# 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 which weighs lb. is supported by two strings , attached to a fixed peg at . If , , , 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 in with a uniform acceleration of ft. per sec.. 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 ft. per sec. to ft. per sec. in a distance of feet, find the slope of the plane.

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

#### Question 5

A bullet weighing lb. is fired horizontally with a velocity of ft. per sec. from a gun which is free to recoil. If the gun weighs 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 lb., and the total kinetic energy was the same as above, show that the velocity of the bullet would be less by about ft. per sec.

#### Question 6

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

A ounce mass supported from a fixed point by a light inextensible string feet long is describing a horizontal circle at a uniform rate of 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

, are two points at ground level feet apart. An aeroplane is flying at a steady height of feet above the ground in the direction , with a uniform velocity of ft. per sec. When the aeroplane is directly over a shell is fired, at an angle of projection , from a gun at and strikes the aeroplane seconds later. If , find the value of and the initial velocity of the shell.

#### Question 8

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

When is feet from its velocity is ft. per sec. away from , and its acceleration is ft. per sec. towards . Find the amplitude and the period. How many seconds later does reach 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 in which and is immersed in water, with the vertex at the surface and the vertex vertically below . Find the total thrust of the water on the lamina in lbs. wt. correct to one decimal place.

[A cubic foot of water weighs lb.]

**Citation:**

**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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