Leaving Certificate Examination 1934 Honours Applied Mathematics
Question 1
Being given a small nearly frictionless truck of known weight, a weight of $10$ lb., $10$ small weights of $0.05$ lb. each, and a horizontal plane fitted with pulley, describe in detail how you would show experimentally that the acceleration of a system of constant mass is proportional to the force producing the acceleration.
Question 2
The amplitude of the vibration of a simple pendulum, of length $1$ ft, making small oscillations is $a$ ft. Find the maximum velocity of the bob.
A bullet weighing $1$ oz. is fired horizontally into a block weighing $24$ lb. suspended as a ballistic pendulum of length $10$ feet and becomes embedded in it. If the block is displaced through a horizontal distance of $8$ inches, find (i) the velocity of the block immediately after the impact, (ii) the velocity of the shot immediately before it struck the block.
Question 3
A man standing on a train which is moving at $48$ miles per hour shoots at an object which is moving away from the railway track at right angles with a speed of $40$ miles per hour. If the bullet, which is supposed to move in a horizontal straight line, has a velocity of $880$ ft. per second and if the line connecting the man and object makes an angle of $45^\circ$ with the direction in which the train is moving, find at what angle to the train he must aim in order to hit the the object.
Question 4
A sphere weighing $4$ lb. is suspended by a light, inextensible wire so that its c.g. is $4$ ft. below the point of support. It is projected with an initial horizontal velocity $v$ so that the wire ceases to be taut when it makes an angle of $60^\circ$ with the upward-drawn vertical. Find the velocity of projection and the greatest vertical height attained above the point of projection.
Question 5
A cycle track has a radius of $120$ ft. and is to be banked for a speed of $30$ miles per hour. Determine the angle which the track surface must make with the horizontal (i) when the friction between the wheels and the surface is entirely neglected, (ii) when the co-efficient of friction is taken as having a value of $0.2$.
Question 6
Taking the frictional resistance as $12$ lb. per ton, find the Horse Power required to produce a speed of $45$ miles per hour in a train weighing $300$ tons in $4$ minutes (i) on the level, (ii) down an incline of $1$ in $200$.
Question 7
A pin-jointed framework, of the form shown in the diagram, is loaded as indicated. It is supported by a fixed pin at $A$ and is kept in position with $AB$ vertical by a horizontal force at $B$. Determine the stresses in the bars, indicating in each case whether the bar is thrust or in tension.
Question 8
Show that the total work done in raising a body against gravity by any part is equal to the product of the weight and the vertical height through which the c.g. is raised. Does this principle hold if the body is not rigid? A chain weighing $12$ lb. per foot length and $60$ ft. long hangs over a pulley with one end $30$ ft. above the other. Find the work done in bringing the lower end to within $10$ ft. of the level of the other end.
Question 9
Explain what is meant by “the angle of friction.” Of a body be placed on a rough horizontal table show that no thrust, however great, applied to the body at an angle with the normal to the plane less than the angle of friction, can push the boy along the plane.
A ladder $24$ feet long, weighing $56$ lb., has one end resting on a concrete floor and the other against a vertical wall. The c.g. is $10$ ft. from the lower end, and a man weighing $140$ lb. stands on a rung $18$ ft. from the same end. If the co-efficient of friction at each end is $\frac{1}{3}$, find the inclination of the ladder if it is just about to slip.
Question 10
A ball weighing $1$ lb. describes a horizontal circle, the ball being attached to two cords, the other ends of which are attached to two points in the same vertical line. Each of these points is $3$ feet from the centre of the ball and the cords are at right angles. If the ball makes $3$ revolutions per second, find the tension of each cord in lbs.
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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