*I, Stephen Easley-Walsh, retyped these Department of Education Examination questions under section 53.5 of the Copyright and Related Rights Act, 2000 i.e. “copyright in a work is not infringed by anything done for the purposes of an examination by way of setting questions, communicating questions to the candidates or answering questions”. As such I invite any student to use this page in their studies but to keep in mind I take no responsibility for any errors/typos/omissions/etc contained below and would appreciate any such errors brought to my attention in the comment section. Thank you.*

If a particle is moving in a straight line with a constant acceleration $f$ prove that $v^2 = u^2 + 2fs$.

If the brakes of a tram bring it to rest when travelling along a level track at a speed of $12$ miles per hour in a distance of $10$ yards, find the slope of an incline on which the breaks will keep the tram at rest ($g=32$ ft./sec$^2$).

[55 marks.]

An Attwood machine has a $3$ lb. weight on one side and two $2$ lb. weights on the other side, $1$ foot apart. Motion begins when the lower $2$ lb. weight is $1$ foot from the floor. Find how near the second $2$ lb. weight will approach the floor after the first one has struck. What changes occur in the tension of the cord attached to the $3$ lb. weight during this experiment?

[55 marks.]

Give a definition of “work.”

A weight of $10$ lbs. is pushed up a smooth inclined plane of inclination $45^\circ$ along the line of greatest slope by a force acting at an angle of $30^\circ$ with the plane.

Derive the magnitude of this force and the pressure on the plane from the condition of equilibrium. Show that the work done by the “effort” is equal to the work done against the “load” in pushing it $1$ foot up the plane.

[55 marks.]

Explain the meaning of the terms “smooth,” and “reaction” as used in mechanics.

A plank $8$ feet long weighing $20$ lb. rests on a rough floor and against the smooth edge of a table $4$ feet high, making an angle of $\tan^{-1} \frac{4}{3}$ with the horizontal. Find the coefficient of friction if the plank is just on the point of slipping.

[60 marks.]

Show that the centre of gravity of a triangular lamina coincides with the centre of gravity of three equal particles placed at its vertices.

Hence prove that the centre of gravity of a quadrilateral lamina having a particle whose weight is one-third of the weight of the lamina attached at the intersection of the diagonals coincides with the centre of gravity of four equal particles placed at the corners of the quadrilateral.

[60 marks.]

What is a hodograph? Show that for a projectile under gravity where $AB$ is the initial velocity of projections, the hodograph is the vertical line through $B$. If this vertical meets a line drawn through $A$ making an angle $\alpha$ with the horizontal in $C$, show that $AC$ multiplied by the time of flight represents the range on a plane of inclination $\alpha$. What does $BC$ represent? Prove that the range is a maximum when the triangle $ABC$ has a maximum area?

[60 marks.]

Two perfectly elastic spheres of mass $m_1$ and $m_2$ collide along their lines of centres with velocities $u_1$ and $u_2$. Find their velocities $\bar{u_1}$ and $\bar{u_2}$ with respect to their centre of gravity before collision. Show that the velocity of the centre of gravity is unaltered by the collision, and the effect of the collision is simply to reverse the velocity of each sphere with respect to their common centre of gravity.

[60 marks.]

What connection exists between a simple harmonic motion and a uniform circular motion. Explain how it can be used to find an expression for the period of the former.

A spiral spring $AB$ of natural length $9$ inches, whose length would be doubled by a steady pull of $10$ lb. is hung up at $A$ and has a $4$ lb. weight attached to it and then let go. Find the distance the weight will fall before it comes to rest and the time of a complete oscillation.

[60 marks.]

]]>*I, Stephen Easley-Walsh, retyped these Department of Education Examination questions under section 53.5 of the Copyright and Related Rights Act, 2000 i.e. “copyright in a work is not infringed by anything done for the purposes of an examination by way of setting questions, communicating questions to the candidates or answering questions”. As such I invite any student to use this page in their studies but to keep in mind I take no responsibility for any errors/typos/omissions/etc contained below and would appreciate any such errors brought to my attention in the comment section. Thank you.*

If a particle is moving in a straight line with constant acceleration $f$, prove that $s=ut + \frac{1}{2} ft^2$.

A stone $A$ is projected vertically upwards with velocity $48$ ft. per second and 2 seconds later a stone $B$ is projected vertically upwards from the same place at $144$ ft. per sec. Find the distance from the point of projection of their meeting point.

[44 marks]

What is a “couple”?

Show that a couple has the same moment about any point is the plane and deduce that forces that are represented completely by the sides of a polygon taken the same way round may also be represented by a couple.

[44 marks]

Prove that if particles start at the same instant from a given point and slide down smooth straight paths of different slopes, their positions reached :- (i) when each has attained a given speed lie on a straight line, (ii) after the lapse of a given time lie on a circle.

Find the straight path of quickest descent from a given point to a given circle.

[44 marks]

Find for small oscillations an expression for the period of a simple pendulum in terms of its length.

A faulty seconds-pendulum loses 5 seconds per hour: find the required alteration in its length so that it may keep correct time.

[48 marks]

Given that the distance, from the centre of the circle, of the centre of gravity of a circular arc which subtends an angle $2a$. at the centre of the circle is $\frac{a \sin a}{a}$, where $a$ is the radius, find the centres of gravity of the corresponding (i) seconds, (ii) segment of the circle.

[48 marks]

A mass of $10$ lbs. is supported on a rough plane by a force $P$ applied in a direction making an angle $\theta$ with the plane which is inclined at angles 40 degrees to the horizontal. If the coefficient of friction is equal to $\tan 10^\circ$, express the value of $P$ in terms of $\theta$ and hence find the minimum value of $P$ and the corresponding value of $\theta$.

[48 marks]

Explain what is meant by “coefficient of restitution.” A ball of coefficient of restitution $e$ falls from height $h$ to a horizontal plane. To what height will it rise after the first rebound? If $e=\frac{7}{12}$, find what time will elapse from the moment the ball is dropped from a height of 10 feet till it comes to rest permanently?

[48 marks]

A string one metre long can support a body whose weight is not greater than 10 kilogrammes. A mass of 100 grammes is tied to one end and whirled in a horizontal circle: find the greatest number of revolutions per second that can be given to the mass without breaking the string and calculate the kinetic energy of the mass when moving at the greatest possible speed.

[48 marks]

]]>*I, Stephen Easley-Walsh, retyped these Department of Education Examination questions under section 53.5 of the Copyright and Related Rights Act, 2000 i.e. “copyright in a work is not infringed by anything done for the purposes of an examination by way of setting questions, communicating questions to the candidates or answering questions”. As such I invite any student to use this page in their studies but to keep in mind I take no responsibility for any errors/typos/omissions/etc contained below and would appreciate any such errors brought to my attention in the comment section. Thank you.*

A ship $A$ sailing due N. at 15 miles per hour is at a certain instant 41 miles due E. of another ship B which is sailing E. at 12 miles per hour. When will the ships be nearest together?

Explain the variations of the forces between the floor of a lift and the feet of a man standing on it during the upward and downward journeys from rest to rest.

A bullet weighing $1$ oz. is fired horizontally into a block of wood weighing 12 lbs. suspended so that the wood and the bullet embedded in it swing without rotation to a height of 2 ft. 6 in. Find the velocity of the bullet on entering the wood.

What is meant by Simple Harmonic Motion?

Show that the bob of a simple pendulum moves with Simple Harmonic Motion when the angle of swing is small.

Find an expression for the periodic time of the pendulum.

Show that the area under a force distance diagram represents work done.

The force exerted by a spring is proportional to the extension of the spring and a forces of 5 lbs. wt. produces an extension of 2 ins. Show graphically the relation between the tension and the extension and deuce the work done in extending the spring through 10 inches.

Show that the work done on such a spring when stretched $a$ feet is $\frac{Pa}{2}$ ft. lbs., where $P$ lbs. wt. is the tension for extension $a$ feet.

A body of $W$ lbs. can just be maintained at rest on a rough inclined plane by a force $P$ lbs. wt. acting along the plane, of by a force $Q$ lbs. wt. acting horizontally.

Show that $\frac{\sec^2 \varphi}{P^2} = \frac{1}{Q^2} + \frac{1}{W^2}$ where $\varphi$ is the angle of friction.

Find the acceleration of a particle which moves with uniform speed in a circle.

A trin is to travel round a curve of radius $r$ ft. If $a$ ft. is the distance between the rail, find the height $b$ ft. to which one rail should be raised above the level of the other so as to eliminate side pressure on the rails for trains travelling at $v$ ft. per sec.

If a train of $W$ tons weight moves on such a track with speed $nv$ feet per sec. prove that there is now a side pressure on rails of approximately $W \frac{b}{a} \left(n^2-1\right)$ tons wt.

A shot at the instant of projection breaks into two parts of masses $m_1$ and $m_2$ lbs., the first part starting with velocity $v_1$ at an angle $\theta_1$ to the horizontal and the second with veloicty $v_2$ at an angle $\theta_2$ to the horizontal. Show that the centre of gravity moves at if the whole shot started with velocity $v$ at an angle $\theta$ to the horizontal; the find $v$ and $\theta$ in terms of $m_1$, $m_2$, $v_1$, $v_2$, $\theta_1$, $\theta_2$.

]]>**Department of Education** Examination questions under section 53.5 of the Copyright and Related Rights Act, 2000 i.e. “copyright in a work is not infringed by anything done for the purposes of an examination by way of setting questions, communicating questions to the candidates or answering questions”. As such I invite any student to use this page in their studies but to keep in mind I take no responsibility for any errors/typos/omissions/etc contained below and would appreciate any such errors brought to my attention in the comment section. Thank you.

State the conditions of equilibrium of three forces acting on an extended body in one plane.

$C$ is a point vertically above $A$, the pivot of light rod $AB$, and $AC=AB$. $B$ is attached to $C$ to $C$ by a cord and a mass $m$ is attached to $B$. Show that the thrust on the rod is Independent of the length of the cord and find the tension of the cord when $AB$ makes an angle $\theta$ with the horizontal.

State the laws of friction.

A ladder with its centre of gravity at its mid-point rests with one end of the ground and the other against a vertical wall. Show that the greatest inclination to the wall consistent with equilibrium is $\tan^{-1} \frac{2\mu}{1-\mu^2}$ where $\mu$ is the coefficient of friction both with ground and wall.

A body revolves with initial angular velocity $\omega_0$ and uniform angular acceleration $\alpha$: write down equations giving the angular velocity and the angle described in time $t$ and derive and equation not involving $t$.

A wheel is making $n$ revolutions per minute and $t$ seconds later it is found to be making $n’$ revolutions per minute: what is its acceleration (supposed uniform)? How many revolutions has the wheel made in the interval.

Th resistance to a train weighing $W$ tons and travelling at $v$ miles per hour is $R$ lbs. wt. per tone: find the rate of working of the engine.

The train consumes $w$ tons of coal per hour and the burning of $1$ lb. of coal produces $s$ ft.-lbs. of energy: what proportion of the energy is usefully employed by the engine?

Evaluate when $W=100$, $v=60$, $R=10$, $w=\frac{1}{2}$, $s=10^7$.

The equation of the path of a projectile referred to horizontal and vertical axes is $y=x-\frac{x^2}{64}$; find the angle at which it was projected and the initial velocity. Find also the direction of motion after $t$ seconds. (Note.-$g$ may be taken as $32$).

A number of unequal particles are distributed in a straight line: find a formula for the position of the centre of gravity. Two masses $m_1$ and $m_2$ are attached to the ends of a light string passing over a smooth peg: show that the acceleration of the centre of gravity is $\left(\frac{m_1-m_2}{m_1+m_2}\right)^2g$.

A mass $m$ hangs from a light spiral spring. Show that, if $m$ is pulled down slightly and released, it will move with simple harmonic motion. Find the greatest velocity and the periodic time of $m$, showing clearly on what they depend.

In a pulley system a weight $W$ is raised with uniform acceleration by means of a load $Q$: show that the ratio of the accelerations of $P$ and $W$ is equal to the velocity ratio of the machine, friction and the weight of the pulleys being neglected.

If in any system of pulleys there is equilibrium when a weight $Q$ is supported by a load $P$, show that if $P$ be increased to $Q$, $W$ will ascend with acceleration $\frac{gP\left(Q-P\right)}{P^2+QW}$.

]]>Happy studying

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**Department of Education** Examination questions under section 53.5 of the Copyright and Related Rights Act, 2000 i.e. “copyright in a work is not infringed by anything done for the purposes of an examination by way of setting questions, communicating questions to the candidates or answering questions”. As such I invite any student to use this page in their studies but to keep in mind I take no responsibility for any errors/typos/omissions/etc contained below and would appreciate any such errors brought to my attention in the comment section. Thank you.

What is meant by ‘work’, ‘power’, ‘one-horse power’?

A locomotive of $w$ tons weight draws a train of $W$ tons at $v$ miles per hour, the resistance to motion being $R$ lbs. per ton. At what horse-power is it working when moving

(a) on the level,

(b) up an incline of $1$ in $n$,

(c) down an incline of $1$ in $n$?

Evaluate when $w=8$, $W=142$, $v=40$, $R=10$, $n=250$.

A body moves in a plane, its position referred to rectangular axes in the plane after $t$ seconds being given by $$x=a+bt \text{, } \; y=c+dt+ft^2$$.

Find its velocity and acceleration in magnitude and direction at any time.

A body is projected from the top of a cliff with a velocity of 45 feet per second, in a direction making $35^\circ$ upward with horizontal, and outwards towards the sea, which lies 150 feet below. Where will the body strike the sea?

Show that the acceleration of a point in its path is the velocity of the corresponding point on the hodograph.

Find the magnitude and direction of the acceleration of a particle moving in a circle with uniform speed, and derive the period of a conical pendulum of length $l$ inclined at an angle $\alpha$ to the vertical.

What are the equations that hold for uniformly accelerated linear motion? Prove the corresponding equations for uniformly accelerated angular motion.

A flywheel, whose mass may be supposed concentrated in the rim of mean radius $r$ feet, possesses $K$ ft.-lb units of energy when its speed is $N$ revolutions per minute. What is its weight? With how much energy does it part in slowing down to $n$ revolutions per minute? If during the interval of slowing down it has made $R$ revolutions, what was its average angular retardation?

Work out your answers for the case

$R=10$, $r=3$, $K=120,000$, $N=120$, $n=117$.

A motor car of weight $W$ lbs and axles of length $2a$ has its centre of gravity at a height $h$ feet above the ground. Show that when the car turns a corner in a circular arc of radius $r$ feet, with speed $v$ feet per second, the reactions of the outer wheels is altered by an amount $\frac{Wv^2h}{2gra}$ lbs, if the outer wheels carry half the weight of the car when moving in a straight path. What is the greatest speed at which the car could so turn, so that the inner wheels should not lift from the ground?

What is meant by velocity ratio, mechanical advantage, and efficiency as applied to machines?

A block and tackle has three pulleys on each block. When a rope passes over a pulley, the tension on one side is $.85$ times that on the other. If $P$ is the effort, find the tension in the part of the rope which is fastened to the upper block

(1) when the load is slowly ascending,

(2) when the load is slowly descending.

If the load raised is $W$, and the weight of the lower block is $w$, measured in the same units, prove that the efficiency is approximately $=\frac{73W}{W+w}$.

What is simple harmonic motion?

A particle is fastened to the mid-point $P$ of a stretched elastic string, fixed at its ends to two points $A$ and $B$ on a smooth horizontal plane. If $P$ is pulled a small distance $x$ at right angles to $AB$, prove $AP$ now equals $l\left(1 + \frac{x^2}{2l^2}\right)$ nearly, where $AB=2l$. Show that the forces tending to restore the particle to its original position is $\frac{2Tx}{l}$, approximately, where $T$ is the original tension of the string. Find the time of a small oscillation.

Three particles are in motion in a plane. Their masses and positions at time $t$ seconds, referred to rectangular axes in the plane, are given in the following table :=

Mass in lbs. | $x$ feet. | $y$ feet. |
---|---|---|

3 | $3+4t+2t^2$ | $7+4t+9t^2$ |

4 | $1+2t+4t^2$ | $3+3t+4t^2$ |

5 | $5+4t-2t^2$ | $6-3t+t^2$ |

Find expressions for the position, velocity, and acceleration of their centre of gravity at any time.

Verify, for components parallel to the $x-$axis, that the system of particles moves in that direction as if the whole mass were concentrated at the centre of gravity, and the $x-$components of the forces acted there.

Define an ‘impulse’, and show how it is measured.

A jet of water is directed from a circular nozzle $1$ inch in diameter with a velocity which would carry it vertically $100$ feet, so as to strike horizontal a wall at a height $50$ feet above the nozzle: find approximately the pressure on the wall, assuming that the water rebounds with half the velocity of impact.

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