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### Leaving Certificate Examination 1925 Honours Applied Mathematics

26 Mar 2018
##### Question 1

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

##### Question 2

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?

##### Question 3

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.

##### Question 4

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

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

##### Question 5

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?

##### Question 6

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 $=\frac{73W}{W+w}$.

##### Question 7

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.

##### Question 8

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.

##### Question 9

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.

##### Citation:

State Examinations Commission (2023). State Examination Commission. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru

Malone, D and Murray, H. (2023). Archive of Maths State Exams Papers. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/

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