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### My Journal

All things Mathematical
29Aug2019

#### Question 1

i) A body starts from rest at , travels in a straight line and then comes to rest at which is km from . The time taken is seconds. For the first seconds it has uniform acceleration . It then travels at constant speed and is finally brought to rest by a uniform deceleration acting for seconds. Calculate and .

ii) If the journey from rest to at to rest at had been travelled with no interval of constant speed, but subject to for a time followed by for a time , show that the time for the journey is seconds.

#### Question 2

a i) If a body is in equilibrium under the action of two and only two forces, what can be deduced about the forces? I

ii) If a body is in equilibrium under the action of three non parallel forces prove that their lines of action must be concurrent and that the forces may be represented in magnitude and direction by the sides of a triangle taken in order.

b) A straight, rigid non-uniform rod of weight rests in equilibrium inside a smooth hollow sphere of radius . The distances of its centre of gravity, , from and are and cm respectively.

i) Show that , the angle the rod makes with the vertical, is given by .

ii) Prove that the magnitude of the reaction at is given by .

#### Question 3

a) Establish an expression, in terms of initial speed and angle of inclination to the horizontal , for the range of a projectile on a horizontal plane through the point of projection. Deduce that the maximum range for a given is .

b) A particle is projected at initial speed from the top of a cliff of height , the trajectory being out to sea in a plane perpendicular to the cliff. The particle strikes the sea at a distance from the foot of the cliff. Show that the possible times of light can be obtained from the equation

Hence, or otherwise, prove that the maximum value of for a particular and is

#### Question 4

i) A sphere , mass , moving with velocity impinges directly on an equal sphere , moving in the same direction with velocity . Show that the loss in kinetic energy due to the impact is

where is the coefficient of restitution between the spheres.

ii) If had been at rest and impinged obliquely, so that after impact, moved with velocity in a direction making an angle of with the line of centres of the spheres, show that the loss in kinetic energy is three times greater than in (i).

#### Question 5

a) Prove that the centre of gravity of a uniform triangular lamina is at the point of intersection of its medians.

b) A uniform triangular lamina has weight newtons.

cm, cm, cm and .

The lamina is suspended in a horizontal position by three inextensible, vertical strings, one at each vertex. A particle of weight newtons is positioned on the lamina cm from and one cm from . Calculate the tension on each string in terms of .

#### Question 6

a) Establish the formula for the periodic time of a simple pendulum of length . The length of a seconds pendulum is altered so as to execute complete oscillations per minute. Calculate the percentage change in length.

b) A heavy particle is describing a circle on a smooth horizontal table with uniform angular velocity . It is partially supported by a light inextensible string attached to a fixed point metres above the table. Calculate the value of if the normal reaction of the table on the particle is half the weight of the particle.

#### Question 7

a) i) A uniform circular disk has mass and radius . Prove that is moment of inertia, , about an axis through its centre perpendicular to its plane is .

ii) Deduce the moment of inertia about an axis through a point on its rim perpendicular to its plane.

b) A uniform circular disk has mass and radius . It is free to rotate about a fixed horizontal axis through a point on its rim perpendicular to its plane. A particle of mass is attached to the disk at a point on its rim diametrically opposite . The disk is held with horizontal and released from rest.

i) Find, in terms of , the angular velocity when is vertically below .

ii) If the system were to oscillate as a compound pendulum, prove that it would have a periodic time equal to that of a simple pendulum of length .

#### Question 8

a) A heavy particle is hung from two points on the same horizontal line and a distance apart by means of two light, elastic strings of natural length , and elastic constants , respectively. In the equilibrium position the two strings make equal angles with the vertical. Prove that

b) A horizontal platform, on which bodies are resting, oscillates vertically with simple harmonic motion of amplitude m. What is the maximum integral number of complete oscillations per minute it can make, if the bodies are not to leave the platform?

#### Question 9

a) A body is weighed in water and in each of two liquids of specific gravity and . If the resulting weights, in order, are , , and , verify that

b) A uniform rectangular board , , hangs vertically in fresh water with the diagonal on the surface. The board is held in that position by a vertical string at .

i) Show on a diagram all forces acting on the board.

ii) Calculate the tension () in the string and buoyancy force () in terms of , the weight of the board.

iii) Calculate the specific gravity of the board.

#### Question 10

a) Solve the differential equation

given that when .

b) Find the general solution to

where is a constant.

A particle moves in a straight lien so that at any instant its acceleration is, in magnitude, half its velocity. If its initial velocity is m/s, find an expression for the distance it describes in the fifth second.

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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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