Leaving Certificate Examination 1967 Honours Applied Mathematics

Question 1

A body is projected under gravity with initial velocity at an angle to the horizontal. Find in terms of , , (a) the maximum height attained, (b) the time required to reach that height, and (c) the distance it has travelled in a horizontal direction on reaching the maximum height.

A man strikes a gold ball which, at the instant it reaches maximum height of feet above the ground, strikes a tree  feet away. Assuming the ground to be level, find the magnitude and direction of the initial velocity of the gold ball, to the nearest ft. per sec. (neglecting effects of air resistance).

Question 2

Show that is the work done on a stationary mass in order to give it a velocity .

A mass of kg. moving East with a velocity of cm. per sec. makes direct impact on a mass of kg. which is at rest. After the impact the kg. mass moves East with a velocity of cm. per sec. Assuming conservation of momentum find

(a) the velocity of the kg. mass after impact,

(b) the kinetic energy lost on impact.

Question 3

Define Simple Harmonic Motion.

(i) If the bob of a pendulum is given a very small displacement to one side, the string being kept taut, and then released, show that the resulting motion is approximately simple harmonic.

(ii) A bob of mass grams hangs motionless at one end of an elastic string the other end of which is fixed in the ceiling. If the bob is pulled vertically down a distance until the tension in the string is grams weight and then released, with what velocity will it pass the position of equilibrium assuming its motion about that position is simple harmonic?

Question 4

is a velocity fixed in magnitude and direction. is a velocity fixed in magnitude only. If , use vector diagrams to illustrate the situations where

(a) the magnitude of is greatest.

(b) the magnitude of is least.

(c) the magnitude of is neither greatest nor least.

A boat crosses a straight river ft. wide which flows with a uniform velocity of miles per hour. The boat’s velocity relative to the water is miles per hour. Calculate to the nearest foot how far down-stream the boat lands if it crosses the stream in such a direction that it reaches the other side in exactly one minute.

Question 5

(i) A rectangular lamina inches long and inches wide is marked off in six equal squares. One corner square is removed. Calculate the position of the centre of gravity of the remainder.

(ii) is a point outside a triangle whose centre of gravity is . Three forces , , act at . Show that their resultant is .

Question 6

A uniform circular hoop of radius rolls on the outer rim of a fixed wheel of radius , hoop and wheel being coplanar. (See diagram.) If the angular velocity of the hoop about its centre is constant find

(i) what distance the hoop travels along the circumference of the wheel in time ;

(ii) the velocity and acceleration of the centre of the hoop;

(iii) the velocity of that point of the hoop which is farthest from the wheel.

Question 7

Say what you know about the pressure in a liquid, and the transmission of pressure in liquid.

The diameter of the small piston of a hydraulic press (see diagram) is inches and the diameter of the large piston is inches. What weight on the small piston will support two tons weight on the large piston at the same horizontal level?

Question 8

Give three properties of friction.

Two light rings can slide on a rough horizontal rod. The rings are connected by a light inextensible string of length to the mid point of which is attached a weight . If is the coefficient of friction between either ring and the rod, and is the distance between the rings, show that for equilibrium

   

Question 9

A straight light rigid rod of length terminated by heavy particles of masses and respectively is placed inside a smooth hemispherical bow of radius , which is fixed with its rim horizontal. If the particle of mass rests just below the rim show that

   

OR

Question 9

A train weighing tons is being pulled down a slope of in by an engine weighing tons. The train has velocity m.p.h. and is accelerating at the rate of ft. per sec. If the frictional resistance to its motion is lbs. weight per ton find the horse-power at which the engine is working.

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Citation:

State Examinations Commission (2018). State Examination Commission. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru

Malone, D and Murray, H. (2016). Archive of Maths State Exams Papers. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/

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