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### My Journal

All things Mathematical
30Aug2019

#### Question 1

a) A car passes a point on a straight road at a constant speed of m/s. At the same time another car starts from rest at with uniform acceleration m/s.

i) When and how far from with overtake ?

ii) If ceases to accelerate on overtaking, what time elapses between the two cars passing a point three kilometres from ?

b) A particle of mass grammes falls from rest from a height of m on to a soft material into which it sinks m. Neglecting air resistance, calculate the constant resistance of the material.

#### Question 2

Two straight roads intersect at an angle of . Cyclist and move towards the point of intersection at km/h and km/h respectively.

i) Calculate the velocity of relative to .

ii) If is km and is km from the intersection at a given moment, calculate the shortest distance between them in their subsequent motion.

#### Question 3

A particle is projected with a speed of m/s at an angle to the horizontal up a plane inclined at to the horizontal.

i) If the particle strikes the plane at right angles, show that the time of flight can be represented by the two expressions and .

ii) Hence deduce a value for .

iii) Calculate the range of the particle along the plane.

#### Question 4

a) A smooth sphere of mass kg moving at m/s impinges directly on another smooth sphere of mass kg moving in the opposite direction at m/s. If the coefficient of restitution is , calculate the speeds after impact and the magnitude of the impulse during impact.

b) A smooth metal sphere falls vertically and strikes a fixed smooth plane inclined at an angle of to the horizontal. If the coefficient of restitution is and the sphere rebounds horizontally, calculate the fraction of kinetic energy lost during impact.

#### Question 5

a) The diagram shows a string which is fixed at and where vertically below . is a small ring threaded on the string which is made to rotate at an angular velocity, rad/s, in a horizontal circle, centre , the string being taut. If m, m, show that rad/s.

b) A small bead of mass is threaded on a smooth circular wire of radius , fixed with its plane vertical. The bead is projected from the lowest poitn of the wire with speed . Show that the reaction between the bead and the wire, when the radius to the bead makes an angle of with the downward vertical is

#### Question 6

a) Define (i) limiting friction, (ii) coefficient of friction.

b) A lamina of weight in the shape of a thin equilateral triangle is positioned vertically with the vertex against a smooth vertical wall, and on rough horizonal floor. [] is parallel to the wall.

i) Find in terms of , the horizontal and vertical reactions at .

ii) Find the least value of , the coefficient of friction, so that slipping will not occur.

#### Question 7

a) Show that the moment of inertia of a uniform square lamina of side and mass about an axis perpendicular to the lamina through its centre of mass is .

b) i) A thin uniform rod of length and of mass is attached to the mid point of the rim of the square. Find the moment of inertia of the system about an axis through perpendicular to the common plane of lamina and rod.

ii) When this system makes small oscillations in a vertical plane about the axis through , show that the period of the oscillations is .

#### Question 8

i) Define simple harmonic motion.

ii) The distance, , of a particle from a fixed point, , is given by

where , , are positive constants. Show that the particle is describing simple harmonic motion about and calculate and if the velocity and where .

iii) After how many seconds from the start of the motion is for the first time?

#### Question 9

a) A bucket has the form of a frustum of a right circular cone. When it is completely filled with water, find

i) the pressure at a point on the base

ii) the thrust, , on the base

iii) the ratio .

b) A cubical block of wood of mass kg floats in water with three quarters of its volume immersed. In oil, when a mass of kg is placed on the same block, it floats just totally immersed, the kg mass being above the oil. Find the specific gravity of the oil.

#### Question 10

a) Find the solution of the differential equation

where at .

b) i) Find the solution of the differential equation

when at and at .

ii) If its initial velocity is m/s, calculate the distance travelled one second later.

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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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