$ and velocity $2 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find an expression for $\mathbf { r }$ in terms of $t$.

Edexcel M5 2009 June Q3

A spaceship is moving in a straight line in deep space and needs to increase its speed. This is done by ejecting fuel backwards from the spaceship at a constant speed $c$ relative to the spaceship. When the speed of the spaceship is $v$, its mass is $m$.
1. Show that, while the spaceship is ejecting fuel,
$$\frac { \mathrm { d } v } { \mathrm {~d} m } = - \frac { c } { m } .$$ The initial mass of the spaceship is $m _ { 0 }$ and at time $t$ the mass of the spaceship is given by $m = m _ { 0 } ( 1 - k t )$, where $k$ is a positive constant.
Find the acceleration of the spaceship at time $t$.

Edexcel M5 2009 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a11940cb-73a8-4f33-bfbc-73841320c1dc-07_515_415_210_758} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform lamina of mass $M$ is in the shape of a right-angled triangle $O A B$. The angle $O A B$ is $90 ^ { \circ } , O A = a$ and $A B = 2 a$, as shown in Figure 1.

Prove, using integration, that the moment of inertia of the lamina $O A B$ about the edge $O A$ is $\frac { 2 } { 3 } M a ^ { 2 }$.
(You may assume without proof that the moment of inertia of a uniform rod of mass $m$ and length $2 l$ about an axis through one end and perpendicular to the rod is $\frac { 4 } { 3 } m l ^ { 2 }$.) The lamina $O A B$ is free to rotate about a fixed smooth horizontal axis along the edge $O A$ and hangs at rest with $B$ vertically below $A$. The lamina is then given a horizontal impulse of magnitude $J$. The impulse is applied to the lamina at the point $B$, in a direction which is perpendicular to the plane of the lamina. Given that the lamina first comes to instantaneous rest after rotating through an angle of $120 ^ { \circ }$,
find an expression for $J$, in terms of $M , a$ and $g$.

Edexcel M5 2009 June Q5

Two forces $\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { 2 } = ( - 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }$ act on a rigid body. The force $\mathbf { F } _ { 1 }$ acts at the point with position vector $\mathbf { r } _ { 1 } = ( 3 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }$ and the force $\mathbf { F } _ { 2 }$ acts at the point with position vector $\mathbf { r } _ { 2 } = ( \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }$. A third force $\mathbf { F } _ { 3 }$ acts on the body such that $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ are in equilibrium.
1. Find the magnitude of $\mathbf { F } _ { 3 }$.
2. Find a vector equation of the line of action of $\mathbf { F } _ { 3 }$.
The force $\mathbf { F } _ { 3 }$ is replaced by a fourth force $\mathbf { F } _ { 4 }$, acting through the origin $O$, such that $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 4 }$ are equivalent to a couple.
Find the magnitude of this couple.

Edexcel M5 2009 June Q6

A pendulum consists of a uniform rod $A B$, of length $4 a$ and mass $2 m$, whose end $A$ is rigidly attached to the centre $O$ of a uniform square lamina $P Q R S$, of mass $4 m$ and side $a$. The $\operatorname { rod } A B$ is perpendicular to the plane of the lamina. The pendulum is free to rotate about a fixed smooth horizontal axis $L$ which passes through $B$. The axis $L$ is perpendicular to $A B$ and parallel to the edge $P Q$ of the square.
1. Show that the moment of inertia of the pendulum about $L$ is $75 m a ^ { 2 }$.
The pendulum is released from rest when $B A$ makes an angle $\alpha$ with the downward vertical through $B$, where $\tan \alpha = \frac { 7 } { 24 }$. When $B A$ makes an angle $\theta$ with the downward vertical through $B$, the magnitude of the component, in the direction $A B$, of the force exerted by the axis $L$ on the pendulum is $X$.
Find an expression for $X$ in terms of $m , g$ and $\theta$. Using the approximation $\theta \approx \sin \theta$,
find an estimate of the time for the pendulum to rotate through an angle $\alpha$ from its initial rest position.

Edexcel M5 2010 June Q1

At time $t = 0$, the position vector of a particle $P$ is $- 3 \mathbf { j } \mathrm {~m}$. At time $t$ seconds, the position vector of $P$ is $\mathbf { r }$ metres and the velocity of $P$ is $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$. Given that

$$\mathbf { v } - 2 \mathbf { r } = 4 \mathrm { e } ^ { t } \mathbf { j }$$ find the time when $P$ passes through the origin.

Edexcel M5 2010 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9fa0d04e-ff7b-46f8-a8ec-44393e383cdf-03_504_584_267_671} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A uniform circular disc has mass $4 m$, centre $O$ and radius $4 a$. The line $P O Q$ is a diameter of the disc. A circular hole of radius $2 a$ is made in the disc with the centre of the hole at the point $R$ on $P Q$ where $Q R = 5 a$, as shown in Figure 1. The resulting lamina is free to rotate about a fixed smooth horizontal axis $L$ which passes through $Q$ and is perpendicular to the plane of the lamina.

Show that the moment of inertia of the lamina about $L$ is $69 m a ^ { 2 }$. The lamina is hanging at rest with $P$ vertically below $Q$ when it is given an angular velocity $\Omega$. Given that the lamina turns through an angle $\frac { 2 \pi } { 3 }$ before it first comes to instantaneous rest,
find $\Omega$ in terms of $g$ and $a$.

Edexcel M5 2010 June Q3

A uniform lamina $A B C$ of mass $m$ is in the shape of an isosceles triangle with $A B = A C = 5 a$ and $B C = 8 a$.
1. Show, using integration, that the moment of inertia of the lamina about an axis through $A$, parallel to $B C$, is $\frac { 9 } { 2 } m a ^ { 2 }$.
The foot of the perpendicular from $A$ to $B C$ is $D$. The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through $D$ and is perpendicular to the plane of the lamina. The lamina is released from rest when $D A$ makes an angle $\alpha$ with the downward vertical. It is given that the moment of inertia of the lamina about an axis through $A$, perpendicular to $B C$ and in the plane of the lamina, is $\frac { 8 } { 3 } m a ^ { 2 }$.
Find the angular acceleration of the lamina when $D A$ makes an angle $\theta$ with the downward vertical. Given that $\alpha$ is small,
find an approximate value for the period of oscillation of the lamina about the vertical.

Edexcel M5 2010 June Q4

Two forces $\mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { N }$ and $\mathbf { F } _ { 2 } = ( 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }$ act on a rigid body.

The force $\mathbf { F } _ { 1 }$ acts through the point with position vector ( $2 \mathbf { i } + \mathbf { k }$ ) m and the force $\mathbf { F } _ { 2 }$ acts through the point with position vector $( \mathbf { j } + 2 \mathbf { k } ) \mathrm { m }$.

If the two forces are equivalent to a single force $\mathbf { R }$, find
1. $\mathbf { R }$,
2. a vector equation of the line of action of $\mathbf { R }$, in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }$.
If the two forces are equivalent to a single force acting through the point with position vector $( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }$ together with a couple of moment $\mathbf { G }$, find the magnitude of $\mathbf { G }$.

Edexcel M5 2010 June Q5

A raindrop falls vertically under gravity through a cloud. In a model of the motion the raindrop is assumed to be spherical at all times and the cloud is assumed to consist of stationary water particles. At time $t = 0$, the raindrop is at rest and has radius $a$. As the raindrop falls, water particles from the cloud condense onto it and the radius of the raindrop is assumed to increase at a constant rate $\lambda$. A time $t$ the speed of the raindrop is $v$.
1. Show that
$$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 3 \lambda v } { ( \lambda t + a ) } = g$$
Find the speed of the raindrop when its radius is $3 a$.

Edexcel M5 2010 June Q6

A uniform circular disc has mass $m$, centre $O$ and radius $2 a$. It is free to rotate about a fixed smooth horizontal axis $L$ which lies in the same plane as the disc and which is tangential to the disc at the point $A$. The disc is hanging at rest in equilibrium with $O$ vertically below $A$ when it is struck at $O$ by a particle of mass $m$. Immediately before the impact the particle is moving perpendicular to the plane of the disc with speed $3 \sqrt { } ( a g )$. The particle adheres to the disc at $O$.
1. Find the angular speed of the disc immediately after the impact.
2. Find the magnitude of the force exerted on the disc by the axis immediately after the impact.

Edexcel M5 2011 June Q1

A particle moves from the point $A$ with position vector $( 3 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }$ to the point $B$ with position vector $( \mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k } ) \mathrm { m }$ under the action of the force $( 2 \mathbf { i } - 3 \mathbf { j } - \mathbf { k } ) \mathrm { N }$. Find the work done by the force.
(4)
A particle $P$ moves in the $x - y$ plane so that its position vector $\mathbf { r }$ metres at time $t$ seconds satisfies the differential equation

$$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } - 4 \mathbf { r } = - 3 \mathrm { e } ^ { t } \mathbf { j }$$ When $t = 0$, the particle is at the origin and is moving with velocity ( $2 \mathbf { i } + \mathbf { j }$ ) $\mathrm { ms } ^ { - 1 }$.
Find $\mathbf { r }$ in terms of $t$.

Edexcel M5 2011 June Q3

A rocket propels itself by its engine ejecting burnt fuel. Initially the rocket has total mass $M$, of which a mass $k M , k < 1$, is fuel. The rocket is at rest when its engine is started. The burnt fuel is ejected with constant speed $c$, relative to the rocket, in a direction opposite to that of the rocket's motion. Assuming that there are no external forces, find the speed of the rocket when all its fuel has been burnt.
Two forces $\mathbf { F } _ { 1 } = ( 3 \mathbf { j } + \mathbf { k } ) \mathrm { N }$ and $\mathbf { F } _ { 2 } = ( 4 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { N }$ act on a rigid body.

The force $\mathbf { F } _ { 1 }$ acts at the point with position vector ( $2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }$ ) m and the force $\mathbf { F } _ { 2 }$ acts at the point with position vector ( $- 3 \mathbf { i } + 2 \mathbf { k }$ ) m.
The two forces are equivalent to a single force $\mathbf { R }$ acting at the point with position vector $( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }$ together with a couple of moment $\mathbf { G }$. Find,

$\mathbf { R }$,
$\mathbf { G }$. A third force $\mathbf { F } _ { 3 }$ is now added to the system. The force $\mathbf { F } _ { 3 }$ acts at the point with position vector ( $2 \mathbf { i } - \mathbf { k }$ ) m and the three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ are equivalent to a couple.
Find the magnitude of the couple.

Edexcel M5 2011 June Q5

5. A uniform rod $P Q$, of mass $m$ and length $2 a$, is made to rotate in a vertical plane with constant angular speed $\sqrt { } \left( \frac { g } { a } \right)$ about a fixed smooth horizontal axis through the end $P$ of the rod. Show that, when the rod is inclined at an angle $\theta$ to the downward vertical, the magnitude of the force exerted on the axis by the rod is $2 m g \left| \cos \left( \frac { 1 } { 2 } \theta \right) \right|$.

Questions M5 (158 questions)

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