Leaving Cert Applied Maths Higher Level 1976
Question 1
(a) Show that, if a particle is moving in a straight line with constant acceleration $k$ and initial speed $u$, the distance travelled in time $t$ is given by $s=ut+\frac{1}{2}kt^2$.
(b) Two points $a$ and $b$ are a distance $l$ apart. A particle starts from $a$ and moves towards $b$ in a straight line with initial velocity $u$ and constant acceleration $k$ A second particle starts at the same time from $b$ and moves towards $a$ with initial velocity $2u$ and constant deceleration $k$. Find the time in terms of $u$, $l$ at which the particle collide, and the condition satisfied by $u$, $k$, $l$ if this occurs before the second particle returns to $b$.
Question 2
A particle is projected upwards with a speed of $35$ m/s from a point $O$ on a plane inclined at $45^\circ$ 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 $\phi$.
(i) Write down the velocity of the particle and
(ii) its displacement from $O$, in terms of $\vec{i}$ and $\vec{j}$, after time $t$ seconds.
(iii) If the particle is moving horizontally when it strikes the plane at $q$ prove that $\cot \phi = 3$ and
(iv) calculate $|oq|$.
Question 3
The diagram shows a light inelastic string, passing over a fixed pulley $B$, connecting a particle $A$ of mass $3M$ to a light movable pulley $C$. Over this pulley passes a second light inelastic string to the ends of which are attached particles $D$, $E$ of masses $2M$, $M$ respectively.
(i) Show in separate diagrams the forces acting on $A$, $D$ and $E$.
(ii) Write down the three equations of motion involving the tensions $T$, $S$ in the strings, the acceleration of $A$ and the common acceleration of $D$, $E$ relative to $C$.
(iii) Show that $T=2S=\frac{48Mg}{17}$.
Question 4
A light smooth ring of mass $M$ 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 $l$ from the write, to a particle of mass $2M$ 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 $x$ while the particle rises through a distance $y$
(i) show that
$x^2 = y^2 + 2ly$ and $\left(l+y\right)\dot{y} = x \dot{x}$
where $\dot{x}=\frac{dx}{dt}$, $\dot{y}=\frac{dy}{dt}$ are the speeds of the ring and particle respectively.
Find $\dot{x}$ when
(ii) $x=l$ and
(iii) when $x=\frac{4l}{3}$.
Question 5
(a) State the laws governing the oblique collisions of elastic spheres.
(b) A sphere of mass $M$ moving with speed $u$ collides obliquely with a second smooth sphere at rest. The direction of motion of the moving sphere is inclined at $45^\circ$ to the line of centres at impact, and the coefficient of restitution is $\frac{1}{2}$. After impact the directions of motion of the spheres are at right angles.
Find the mass of the second sphere in terms of $M$, and the velocities of the two spheres after impact in terms of $u$. Hence show that one quarter of the kinetic energy is lost.
Question 6
Two uniform rods $ab$, $bc$ of lengths $2l$, $2r$ and of weights $2W$, $3W$ respectively are smoothly hinged together at $b$. They stand in equilibrium in a vertical plane with the end $a$ resting on rough horizontal ground and the end $c$ resting against a smooth vertical wall. The point $a$ is farther from the wall than $b$ and the rods $ab$, $bc$ are inclined at angles $\alpha$, $45^\circ$ respectively to the horizontal where $\alpha > 45^\circ$.
(i) Show in separate diagrams the forces acting on each rod.
(ii) By considering separately the equilibrium of the system $abc$ and the rod $bc$, find the coefficient of friction at $a$ and
(iii) show that $\tan \alpha = \frac{8}{3}$.
Question 7
(a) Define simple harmonic motion.
(b) A particle of mass $2$ kg is attached to the ends of two light elastic strings, each of natural length $1$ m and elastic constant $49$ N/m. The other ends of the two strings are attached to two fixed points $a$ and $b$ in the same vertical line, where $a$ is $4$ m above $b$. The particle when is released from rest from the midpoint of $ab$.
(i) By considering the forces acting on the particle when it is $x$ metres from $a$, where $2 < x < 2.4$, show that it is moving with simple harmonic motion.
(ii)Find the least time taken for the particle to reach the point $x=2.3$, and find its speed there.
Question 8
A pendulum of a clock consists of a thin uniform rod $ab$ of mass $M$ and length $6l$ to which is rigidly attached a uniform circular disc of mass $4M$ and radius $l$ with the centre of the disc being at the point $c$ on $ab$ where $bc=l$.
(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 $a$.
(ii) It is released from rest with $ab$ horizontal. Find the speed of $b$ when $ab$ is vertical.
Question 9
An atomic nucleus of mass $M$ is repelled from a fixed point $o$ by a force $M k^2 x^{-5}$, where $x$ is the distance of the nucleus from $o$ and $k$ is a constant. It is projected directly towards $o$ with speed $\frac{2k\sqrt{3}}{d^2}$ from a point $a$ where $|oa|=d$. Find the speed of the nucleus when it reaches the midpoint of $oa$ and find how near it gets to $o$.
Question 10
(a) Using Taylor’s theorem find the first two terms in the Taylor series for $e^{x^2}$ in the neighbourhood of $x=0$, i.e. the Maclaurin series for $e^{x^2}$.
(b) State Archimedes principle for a body wholly or partly immersed in a liquid.
(c) A uniform thin rod is of length $2a$, of weight $4W$ and specific gravity $\frac{4}{9}$. 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 $\frac{2a}{3}$ 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 (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/
