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14Nov2018

Leaving Certificate Examination 1957 Honours Applied Mathematics

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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:

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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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).

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Links:

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

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

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