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28Nov2018

Leaving Certificate Examination 1968 Honours Applied Mathematics

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Question 1

Show how to find the centre of gravity of a non-rectangular parallelogram.

A thin uniform square sheet of metal ABCD of area 4 square inches weighs 4 ounces. At the corners A, B, C, D weights of 1 oz., 2 oz., 3 oz., and 4 oz., respectively are placed. Locate the centre of gravity of the system.

Question 2

 

Establish the formulae v=r\omega and a=r\omega^2 for the velocity and acceleration of a particle moving with uniform speed in a circle or radius r.

Assuming that the moon travels in a circular orbit whose radius is 3.8 \times 10^10 cm. and whose centre ie the center of the earth, and completes a revolution every 2.4 \times 10^6 seconds, calculate the ratio of its acceleration g^1 towards the centre of the earth to gravity g at the earth’s surface.

Question 3

Two indentical cars A and B each 16 feet in length travel with constant velocity 60 feet per second along a straight level road, the front of A being 44 feet directly behind the rear of B. Immediately on reaching a line drawn across the road perpendicular to the direction of motion of the cars each car decelerates at 12 feet per sec^2.

(i) Show that A crashes into B.

(ii) At what distance from the line does the crash occur?

Question 4

An aeroplane A about to drop a bomb on a small stationary target T. on the sea surface descends along a line which is inclined at an angle of 30^\circ to the horizontal and which passes directly over the target at a vertical height of \frac{1}{2} mile. If the aeroplane travels at 600 m.p.h. what duration of time must elapse between the release of the bomb and its impact on the target. (See diagram).

Question 5

Show that the work done by a body in coming to rest is the kinetic energy of the moving body. A body of mass 20 lbs. sliding on a rough horizontal plane with initial velocity 22 feet sec^{-1}. is brought to rest by friction in 15 feet. Calculate the coefficient of friction.

At what average rate in horse-power was work done on the body in bringing it to rest?

Question 6

A particle executes simple harmonic motion with a frequency of 60 vibrations per second. The amplitude is 1.5 millimetres.

Find (i) the maximum speed of the particle,

(ii) the maximum acceleration of the particle.

If the cutting-head of an electric razor has mass 10 grams and oscillates with simple harmonic motion at a frequency of 20 vibrations per second, and with an amplitude of 1.5 millimetres, calculate the maximum value of the driving force.

Question 7

A train is rounding a circular curve of radius 440 yards. A bob hangs at one end of a light string which is fixed in the ceiling of a carriage and which is inclined at an angle of 4^\circ18' to the vertical. Estimate the speed of the train.

Question 8

A fire-engine raises 200 gallons of water per minute from a well 16 feet deep and then pumps it as a jet issuing with a nozzle velocity of 44 feet per sec. At what horsepower is the engine working?

Question 9

The density of lead is 11.3 gram cm^{-3}. and that of mercury is 13.6 grams cm^{-3}. Explain why a block of lead will float on mercury.

A rectangular uniform block of wood floats on sea water with \frac{1}{3} of its volume above the water-line. What is the volume of the smallest rectangular block of the wood which will float without being totally submerged while supporting a weight of 100 lb. placed vertically above its centre of gravity?

(Take $1.03 as the specific gravity of sea-water.)


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/

Licence:

“Contains Irish Public Sector Information licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) licence”.

The EU Directive 2003/98/EC on the re-use of public sector information, its amendment EU Directive 2013/37/EC, its transposed Irish Statutory Instruments S.I. No. 279/2005, S.I No. 103/2008, and S.I. No. 525/2015, and related Circulars issued by the Department of Finance (Circular 32/05), and Department of Public Expenditure and Reform (Circular 16/15 and Circular 12/16).

Note. Circular 12/2016: Licence for Re-Use of Public Sector Information adopts CC-BY as the standard PSI licence, and notes that the open standard licence identified in this Circular supersedes PSI General Licence No: 2005/08/01.

Links:

https://circulars.gov.ie/pdf/circular/per/2016/12.pdf

https://creativecommons.org/licenses/by/4.0/legalcod

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