Leaving Certificate Examination 1933 Honours Applied Mathematics
Question 1
The following table gives the speed $v$ of a train $t$ seconds after leaving a station $A$ for portion of the journey between two stations $A$ and $B$, $4800$ ft. apart.
| Speed in feet per second. | $0$ | $18$ | $34$ | $40$ | $44$ | $45.5$ | $45.5$ | $42$ | $35$ |
| Time in seconds. | $0$ | $10$ | $20$ | $30$ | $40$ | $50$ | $60$ | $70$ | $80$ |
Estimate the acceleration after $15$ seconds and the distance travelled in the $80$ seconds. If the train continues to move with the uniform speed of $35$ ft. per second until the brakes are applied, producing a uniform retardation of $2$ ft. per sec. per sec., find the total time taken in going from $A$ to $B$.
Question 2
(a) A particle describes simple harmonic motion of amplitude $a$, the period being $T$. Prove that at distance $x$ from its mean position its speed $v$ is given by $\frac{2\pi}{T}\sqrt{a^2-x^2}$.
(b) If the amplitude of the motion of bob of a simple pendulum is $1$ foot and the period $3$ seconds, find the time occupied by the bob in passing between points which are distance $8$” and $4$” from the mean position and are on the same side of it.
Question 3
Prove that two bodies moving in different straight lines with uniform velocities will meet if their relative velocity is in the straight line joining them.
A ship steaming in a direction $17^\circ$ North of East is seen by a submarine which is in a direction $10^\circ$ East of South from the ship. Find in what direction a torpedo travelling at $20$ miles per hour must be fired in order to strike ship.
Question 4
A bullet of mass $1$ oz. travelling horizontally with a speed of $1200$ ft. per second becomes embedded in a block of wood of mass $4$ lb. suspended by a light steel wire, the C.G. of the block being $4$ ft. below the point of suspension. Find the vertical distance through which the block rises.
Question 5
When the motors of an electric train of effective weight $100$ tons are working at the rate of $80$ H.P., a uniform speed of $30$ miles per hour is attained on the level. Assuming the frictional resistances to remain the same, what is the maximum speed attainable with $150$ H.P. when ascending an incline of $1$ in $250$?
Question 6
Show that the centre of gravity of $3$ equal particles placed at the three vertices of a triangle respectively coincide with the centre of gravity of the area of the triangle. The parallel side of a trapezium are of lengths $a$ and $b$ and the perpendicular distance between them is $h$. Find the distance of the C.G. from the side $a$.
Question 7
What are the laws of solid friction?
A body of weight $w$ is dragged along a horizontal plane, coefficient of friction $\mu$, by a cord inclined at an angle $\theta$ to the horizontal; find the tension in the cord. For what value of $\theta$ is the tension a minimum? A body is placed on a plane, coefficient of friction $0.15$ inclined at $50^\circ$ to the horizontal; find the velocity acquired in one second.
Question 8
A triangular roof-truss $ABC$ has a horizontal span $AC$ of $36$ feet and angle $ABC$ is $120^\circ$, $AB$ and $BC$ being of equal length. The roof-truss is hinged at $A$ and simply supported on rollers at $C$ (so that the supporting force at $C$ is vertical). It is loaded as follows: $1\frac{1}{2}$ tons perpendicular to $AB$ at its mid-point at $BC$. Find the reactions of the supporting forces at $A$ and $C$ and the tensions in $AC$.
Question 9
A body of weight $20$ lb. is moving due East with a velocity of $12$ ft. per second, and forty seconds later it is moving North-East with a velocity of $20$ ft. per second. What is the change in velocity? What constant force acting during the interval would produce the change? Show that in the latter case the hodograph is a straight line.
Question 10
Assuming that the wheels of a motor-car are $2\frac{1}{2}$ feet in diameter, what is the angular velocity of a wheel when the car s travelling at the rate of $50$ miles per hour? What is the velocity of a point on the surface of tyre (a) when vertical above the centre, (b) when the radius from the point to the centre is inclined at an angle of $60^\circ$ to the upward vertical? What would be the force acting on a stud of mass $\frac{1}{2}$ oz. embedded in the type when at its greatest height?
Citation:
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:
About Post Author
