085-1263404

### My Journal

All things Mathematical
14Nov2018

#### 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:

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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