Leaving Cert Applied Maths Higher Level 1976
Question 1
(a) Show that, if a particle is moving in a straight line with constant acceleration and initial speed
, the distance travelled in time
is given by
.
(b) Two points and
are a distance
apart. A particle starts from
and moves towards
in a straight line with initial velocity
and constant acceleration
A second particle starts at the same time from
and moves towards
with initial velocity
and constant deceleration
. Find the time in terms of
,
at which the particle collide, and the condition satisfied by
,
,
if this occurs before the second particle returns to
.
Question 2
A particle is projected upwards with a speed of m/s from a point
on a plane inclined at
to the horizontal. The plane of projection meets the inclined plane in a line of greatest slope and the angle of of projection, measured to the inclined plane, is
.
(i) Write down the velocity of the particle and
(ii) its displacement from , in terms of
and
, after time
seconds.
(iii) If the particle is moving horizontally when it strikes the plane at prove that
and
(iv) calculate .
Question 3
The diagram shows a light inelastic string, passing over a fixed pulley , connecting a particle
of mass
to a light movable pulley
. Over this pulley passes a second light inelastic string to the ends of which are attached particles
,
of masses
,
respectively.
(i) Show in separate diagrams the forces acting on ,
and
.
(ii) Write down the three equations of motion involving the tensions ,
in the strings, the acceleration of
and the common acceleration of
,
relative to
.
(iii) Show that .
Question 4
A light smooth ring of mass is threaded on a smooth fixed vertical wire and is connected by a light inelastic string, passing over a fixed smooth peg at a distance
from the write, to a particle of mass
hanging freely. The system is released from rest when the string is horizontal. Explain why the conservation of energy can be applied to the system. If the ring descends a distance of
while the particle rises through a distance
(i) show that
and
where ,
are the speeds of the ring and particle respectively.
Find when
(ii) and
(iii) when .
Question 5
(a) State the laws governing the oblique collisions of elastic spheres.
(b) A sphere of mass moving with speed
collides obliquely with a second smooth sphere at rest. The direction of motion of the moving sphere is inclined at
to the line of centres at impact, and the coefficient of restitution is
. After impact the directions of motion of the spheres are at right angles.
Find the mass of the second sphere in terms of , and the velocities of the two spheres after impact in terms of
. Hence show that one quarter of the kinetic energy is lost.
Question 6
Two uniform rods ,
of lengths
,
and of weights
,
respectively are smoothly hinged together at
. They stand in equilibrium in a vertical plane with the end
resting on rough horizontal ground and the end
resting against a smooth vertical wall. The point
is farther from the wall than
and the rods
,
are inclined at angles
,
respectively to the horizontal where
.
(i) Show in separate diagrams the forces acting on each rod.
(ii) By considering separately the equilibrium of the system and the rod
, find the coefficient of friction at
and
(iii) show that .
Question 7
(a) Define simple harmonic motion.
(b) A particle of mass kg is attached to the ends of two light elastic strings, each of natural length
m and elastic constant
N/m. The other ends of the two strings are attached to two fixed points
and
in the same vertical line, where
is
m above
. The particle when is released from rest from the midpoint of
.
(i) By considering the forces acting on the particle when it is metres from
, where
, show that it is moving with simple harmonic motion.
(ii)Find the least time taken for the particle to reach the point , and find its speed there.
Question 8
A pendulum of a clock consists of a thin uniform rod of mass
and length
to which is rigidly attached a uniform circular disc of mass
and radius
with the centre of the disc being at the point
on
where
.
(i) Using the parallel axes theorem for the disc, show that the moment of inertia of the pendulum is free to oscillate in a vertical plane about such a fixed horizontal axis at .
(ii) It is released from rest with horizontal. Find the speed of
when
is vertical.
Question 9
An atomic nucleus of mass is repelled from a fixed point
by a force
, where
is the distance of the nucleus from
and
is a constant. It is projected directly towards
with speed
from a point
where
. Find the speed of the nucleus when it reaches the midpoint of
and find how near it gets to
.
Question 10
(a) Using Taylor’s theorem find the first two terms in the Taylor series for in the neighbourhood of
, i.e. the Maclaurin series for
.
(b) State Archimedes principle for a body wholly or partly immersed in a liquid.
(c) A uniform thin rod is of length , of weight
and specific gravity
. The rod rests in equilibrium in an inclined position partly immersed in water with its lower end freely pivoted to a fixed point at depth
below the surface of the water. Show in a diagram the forces acting on the rod and calculate the inclination of the rod to the vertical.
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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/