# Leaving Certificate Examination 1953 Honours Applied Mathematics

##### Question 1

The ends of a light inextensible string $5$ feet long are attached to two pegs which are $4$ feet apart and in the same horizontal line. A mass of $20$ lb. is suspended from the string at a point which is $2$ feet from one peg and $3$ feet from the other. Find the tension in each part of the string.

##### Question 2

A block weighing $10$ lb. can just rest without slipping on a rough plane when the plane is inclined at an angle of $40^\circ$ to the horizontal. Find

(a) the magnitude of the least force in the horizontal direction,

(b) the magnitude and direction of the least force, that will keep the block from slipping when the plane is inclined at an angle of $50^\circ$ to the horizontal.

##### Question 3

An engine weighing $50$ tons is pulling a train of weight $150$ tons down an incline of $1$ in $80$. The speed of the train is $30$ m.p.h. and it is accelerating at the rate of $\frac{1}{2}$ ft. per sec$^2$. If the frictional resistances to motion are equivalent to $10$ lb. wt. per ton, find the horse-power at which the engine is working.

##### Question 4

A pile driver of mass $12$ cwt. falls freely from rest through a distance of $9$ feet and strikes a pile of mass $4$ cwt. The pile driver and the pile move together as a single body after the impact and the pile is driven $6$ inches into the ground. Find the average resistance of the ground, in tons weight.

##### Question 5

What is meant by the velocity of one body relative to another? Explain with the aid of a diagram how the velocity of $A$ relative to $B$ may be found if the velocities of $A$ and $B$ are known.

To a person on a ship travelling due East at $20$ m.p.h., another ship two miles due South appears to be travelling at $8$ m.p.h. in a direction $30^\circ$ West of North. Find the velocity of the second ship in magnitude and direction.

What is the distance between the ships when they are nearest to each other?

##### Question 6

If $h$ if the greatest height reached by a projectile, prove that

$$h=\frac{u^2\sin^2\alpha}{2h}=\frac{1}{8}gt^2$$

where $u$ is the initial velocity, $\alpha$ the angle of projection, and $t$ the total time of flight.

Two vertical posts, each of height $l$ feet, stand $\frac{1}{2}l$ feet apart on a horizontal plane. A particle is projected from a point on the plane at an angle of $\tan^{-1}3$ with the horizontal, and just clears the top of each post. If $H$ is the greatest height reached by the particle, and v$ its initial velocity, show that

$$H=\frac{9v^2}{20g}=l+\frac{5gl^2}{16v^2}$$

and hence show that $v^2=\frac{5}{2}gl$$.

##### Question 7

A mass of $10$ ounces attached to a fixed point by a light inextensible string of length $2$ feet is describing a horizontal circle at the uniform rate of $60$ revolutions per minute. Find the radius of teh circle in feet and the tension in the string in lb. wt., correct to one place of decimals in each case.

By how much is the tension increased if the speed is increased to $80$ revolutions per minute?

##### Question 8

Define Simple Harmonic Motion.

A particle moves in a straight line so that is displacement, $c$, from a fixed point at any time, $t$, is given by the formula

$$x=a \sin \omega t – b \cos \omega t$$.

Show that the motion is Simple Harmonic, and find an expression for (i) the amplitude, (ii) the maximum velocity.

Find also an expression for the least time taken by the particle to travel a distance $d$ from its mean position.

##### Question 9

A triangular lamina $ABC$ is immersed in a vertical position in water with its vertex $A$ at the surface and its base $BC$ parallel to the surface. The base is $5$ inches in length, and the height of the triangle is $3$ inches. Find the total thrust of the water on $ABC$.

$X$, $Y$ are two points on $AB$, $AC$ respectively such that the straight line $XY$ is horizontal. If the thrust on $AXY$ is one-eight of the thrust on $ABC$, find the depth of $XY$ below the surface.

[One cubic foot of water weight $62.5$ lb.]

**Citation:**

**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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