Force at pivot/axis - Moments of inertia

OCR M4 2004 January Q6

13 marks Standard +0.3

6 A rigid body consists of a uniform rod $A B$, of mass 15 kg and length 2.8 m , with a particle of mass 5 kg attached at $B$. The body rotates without resistance in a vertical plane about a fixed horizontal axis through $A$.

Find the distance of the centre of mass of the body from $A$.
Find the moment of inertia of the body about the axis.
\includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-3_475_682_680_719} At one instant, $A B$ makes an acute angle $\theta$ with the downward vertical, the angular speed of the body is $1.2 \mathrm { rad } \mathrm { s } ^ { - 1 }$ and the angular acceleration of the body is $3.5 \mathrm { rad } \mathrm { s } ^ { - 2 }$ (see diagram).
Show that $\sin \theta = 0.8$.
Find the components, parallel and perpendicular to $B A$, of the force acting on the body at $A$.
[0pt] [Question 7 is printed overleaf.]
\includegraphics[max width=\textwidth, alt={}, center]{4cac1898-8251-4cda-bbbc-c2c30fde5a6e-4_949_1112_281_550} A small bead $B$, of mass $m$, slides on a smooth circular hoop of radius $a$ and centre $O$ which is fixed in a vertical plane. A light elastic string has natural length $2 a$ and modulus of elasticity $m g$; one end is attached to $B$, and the other end is attached to a light ring $R$ which slides along a smooth horizontal wire. The wire is in the same vertical plane as the hoop, and at a distance $2 a$ above $O$. The elastic string $B R$ is always vertical, and $O B$ makes an angle $\theta$ with the downward vertical (see diagram).
Show that $\theta = 0$ is a position of stable equilibrium.
Find the approximate period of small oscillations about the equilibrium position $\theta = 0$.

OCR M4 2003 June Q6

13 marks Standard +0.3

6
\includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-3_468_550_1201_824} A wheel consists of a uniform circular disc, with centre $O$, mass 0.08 kg and radius 0.35 m , with a particle $P$ of mass 0.24 kg attached to a point on the circumference. The wheel is rotating without resistance in a vertical plane about a fixed horizontal axis through $O$ (see diagram).

Find the moment of inertia of the wheel about the axis.
Find the distance of the centre of mass of the wheel from the axis. At an instant when $O P$ is horizontal and the angular speed of the wheel is $5 \mathrm { rad } \mathrm { s } ^ { - 1 }$, find
the angular acceleration of the wheel,
the magnitude of the force acting on the wheel at $O$.

OCR M4 2010 June Q7

16 marks Challenging +1.8

7
\includegraphics[max width=\textwidth, alt={}, center]{ea62d6d9-ac2f-44e7-8bfb-ae9aeea7109b-4_524_732_258_705} The diagram shows a uniform rectangular lamina $A B C D$ with $A B = 6 a , A D = 8 a$ and centre $G$. The mass of the lamina is $m$. The lamina rotates freely in a vertical plane about a fixed horizontal axis passing through $A$ and perpendicular to the lamina.

Find the moment of inertia of the lamina about this axis. The lamina is released from rest with $A D$ horizontal and $B C$ below $A D$.
For an instant during the subsequent motion when $A D$ is vertical, show that the angular speed of the lamina is $\sqrt { \frac { 3 g } { 50 a } }$ and find its angular acceleration. At an instant when $A D$ is vertical, the force acting on the lamina at $A$ has magnitude $F$.
By finding components parallel and perpendicular to $G A$, or otherwise, show that $F = \frac { \sqrt { 493 } } { 20 } \mathrm { mg }$.
[0pt] [8]

OCR M4 2011 June Q7

16 marks Challenging +1.8

7
\includegraphics[max width=\textwidth, alt={}, center]{337dd1f9-a691-4e99-9aa7-7a93d8bb13be-3_479_1225_1484_461} A uniform rectangular block of mass $m$ and cross-section $A B C D$ has $A B = C D = 6 a$ and $A D = B C = 2 a$. The point $X$ is on $A B$ such that $A X = a$ and $G$ is the centre of $A B C D$. The block is placed with $A B$ perpendicular to the straight edge of a rough horizontal table. $A X$ is in contact with the table and $X B$ overhangs the edge (see diagram). The block is released from rest in this position, and it rotates without slipping about a horizontal axis through $X$.

Find the moment of inertia of the block about the axis of rotation. For the instant when $X G$ is horizontal,
show that the angular acceleration of the block is $\frac { 3 \sqrt { 5 } g } { 25 a }$,
find the angular speed of the block,
show that the force exerted by the table on the block has magnitude $\frac { 2 \sqrt { 70 } } { 25 } m g$.

OCR M4 2013 June Q7

14 marks Challenging +1.8

7
\includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-5_584_686_264_678}
$A B C D$ is a uniform rectangular lamina with mass $m$ and sides $A B = 6 a$ and $A D = 8 a$. The lamina rotates freely in a vertical plane about a fixed horizontal axis passing through $A$, and it is released from rest in the position with $D$ vertically above $A$. When the diagonal $A C$ makes an angle $\theta$ below the horizontal, the force acting on the lamina at $A$ has components $R$ parallel to $C A$ and $S$ perpendicular to $C A$ (see diagram).

Find the moment of inertia of the lamina about the axis through $A$, in terms of $m$ and $a$.
Show that the angular speed of the lamina is $\sqrt { \frac { 3 g ( 4 + 5 \sin \theta ) } { 50 a } }$.
Find the angular acceleration of the lamina, in terms of $a , g$ and $\theta$.
Find $R$ and $S$, in terms of $m , g$ and $\theta$.

OCR M4 2014 June Q4

13 marks Challenging +1.8

4 A uniform square lamina has mass $m$ and sides of length $2 a$.

Calculate the moment of inertia of the lamina about an axis through one of its corners perpendicular to its plane.
\includegraphics[max width=\textwidth, alt={}, center]{639c658e-0aca-4161-9e77-0f4c494b0b55-3_693_640_434_715} The uniform square lamina has centre $C$ and is free to rotate in a vertical plane about a fixed horizontal axis passing through one of its corners $A$. The lamina is initially held such that $A C$ is vertical with $C$ above $A$. The lamina is slightly disturbed from rest from this initial position. When $A C$ makes an angle $\theta$ with the upward vertical, the force exerted by the axis on the lamina has components $X$ parallel to $A C$ and $Y$ perpendicular to $A C$ (see diagram).
Show that the angular speed, $\omega$, of the lamina satisfies $a \omega ^ { 2 } = \frac { 3 } { 4 } g \sqrt { 2 } ( 1 - \cos \theta )$.
Find $X$ and $Y$ in terms of $m , g$ and $\theta$. \section*{Question 5 begins on page 4.}
\includegraphics[max width=\textwidth, alt={}]{639c658e-0aca-4161-9e77-0f4c494b0b55-4_767_337_248_863}
A pendulum consists of a uniform rod $A B$ of length $4 a$ and mass $4 m$ and a spherical shell of radius $a$, mass $m$ and centre $C$. The end $B$ of the rod is rigidly attached to a point on the surface of the shell in such a way that $A B C$ is a straight line. The pendulum is initially at rest with $B$ vertically below $A$ and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through $A$ (see diagram).
Show that the moment of inertia of the pendulum about the axis of rotation is $47 m a ^ { 2 }$. A particle of mass $m$ is moving horizontally in the plane in which the pendulum is free to rotate. The particle has speed $\sqrt { k g a }$, where $k$ is a positive constant, and strikes the rod at a distance $3 a$ from $A$. In the subsequent motion the particle adheres to the rod and the combined rigid body $P$ starts to rotate.
Show that the initial angular speed of $P$ is $\frac { 3 } { 56 } \sqrt { \frac { k g } { a } }$.
For the case $k = 4$, find the angle that $P$ has turned through when $P$ first comes to instantaneous rest.
Find the least value of $k$ such that the rod reaches the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{639c658e-0aca-4161-9e77-0f4c494b0b55-5_437_903_269_573} A uniform rod $A B$ has mass $m$ and length $2 a$. The rod can rotate in a vertical plane about a smooth fixed horizontal axis passing through $A$. One end of a light elastic string of natural length $a$ and modulus of elasticity $\sqrt { 3 } m g$ is attached to $A$. The string passes over a small smooth fixed pulley $C$, where $A C$ is horizontal and $A C = a$. The other end of the string is attached to the rod at its mid-point $D$. The rod makes an angle $\theta$ below the horizontal (see diagram).
Taking $A$ as the reference level for gravitational potential energy, show that the total potential energy $V$ of the system is given by $$V = m g a ( \sqrt { 3 } - \sin \theta - \sqrt { 3 } \cos \theta ) .$$
Show that $\theta = \frac { 1 } { 6 } \pi$ is a position of stable equilibrium for the system. The system is making small oscillations about the equilibrium position.
By differentiating the energy equation with respect to time, show that $$\frac { 4 } { 3 } a \ddot { \theta } = g ( \cos \theta - \sqrt { 3 } \sin \theta ) .$$
Using the substitution $\theta = \phi + \frac { 1 } { 6 } \pi$, show that the motion is approximately simple harmonic, and find the approximate period of the oscillations. \section*{END OF QUESTION PAPER}

Edexcel M5 Q6

17 marks Challenging +1.8

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9e3d76a7-b997-4e46-a5dd-aeeaa5abfa4e-02_211_611_388_1845}

\end{figure} A rough uniform rod, of mass $m$ and length $4 a$, is rod is held on a rough horizontal table. The rod is perpendicular to the edge of the table and a length $3 a$ projects horizontally over the edge, as shown in Fig. 1.

Show that the moment of inertia of the rod about the edge of the table is $\frac { 7 } { 3 } m a ^ { 2 }$. The rod is released from rest and rotates about the edge of the table. When the rod has turned through an angle $\theta$, its angular speed is $\dot { \theta }$. Assuming that the rod has not started to slip,
show that $\dot { \theta } ^ { 2 } = \frac { 6 g \sin \theta } { 7 a }$,
find the angular acceleration of the rod,
find the normal reaction of the table on the rod. The coefficient of friction between the rod and the edge of the table is $\mu$.
Show that the rod starts to slip when $\tan \theta = \frac { 4 } { 13 } \mu$

Edexcel M5 2006 January Q8

17 marks Challenging +1.2

8. Four uniform rods, each of mass $m$ and length $2 a$, are joined together at their ends to form a plane rigid square framework $A B C D$ of side $2 a$. The framework is free to rotate in a vertical plane about a fixed smooth horizontal axis through $A$. The axis is perpendicular to the plane of the framework.

Show that the moment of inertia of the framework about the axis is $\frac { 40 m a ^ { 2 } } { 3 }$. The framework is slightly disturbed from rest when $C$ is vertically above $A$. Find
the angular acceleration of the framework when $A C$ is horizontal,
the angular speed of the framework when $A C$ is horizontal,
the magnitude of the force acting on the framework at $A$ when $A C$ is horizontal.

Edexcel M5 2004 June Q3

9 marks Standard +0.8

3. A uniform lamina of mass $m$ is in the shape of a rectangle $P Q R S$, where $P Q = 8 a$ and $Q R = 6 a$.

Find the moment of inertia of the lamina about the edge $P Q$. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{26fef791-e6fb-45a8-89e9-16c4b4a1a4e4-3_336_772_528_642}
\end{figure} The flap on a letterbox is modelled as such a lamina. The flap is free to rotate about an axis along its horizontal edge $P Q$, as shown in Fig. 1. The flap is released from rest in a horizontal position. It then swings down into a vertical position.
Show that the angular speed of the flap as it reaches the vertical position is $\sqrt { \left( \frac { g } { 2 a } \right) }$.
Find the magnitude of the vertical component of the resultant force of the axis $P Q$ on the flap, as it reaches the vertical position.

Edexcel M5 2009 June Q6

19 marks Challenging +1.8

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 2013 June Q6

15 marks Challenging +1.2

6. A uniform circular disc, of radius $r$ and mass $m$, is free to rotate in a vertical plane about a fixed smooth horizontal axis $L$ which is perpendicular to the plane of the disc and passes through a point which is $\frac { 1 } { 4 } r$ from the centre of the disc. The disc is held at rest with its centre vertically above the axis. The disc is then slightly disturbed from its rest position. You may assume without proof that the moment of inertia of the disc about $L$ is $\frac { 9 m r ^ { 2 } } { 16 }$.

Show that the angular speed of the disc when it has turned through $\frac { \pi } { 2 }$ is $\sqrt { } \left( \frac { 8 g } { 9 r } \right)$.
Find the magnitude of the force exerted on the disc by the axis when the disc has turned through $\frac { \pi } { 2 }$.

Edexcel M5 2014 June Q5

15 marks Challenging +1.2

A uniform rod $A B$, of mass $m$ and length $2 a$, is free to rotate in a vertical plane about a fixed smooth horizontal axis $L$. The axis $L$ is perpendicular to the rod and passes through the point $P$ of the rod, where $A P = \frac { 2 } { 3 } a$.
1. Find the moment of inertia of the rod about $L$.
The rod is held at rest with $B$ vertically above $P$ and is slightly displaced.
Find the angular speed of the rod when $P B$ makes an angle $\theta$ with the upward vertical.
Find the magnitude of the angular acceleration of the rod when $P B$ makes an angle $\theta$ with the upward vertical.
Find, in terms of $g$ and $a$ only, the angular speed of the rod when the force acting on the rod at $P$ is perpendicular to the rod.

Edexcel M5 2018 June Q6

15 marks Challenging +1.2

6. Three equal uniform rods, each of mass $m$ and length $2 a$, form the sides of a rigid equilateral triangular frame $A B C$. The frame 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 frame.

Show that the moment of inertia of the frame about $L$ is $6 m a ^ { 2 }$. The frame is held with $A B$ horizontal and $C$ below $A B$, and released from rest. Given that the centre of mass of the frame is two thirds of the way along a median from a vertex,
find the magnitude of the force exerted by the axis on the frame at $A$ at the instant when the frame is released.

OCR M4 2008 June Q6

15 marks Challenging +1.3

Show that the moment of inertia of the lamina about the axis through $X$ is $\frac { 4 } { 3 } m a ^ { 2 }$.
At an instant when $\cos \theta = \frac { 3 } { 5 }$, show that $\omega ^ { 2 } = \frac { 6 g } { 5 a }$.
At an instant when $\cos \theta = \frac { 3 } { 5 }$, show that $R = 0$, and given also that $\sin \theta = \frac { 4 } { 5 }$ find $S$ in terms of $m$ and $g$.

Edexcel M5 2002 June Q6

17 marks Challenging +1.8

Show that the moment of inertia of the rod about the edge of the table is $\frac { 7 } { 3 } m a ^ { 2 }$. The rod is released from rest and rotates about the edge of the table. When the rod has turned through an angle $\theta$, its angular speed is $\dot { \theta }$. Assuming that the rod has not started to slip,
show that $\dot { \theta } ^ { 2 } = \frac { 6 g \sin \theta } { 7 a }$,
find the angular acceleration of the rod,
find the normal reaction of the table on the rod. The coefficient of friction between the rod and the edge of the table is $\mu$.
Show that the rod starts to slip when $\tan \theta = \frac { 4 } { 13 } \mu$
(6)

Edexcel M5 Q6

12 marks Challenging +1.8

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 $u = 7 v$.