Question 6 - A-Level Maths

Edexcel M5 — Question 6

Exam Board	Edexcel
Module	M5 (Mechanics 5)
Topic	Circular Motion 2

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. Turn over
1. 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.
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{19c5a621-c175-4d58-9002-4bcdefd02b71-16_504_586_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 $a$.
1. 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 $B C = 8 a$.
2. 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.
1. 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 $\mathbf { G }$.
1. 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 $v$.
2. 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 $3 a$.
1. 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 $O$.
2. Find the angular speed of the disc immediately after the impact.
3. Find the magnitude of the force exerted on the disc by the axis immediately after the impact.
Turn over
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1. 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.
2. 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 $t$.
1. 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.
2. 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 $( - 3 \mathbf { i } + 2 \mathbf { k } ) \mathrm { m }$.
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,
$\mathbf { R }$,
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 $( 2 \mathbf { i } - \mathbf { k } ) \mathrm { m }$ 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.
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 $2 m g \left| \cos \left( \frac { 1 } { 2 } \theta \right) \right|$.
6. A uniform $\operatorname { rod } A B$ of mass $4 m$ is free to rotate in a vertical plane about a fixed smooth horizontal axis, $L$, through $A$. The rod is hanging vertically at rest when it is struck at its end $B$ by a particle of mass $m$. The particle is moving with speed $u$, in a direction which is horizontal and perpendicular to $L$, and after striking the rod it rebounds in the opposite direction with speed $v$. The coefficient of restitution between the particle and the rod is 1 . Show that $u = 7 v$.

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