1Dec2018

Leaving Cert Applied Maths Higher Level 1971

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Special Thanks: A copy of this Leaving Cert Applied Maths Higher Level 1971 exam was kindly provided by Noel Cunningham.

Question 1

a) Explain how a graph of velocity plotted against time can be used to calculate acceleration and distance travelled, with particular reference to motion with constant acceleration.

[Video Solution]

b) A pigeon in flight releases a small stone from its beak at a height of $50$ metres when its velocity is $u$ . If the stone takes $3\frac{1}{2}$ seconds to reach the ground, show that the direction of $u$ is not horizontal and compute the greatest height reached by the stone after release. (Give your answer correct to the nearest tenth of a metre.)

[Video Solution]

Question 2

A particle is projected from a point $O$ on a plane inclined at $60^\circ$ to the horizontal with a velocity $u=7\sqrt{3}\vec{i}+4.9\vec{j}$ metres/second where $\vec{i}$ is a unit vector through $O$ pointing upward along the line of greatest slope in the plane and $\vec{j}$ is a unit vector perpendicular to the plane.

i) Show that after time $t$ seconds the position vector, $\vec{r}$ of the particle relative to $O$ is given by

$\vec{r}=\frac{7}{20}\left[\left(20\sqrt{3}t-7\sqrt{3}t^2\right)\vec{i}+\left(14t-7t^2\right)\vec{j}\right]\mbox{ metres.}$

ii) Prove that the range on the inclined plane is $\frac{21\sqrt{3}}{5}$ metres,

iii) and find the velocity of the particle when it strikes the plane.

[Video Solution]

Question 3

Two smooth spheres $A$ and $B$ of equal radii but of masses $20$ kg and $10$ kg respectively collide on a smooth horizontal table. Before collision the velocity of $A$ is $\left(5\vec{i}+3\vec{j}\right)$ m/s and the velocity of $B$ is $2\vec{i}$ m/s where $\vec{i}$ and $\vec{j}$ are unit perpendicular vectors in the plane of the table and $\vec{i}$ lies along the line of centres at impact. If the collision is perfectly elastic find the velocity of $A$ and the velocity of $B$ immediately afterwards.

[Video Solution]

Question 4

A light inextensible string which is fastened at one end to a point in the ceiling passes under a smooth movable pulley $A$ of mass $13$ kg, then over a smooth fixed pulley $B$ , and a particle $C$ of mass $9$ kg hangs freely from its other side. All parts of the string which are not touching the pulleys are vertical.

i) When the system is released from rest show that the acceleration of the particle is $2$ m/s $^2$ and that the acceleration of the pulley is half this.

ii) Find also the tension in the string.

[Video Solution]

Question 5

a) Define simple harmonic motion in a straight line and show it can be described by the differential equation $\frac{d^2 x}{dt^2} = -\omega^2 x$ . Prove that $x=A\cos\omega t$ , where $A$ is a constant, is a solution of this equation.

[Video Solution]

b) A particle describing simple harmonic motion on a straight line has velocities $4$ m/s and $2$ m/s when at a distance of $1$ m and $2$ m respectively from the centre of oscillation.

i) Find the amplitude and periodic time of the motion.

ii) Calculate the least time taken for the particle to travel from the position of rest to the point where the velocity was $2$ m/s.

[Video Solution]

Question 6

a) i) Show that $\vec{r} = A \cos \omega t \vec{i} + A \sin \omega t \vec{j}$ , where $A$ , $\omega$ are constants and $\vec{i}$ , $\vec{j}$ are constant unit perpendicular vectors, is the position vector at time $t$ of a particle moving in a circle.

ii) Show that the velocity $v$ is of magnitude $\omega r$ and is at right angles to $\vec{r}$ .

iii) Prove that the acceleration is of magnitude $A\omega^2$ and acts towards the centre of the circle.

[Video Solution]

b) A particle of mass $10$ kg is describing a circle on a smooth horizontal table. It is connected by a light inelastic string of length $0.5$ m to a point which is $0.4$ m vertically above the centre of the circle. If the reaction of the table on the mass is $18$ N, calculate:

i) the constant angular velocity of the particle,

(ii) the tension of the string.

[Video Solution]

Question 7

a) Two rods $XY$ , $YZ$ , of equal lengths but of weights $\frac{1}{2}W$ and $W$ , respectively, are freely hinged together at $Y$ . They stand in equilibrium in a vertical plane with the ends $X$ and $Z$ on a rough horizontal plane with the angle $XYZ$ equal to $90^\circ$ .

i) Show that the vertical components of the action of the plane at $X$ and $Z$ are $\frac{5}{8}W$ and $\frac{7}{8}W$ respectively.

ii) Show in a separate diagram the forces acting on each rod, using vertical and horizontal components for the reaction at $Y$ . Find all these forces and prove that if one rod is on the point of slipping, the coefficient of friction is $\frac{3}{5}$ .

[Video Solution]

Question 8

i) Prove that the moment of inertia of a uniform circular lamina of mass $M$ and radius $r$ about an axis through its centre $c$ , perpendicular to the plane of the lamina, is $\frac{1}{2} M r^2$ .

[Video Solution]

ii) Deduce that the moment of inertia about a parallel axis through a point of the circumference is $\frac{3}{2} M r^2$ .

[Video Solution]

iii) Such a lamina is free to rotate in a vertical plane about a horizontal axis perpendicular to its plane through a point $a$ on its circumference. It is released from rest with $ac$ horizontal. Find the angular velocity of the lamina where $ac$ makes an angle $\theta$ with the download vertical, show that the speed of $c$ at its lower point is $\left(\frac{4gr}{3}\right)^\frac{1}{2}$ .

[Video Solution]

Question 9

a ) State Archimedes Principle for a body immersed in a liquid.

[Video Solution]

b) A uniform rod $ab$ in equilibrium is inclined to the vertical with one quarter of its length immersed under water and its upper end $a$ supported by a vertical force $F$ .

i) Show that in a diagram the three force acting on the rod, and prove that the specific gravity of the rod is $\frac{7}{16}$ .

ii) Express $F$ in terms of $W$ , the weight of the rod.

[Video Solution]

Question 10

A particle of mass $M$ is projected with speed $u$ along a smooth horizontal table. The air resistance to motion when the speed of the particle is $v$ , is $Mkv$ where $k$ is a constant.

i) By solving the equation of motion for the particle, show that $v=ue^{-kt}$ and

ii) prove that as time increases indefinitely, the distance travelled ultimately approaches $\frac{u}{k}$ .

iii) Find the time taken for the particle to travel half this distance.

[Video Solution]

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