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### My Journal

All things Mathematical
3 Jul 2019

## Question 1

A driver starts from rest at and travels with a uniform acceleration of m/s for seconds. He continues with uniform velocity for seconds, and then decelerates uniformly to rest at   in a further seconds. Express the distance in terms of and .

Another driver can accelerate at m/s and can decelerate at m/s. Find, in terms of , the least time in which this driver can cover the distance from rest to rest

(i) subject to a speed limit of m/s,

(ii) subject to a speed limit of m/s.

[Video Solution]

## Question 2

A plane is inclined at an angle to the horizontal. A particle is projected up the plane with initial velocity at an angle to the plane. The plane of projection is vertical and contains the line of greatest slope.

(i) Write down the displacement and velocity of the particle parallel and perpendicular to the plane at time .

(ii) Show that the time taken by the particle to reach its maximum perpendicular height above the plane is half the time of flight up the plane.

(iii) When the particle is at its maximum perpendicular height above the plane, the distance travelled parallel to the plane is of the range up the plane. Show that in that case .

[Video Solution]

## Question 3

(i) Two vectors and are at right angles. Write down the condition satisfied by the scalars .

(ii) Two smooth spheres of masses and and velocities and , respectively, collide as shown in the diagram, where . The sphere of mass is deflected through an angle of by the collision. If the coefficient of restitution is , show that .

(iii) Find the direction of motion of the other sphere after the collision.

[Video Solution]

## Question 4

A body of mass lies on a smooth horizontal table. It is connected by means of a light string passing over a smooth light pulley at the edge of the table, to a second smooth pulley of mass hanging freely. Over this second pulley passes another light string carrying masses of and (see diagram).

(i) Show in separate diagrams the forces acting on each of the masses.

(ii) Write down the equations of motion involving the tensions and in the strings, the common acceleration of the and masses and the common acceleration of the and masses relative to the mass.

(iii) Show that .

[Video Solution]

## Question 5

Two uniform rods and of equal length and of masses kg and kg, respectively, are freely hinged at . and are in a vertical plane and the ends and are on a rough horizontal plane. The coefficient of friction between each rod and the plane is the same.

(i) Find the normal reactions at and .

(ii) The angle is increased until one of the rods begins to slip. Show that slipping will first occur at rather than at .

(iii) Find the least value of the coefficient of friction if slipping has not occurred before .

[Video Solution]

## Question 6

Two small smooth rings and , each of mass are threaded on a fixed smooth horizontal wire. They are connected by means of two light inextensible strings and , each of length metres, to a particle of mass hanging freely at . , , are in the same vertical plane. The system is released from rest with the angles .

(i) If travels a horizontal distance while falls a vertical distance , show from geometry that .

(ii) By differentiating find in terms of and , where means , and using the conservation of energy find in terms of .

(iii) Show that the velocity of is where .

[Video Solution]

## Question 7

(a) Prove that the moment of inertia of a uniform circular disc about a perpendicular axis through its centre is , where is the radius of the disc and is its mass.

[Video Solution]

(b) A light string is wound around the rim of a uniform disc of radius and mass . One end of the string is attached to the rim of the disc and the other end is attached to a fixed point above the disc, with the plane of the disc vertical (see diagram). When the disc is released from rest it falls vertically and the string unwinds.

If the disc falls a distance while it turns through an angle , show that and deduce that . where is the angular velocity of the disc. ( means , means )

Using the principle of angular momentum, find the tension in the string and the vertical acceleration of the disc.

[Video Solution]

## Question 8

Solve the following differential equations:

(i) if when

(ii) if and when .

(iii) A particle of mass is acted on by a force directed away from a fixed point , where is the distance of the particle from . The particle starts from rest at a distance from . Show that the velocity of the particle tends to a limit .

[Video Solution]

## Question 9

(a) Two particles and are moving along two perpendicular lines towards a point with constant velocities of m/s and m/s respectively. When is metres from , is metres from . Find the distance between them when they are nearest to each other.

[Video Solution]

(b) State the Principle of Archimedes.

[Video Solution]

(c) A uniform circular cylinder of height and relative density floats with its axis vertical in a liquid of relative density .

(i) Find the length of the axis of the cylinder immersed.

(ii) The cylinder is depressed vertically a further small distance and release. Show that it will preform simple harmonic motion, and find the period.

[Video Solution]

## Question 10

(a) A portion in the shape of an equilateral triangle is removed from a circular lamina of radius . A vertex of the triangle was at the centre of the lamina and the sides of the triangle are of length . Find the position of the centre of gravity of the remainder.

[Video Solution]

(b) A train of mass tonnes is maintaining a steady speed of m/s up an incline of in against frictional forces amounting to kN. Calculate the power at which the engine is working. ( tonne = kg).

[Video Solution]

(c) A corner on a level track has a radius of m. Calculate the maximum speed at which a cyclist could take the corner if the coefficient of friction were .

[Video Solution]

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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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