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28 Nov 2018

Leaving Certificate Examination 1968 Honours Applied Mathematics

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Question 1

Show how to find the centre of gravity of a non-rectangular parallelogram.

A thin uniform square sheet of metal $ABCD$ of area $4$ square inches weighs $4$ ounces. At the corners $A$, $B$, $C$, $D$ weights of $1$ oz., $2$ oz., $3$ oz., and $4$ oz., respectively are placed. Locate the centre of gravity of the system.

Question 2

 

Establish the formulae $v=r\omega$ and $a=r\omega^2$ for the velocity and acceleration of a particle moving with uniform speed in a circle or radius $r$.

Assuming that the moon travels in a circular orbit whose radius is $3.8 \times 10^10$ cm. and whose centre ie the center of the earth, and completes a revolution every $2.4 \times 10^6$ seconds, calculate the ratio of its acceleration $g^1$ towards the centre of the earth to gravity $g$ at the earth’s surface.

Question 3

Two indentical cars $A$ and $B$ each $16$ feet in length travel with constant velocity $60$ feet per second along a straight level road, the front of $A$ being $44$ feet directly behind the rear of $B$. Immediately on reaching a line drawn across the road perpendicular to the direction of motion of the cars each car decelerates at $12$ feet per sec$^2$.

(i) Show that $A$ crashes into $B$.

(ii) At what distance from the line does the crash occur?

Question 4

An aeroplane $A$ about to drop a bomb on a small stationary target $T$. on the sea surface descends along a line which is inclined at an angle of $30^\circ$ to the horizontal and which passes directly over the target at a vertical height of $\frac{1}{2}$ mile. If the aeroplane travels at $600$ m.p.h. what duration of time must elapse between the release of the bomb and its impact on the target. (See diagram).

HAM-1968-Q4-300x182 Leaving Certificate Examination 1968 Honours Applied Mathematics Maths Grinds

Question 5

Show that the work done by a body in coming to rest is the kinetic energy of the moving body. A body of mass $20$ lbs. sliding on a rough horizontal plane with initial velocity $22$ feet sec$^{-1}$. is brought to rest by friction in $15$ feet. Calculate the coefficient of friction.

At what average rate in horse-power was work done on the body in bringing it to rest?

Question 6

A particle executes simple harmonic motion with a frequency of $60$ vibrations per second. The amplitude is $1.5$ millimetres.

Find (i) the maximum speed of the particle,

(ii) the maximum acceleration of the particle.

If the cutting-head of an electric razor has mass $10$ grams and oscillates with simple harmonic motion at a frequency of $20$ vibrations per second, and with an amplitude of $1.5$ millimetres, calculate the maximum value of the driving force.

Question 7

A train is rounding a circular curve of radius $440$ yards. A bob hangs at one end of a light string which is fixed in the ceiling of a carriage and which is inclined at an angle of $4^\circ18’$ to the vertical. Estimate the speed of the train.

Question 8

A fire-engine raises $200$ gallons of water per minute from a well $16$ feet deep and then pumps it as a jet issuing with a nozzle velocity of $44$ feet per sec. At what horsepower is the engine working?

Question 9

The density of lead is $11.3$ gram cm$^{-3}$. and that of mercury is $13.6$ grams cm$^{-3}$. Explain why a block of lead will float on mercury.

A rectangular uniform block of wood floats on sea water with $\frac{1}{3}$ of its volume above the water-line. What is the volume of the smallest rectangular block of the wood which will float without being totally submerged while supporting a weight of $100$ lb. placed vertically above its centre of gravity?

(Take $1.03 as the specific gravity of sea-water.)


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:

“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:

https://circulars.gov.ie/pdf/circular/per/2016/12.pdf

https://creativecommons.org/licenses/by/4.0/legalcod

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