Leaving Certificate Examination 1935 Honours Applied Mathematics
Question 1
A particle moves along a straight line and its distance in feet $s$ from a fixed point in the line is given by the formula $s=at^3+bt^2+ct$ where $a$, $b$, $c$ are constants and $t$ denotes the time in seconds. The distance is $16$ when $t=1$ and the velocity is zero when $t=2$ and $t=4$. Find the values of $a$, $b$, $c$ and verify that the acceleration is zero when $t=3$. Draw rough diagrams showing how (a) the acceleration, (b) the velocity varies between $t=0$ and $t=6$.
Question 2
A body weighing $1$ lb. is moving in a straight line with Simple Harmonic Motion. If the greatest velocity is $8\pi$ ft. per second. and the amplitude of the oscillation is $\frac{1}{2}$ ft., find the period of oscillation and the force of attraction towards the centre when the body is at its greatest distance from the centre.
Question 3
A body is projected with velocity $u$, at an inclination $\alpha$ to the horizon, from the foot of a plane inclined at angle $\beta$ to the horizon. Determine its range up the plane. Show that for a given value of $u$, the range is a maximum when the direction of projection bisects the angle between the inclined plane and the vertical. Determine the least velocity with a ball can be thrown to reach the top of a cliff $6$ ft. high and $64\sqrt{3}$ ft. away from the thrower, neglecting air resistance.
Question 4
If the velocities and accelerations of a number of particles be know, how many the velocity and acceleration of their centre of gravity be found?
A $4$ lb. weight is placed on a smooth table and is attached to a $3$ lb. weight, which vertically by a light inextensible string passing over a smooth pulley at the edge of the table. Show that the acceleration of the centre of gravity of the two weights is constant in magnitude and direction, and that the centre of gravity moves in a straight line.
Question 5
A car weighing one tone is rounding a curve of $160$ yards radius on a level road. What is the greatest speed at which this is possible, without causing the car to overturn, if the wheel gauge is $4 \frac{1}{2}$ ft. and the C.G. is mid-way between the wheels transversely and at a height of $3$ ft. from the ground? What is the frictional force between the road and tyres at this speed?
Question 6
An engine pumps one ton of water per minute to a height of $120$ ft. and delivers it through a pipe whose cross-sectional area is $3$ square inches. Find (i) the work done against gravity per minute in ft. lbs.; (ii) the kinetic energy of the water delivered in one minute in ft. lbs.; (iii) the Horse Power of the engine, neglecting all losses due to friction. (A cubic foot of water weights $62\frac{1}{2}$ lbs.)
Question 7
A pin-jointed framework of the form shown in diagram, is pin-jointed to a vertical wall at $A$ and $C$ and carries a vertical load of $1$ ton at $D$. Find the forces exerted by the frame on the wall, and determine the stresses in the bards, indicating whether the bar is in thrust or in tension in each case.
Question 8
A weight of half-a-tone is allowed to fall freely through a height of $12$ ft. to drive a pile weighing $5$ cwt. into the ground. Find the average resistance to the motion of the pile if it is driven $2$ inches by the blow, assuming that the weight moves on with the pile.
Question 9
A body, weighing $12$ lbs., is kept in equilibrium on a rough plane inclined at $30^\circ$ to the horizontal by a cord inclined at $30^\circ$ to the plane. Find the frictional force between the body and the plane and indicate its direction when the tension in the cord is 6lbs. Find the work done in dragging the body slowly through a distance of one foot up the plane, assuming the coefficient of friction to be $\frac{1}{2}$ and the cord to remain constantly at an angle of $30^\circ$ to the plane.
Question 10
The two balls, $A$ and $B$, of a governor, weigh $10$ lbs. each; the arms $AB$ and $AC$ are each inclined at $30^\circ$ to the vertical and are $1$ ft. long. Find the number of revolutions per minute they are making and the tension in each rod; the weights of the rods, $AB$, $AB$ [sic], may be neglected.
Citation:
State Examinations Commission (2023). State Examination Commission. Accessed at: https://www.examinations.ie/?l=en&mc=au&sc=ru
Malone, D and Murray, H. (2023). Archive of Maths State Exams Papers. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/
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