# Leaving Certificate Examination 1960 Honours Applied Mathematics

##### Question 1

A mass of $21$ gm. is supported at $O$ by two light strings $OA$, $OB$ which are attached to fixed pegs at $A$, $B$, respectively, the straight line $AB$ being horizontal. If $AB=7$ ins., $AO=3$ ins., $BO=5$ ins., find the tensions in the strings.

##### Question 2

Explain the terms “limiting friction”, “coefficient of friction.” A uniform ladder of weight $W$ rests with one end in contact with a rough horizontal plane (coefficient of friction $0.6$) and the other end in contact with rough vertical wall. The ladder makes an angle $\tan^{-1}\frac{1}{3}$ with the horizontal. When a man of weight $6W$ is one-third the way up the ladder, the ladder is on the point of slipping. Find the coefficient of friction between the ladder and the wall.

##### Question 3

State the theorem of the Triangle of Forces and its converse. The perpendicular bisectors of the sides of a triangle are the lines of action of three forces. Each force acts outwards and is proportional in magnitude to the length of the side to which it is perpendicular. Prove that the three forces are in equilibrium.

##### Question 4

$ABCD$ is a quadrilateral lamina. $AB=BC=5″$ and $CD=DA=AC=6″$. Find the perpendicular distance from $AC$ of the centre of gravity of $ABCD$.

If $X$ is a point on $AD$ such that the centre of gravity of $ABCX$ is on $AC$, find the perpendicular distance from $AC$ of the centre of gravity of $CDX$.

##### Question 5

Derive an expression for the greatest height reached by a projectile in terms of the angle of projection and the initial velocity. A projectile, fired from ground level, just clears a vertical wall which is $210$ feet from the point of projection and is $84$ feet high, and the greatest height the projectile reaches is $100$ feet. Show that $\tan${-1}\frac{4}{3}$ is one possible angle of projection, and find the other possible angle.

##### Question 6

A car of mass $15$ cwt. is travelling with a uniform acceleration of $2$ ft. per sec.$^2$ The frictional resistances to motion being equivalent to $35$ lb. wt., find the horse-power at which the car is working at the instant that its speed is $30$ m.p.h.

(i) if it is travelling on a horizontal road,

(ii) if it is travelling down an incline of $1$ in $224$.

##### Question 7

Define simple harmonic motion.

A particle is moving with simple harmonic motion. Its velocity is $4$ cm. per sec. when it is $1$ cm. from its mean position, and its maximum acceleration is $6$ cm. per sec.$^2$ Find the amplitude and the period of the motion.

Find how far the particle will be from its mean position one second after it passes through its mean position.

##### Question 8

If two bodies are moving freely under gravity in the same straight line, show that the velocity of one body relative to the other is constant.

A mass of $10$ oz. is projected vertically upwards with an initial velocity of $70$ ft. per sec. and one second later a mass of $8$ oz. is projected vertically upwards from the same point with an initial velocity of $74$ ft. per sec. How many seconds later again will the masses collide?

If the masses coalesce on colliding, find in ft. lbs., the kinetic energy lost by the collision.

##### Question 9

In the case of a vertical surface immersed in a liquid at rest, prove that the total thrust on the surface due to the liquid is equal to the area of the surface multiplied by the pressure at is centre of gravity.

A rectangular swimming-pool is $100$ feet long and $40$ feet wide. The walls are vertical and the floor of the pools slopes uniformly so that the water if $4$ feet deep at one end and $16$ feet deep at the other end. Find in tons, correct to the nearest ton in each case, the total thrust of the water (i) on the wall at the shallow end, (ii) on one of the side-walls.

[A cubic foot of water weighs $62\frac{1}{2}$ 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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