Leaving Certificate Examination 1937 Honours Applied Mathematics
Question 1
A body of mass 
units moves in a straight line so that its distance 
from a fixed point in the line of motion at time 
is given by the equation : 
.
Derive formulae for its velocity, acceleration and momentum at any time.
Given that when 
the velocity, acceleration and momentum at any time.
Given that when the velocity in 
units and that the distances travelled when 
, and 
are 
and 
respectively, determine the values 
, 
and 
in the above equation.
Question 2
A mass weighing 
ton drops without rebound from a height of 
ft. on to the top of a pile weighing 
cwt., and the pile is driven 
inches into the ground by the blow.
Calculate in ft.lbs. the energy lost in the impact, and the average resistance offered by the ground to the motion of the pile.
Question 3
Calculate the effective h.p. delivered by the engine of a ton motor vehicle in each of the following cases :-
(a) The vehicle is ascending an incline of 
in 
with a uniform speed of 
miles per hour;
(b) the vehicle accelerates uniformly from a speed of 
miles per hour to a speed of 
miles per hour while travelling a distance of 
yards along a level road.
The frictional resistances to motion may be taken as the same in both cases and equal to a force of lbs. wt.
Question 4
State the laws of statical friction.
A uniform ladder 
feet long and weighing 
lbs. is on the point of slipping when placed with its upper end against a smooth vertical wall and its lower end on a rough horizontal surface at a distance of feet from the wall.
Calculate the value of the frictional force between the foot of the ladder and the ground and also the value of the coefficient of friction.
Question 5
A point 
describes a circular path of radius 
with a uniform velocity. 
is the foot of the perpendicular from to a fixed diameter of the circle. Show that the velocity and acceleration of at any instant are equal to 
and 
respectively, where 
is the displacement of fron [sic] the centre of the circle, is the time for this displacement and 
is the angular velocity of .
(b) Show that for small oscillations the bob of a simple pendulum may be regarded as having a similar motion of that of and that the time of oscillation is 
where 
is the length of the pendulum.
Question 6
Through what vertical height will the bob of a conical pendulum drop if its speed falls from to revolutions per minute?
Question 7
A train is travelling at the rate of miles per hour round a circular track of 
yards radius. The horizontal distance between the rails is 
inches. How much must the outer rail be raised [above the inner one so that there may be no side thrust on the rails ?
Question 8
The force acting on a body moving in a straight line is given at any instant by the formula
where 
is in lbs.-st. and is the distance in feet of the body at that instant from a fixed point in the line. If the motion is resisted by a constant force of 
lbs.-wt., find the kinetic energy stored in the body in moving from 
to 
.
Question 9
A particle of mass 
lbs. is placed on a smooth plane inclined at 
to the horizon and is connected by a light string passing over the top of the plane to a particle weighing lbs. which hangs vertically.
Find (a) the common acceleration;
(b) the velocity after seconds of the centre of inertia of the particles.
Question 10
A body is projected up a smooth incline of with an initial velocity of feet per second. If the length of the incline is 
feet find the greatest height attained by the body and the range on a horizontal plane passing through the foot of the incline.
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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