# Leaving Certificate Examination 1957 Honours Applied Mathematics

#### Question 1

If three co-planar non-parallel forces acting on a rigid body are in equilibrium, prove that their lines of action are concurrent and that the forces may be represented in magnitude and direction by the sides of a triangle taken in order.

A uniform bar $AB$ which weighs $8$ lb. is supported by two strings $AC$, $BC$ attached to a fixed peg at $C$. If $AB=6″$, $AC=5″$, $BC=3″$, find the tension in each of the strings.

#### Question 2

State the main laws of friction. Explain the term “coefficient of friction,” and describe briefly how its value for two given surfaces may be found by experiment.

A box, lying on the floor of a railway carriage, just begins to slide back along the floor as the carriage is descending an incline of $1$ in $8$ with a uniform acceleration of $12$ ft. per sec.$^2$. Show by diagram the forces acting on the box, and find the coefficient of friction between the box and the floor.

#### Question 3

Show that the moment of a couple is the same about all points in the plane of the couple. Explain how couples may be compounded, and show that a force and a couple acting in the same plane are equivalent to a single force.

Prove that four forces represented in magnitude, direction *and position* by the sides of a quadrilateral taken in order are equivalent to couple.

#### Question 4

(i) A block is sliding freely down the line of greatest slope of a smooth inclined plane. If the velocity of the block increases from $3$ ft. per sec. to $5$ ft. per sec. in a distance of $5$ feet, find the slope of the plane.

(ii) A car which weigh $1$ ton is descending an incline of $1$ in $80$ ; the velocity of the car is $20$ m.p.h. and it is accelerating at the rate of $2$ ft. per sec.$^2$. If the frictional resistances to motion are equivalent to $55$ lb. wt., find the horse-power at which the car is working.

#### Question 5

A bullet weighing $.02$ lb. is fired horizontally with a velocity of $2,000$ ft. per sec. from a gun which is free to recoil. If the gun weighs $20$ lb., find the velocity with which the gun begins to recoil, and find the total kinetic energy, in foot-lbs., of the gun and the bullet.

If the same bullet were fired from a gun weighing $5$ lb., and the total kinetic energy was the same as above, show that the velocity of the bullet would be less by about $3$ ft. per sec.

#### Question 6

A particle is describing a circle of radius $r$, with constant angular velocity $w$: show that is acceleration is $w^2r$ directed towards the centre of the circle.

A $2-$ounce mass supported from a fixed point by a light inextensible string $4$ feet long is describing a horizontal circle at a uniform rate of $2$ revolutions per second. Find the tension in the string in lbs. wt., and find the vertical distance from the fixed point to the plane of the circle.

#### Question 7

$P$,$Q$ are two points at ground level $3,500$ feet apart. An aeroplane is flying at a steady height of $1,600$ feet above the ground in the direction $QP$, with a uniform velocity of $400$ ft. per sec. When the aeroplane is directly over $Q$ a shell is fired, at an angle of projection $\alpha$, from a gun at $P$ and strikes the aeroplane $t$ seconds later. If $\cos\alpha=\frac{3}{5}$, find the value of $t$ and the initial velocity of the shell.

#### Question 8

A point $A$ is describing a circle, centre $O$, at constant speed. If $N$ is the foot of the perpendicular from $A$ to a fixed diameter, show that $N$ is moving with an acceleration which is directly proportional to is distance from $O$.

When $N$ is $4$ feet from $O$ its velocity is $6$ ft. per sec. away from $O$, and its acceleration is $16$ ft. per sec.$^2$ towards $O$. Find the amplitude and the period. How many seconds later does $N$ reach $O$ from that position? (Give your answer correct to the nearest tenth of a second.)

#### Question 9

If a plane lamina is immersed vertically in a liquid at rest, prove that the total thrust on it due to the liquid is equal to the area of the immersed surface multi[lied by the pressure at its centre of gravity.

A quadrilateral lamina $ABCD$ in which $BC=CD=5″$ and $AB=BD=AD=6″$ is immersed in water, with the vertex $A$ at the surface and the vertex $C$ vertically below $A$. Find the total thrust of the water on the lamina in lbs. wt. correct to one decimal place.

[A cubic foot of water weighs $62.5$ 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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