# Leaving Certificate Examination 1931 Honours Applied Mathematics

##### Question 1

The motion of a particle along a straight line is given in the following table :-

Time in seconds. | ||||||||||||||

Distance in cm. |

Obtain approximately its velocity at intervals of seconds from seconds to seconds, and plot against . Hence determine approximately when the velocity of the particle is zero and is acceleration is zero.

**OR**

**OR**

If a particle moves according to the law , explain how you can obtain, by calculus, expressions for its speed, and its acceleration at any instant. When, during the motion, are these magnitudes zero? How is this kind of motion usually described?

##### Question 2

A train acquires a speed of miles an hour in minutes. If the carriage wheels are feet in diameter, what is their angular velocity at this speed? What was their average angular acceleration?

Show, on a diagram, the magnitude and direction of the components of the acceleration of the highest point of the wheel during the accelerated motion of the train, and the acceleration of the same point when the train is moving with uniform speed.

##### Question 3

A stone is projected with velocity and elevation from a point in a horizontal plane, so as to hit a mark at a horizontal distance from , and at a height above the plane. Show that , , and are connected by the relation:

.

Derive the condition that should be a maximum when and are constants, and show that is is satisfied by where is the elevation of from .

##### Question 4

In an Attwood’s machine, a mass of pounds is attached to each end of the cord. An additional mass of pound is placed on one side and is found to produce a velocity of feet per second at end of a descent from rest of feet. Compare this result with that given by the simple theory. Express in ft-lb. the kinetic energy of the masses and the work done by the weights, and account for the difference between these quantities.

##### Question 5

Two uniform rods and weighing grams each, and of lengths cm. and cm. respectively are freely jointed together at . They are maintained in a horizontal straight line by three vertical strings, one attached to a point in , cm. from , and the others at and . Find the tensions in the strings.

##### Question 6

A uniform square lamina, of 9 in. side, is divided into two parts by a line joining to a point in where in. State the distances of the centres of gravity of the triangles and from the sides and . Find the distance of the centre of gravity of from and .

##### Question 7

A uniform rod, cm. long and weight grams, is suspended from a fixed point by strings cm. and cm. long attached to the ends of the rod. Find the tension in each string.

##### Question 8

Two particles and moving along the axes and respectively towards with velocities and collide at . What are the components of the velocity of centre of gravity before the collision? Why do they remain unaltered by the collision? Show that the kinetic energy of the particles is equal to the kinetic energy of a mass moving with velocity of the centre of gravity and a mass and a mass moving with the relative velocity of either particle with respect to the other.

##### Question 9

Define the term Power. A motor-car weighing cwt. is travelling at uniform speed of miles per hour on a level road. On reaching a hill which descends with a uniform gradient of in , it is allowed to run free, and the speed is observed to be the same as before. Calculate the resistance of the road and the horsepower expended on the level.

##### Question 10

A pendulum bob at the end of a string inches long, describes a horizontal circle with the string making an angle of degrees with the vertical. Find its angular velocity and the time of describing the circle. State clearly the principles applied in the solution of this problem.

**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/

**Licence:**

**Licence:**

*“Contains Irish Public Sector Information licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) licence”.*

The EU Directive 2003/98/EC on the re-use of public sector information, its amendment EU Directive 2013/37/EC, its transposed Irish Statutory Instruments S.I. No. 279/2005, S.I No. 103/2008, and S.I. No. 525/2015, and related Circulars issued by the Department of Finance (Circular 32/05), and Department of Public Expenditure and Reform (Circular 16/15 and Circular 12/16).

Note. Circular 12/2016: Licence for Re-Use of Public Sector Information adopts CC-BY as the standard PSI licence, and notes that the open standard licence identified in this Circular supersedes PSI General Licence No: 2005/08/01.

**Links:**