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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