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Maths Grinds
26 Nov 2018
##### Question 1

(a) Represent any two velocities $\vec{v_1}$ and $\vec{v_2}$ by a vector diagram, and illustrate the vectors $(\vec{v_1}+\vec{v_2})$ and $(\vec{v_1}-\vec{v_2})$.

(b) When the sun, S, is $30^\circ$ above the horizon, an aeroplane $A$ glides in for landing with the sun behind, moving with a speed of $80$ m.p.h. along a path sloping $15^\circ$ down from the horizontal, as in the diagram. Find the speed of the plane’s shadow on level ground ($PQ$).

##### Question 2

Write a short note on linear momentum.

A uniform cube of wood of mass $10$ lbs. hangs vertically by a long light string fixed in the centre of one face of the cube. A bullet of mass half an ounce moving horizontally with velocity $\vec{v}$ strikes the centre of another face in a direction perpendicular to the face, embeds itself in the wood and causes the combined mass to swing. If in the first swing the centre of gravity of the combined mass rises $6$ inches, calculate the magnitude of $\vec{v}$.

##### Question 3

For a body moving in a straight line with constant acceleration $\vec{f}$, initial velocity $\vec{u}$ and velocity $\vec{v}$ at time $t$, show that $v=u+ft$ and $a=ut+\frac{1}{2}ft^2$.

A cage travels down a vertical mine-shaft $750$ yards deep in $45$ seconds, starting from rest. For the first quarter of the distance only the velocity undergoes a constant acceleration, and for the last quarter of the distance the velocity undergoes a constant retardation, the acceleration and retardation having equal magnitudes. If the velocity, $\vec{v_1}$, is uniform while traversing the central portion of the shaft, find the magnitude of $\vec{v_1}$.

##### Question 4

The co-ordinates $(x,y)$ of a moving particle at any time $t$ are given by:

$x=2\cos\frac{t}{2}$, $y=2\sin\frac{t}{2}$.

Deduce that its motion is circular.

A particle is moving along the circumference of a circle with constant speed. Show that that components of the motion of the particle along any two perpendicular diameters are Simple Harmonic Motions.

##### Question 5

A particle is projected horizontally from a height with velocity $\vec{v}$ and falls under gravity. If $(x,y)$ are the co-ordinates of the particle at any time, and $(0,0)$ the co-ordinates of the point of projection, show that $y=-\frac{gx^2}{2v^2}$ us the equation of its path, the x-axis being horizontal and the y-axis vertical.

A marble rolls off the top of a stairway horizontally and in a direction perpendicular to the edge of each step, with a speed of $20$ feet per second. Each step is $6$ in. high and $8$ in. wide. Which is the first step struck by the marble (assuming the stairway is sufficiently high)?

##### Question 6

Masses of $1$ lb., $2$ lbs., $3$ lbs., and $4$ lbs. are placed at points the co-ordinate of which are $(4,1)$, $(-3,2)$, $(2,-3)$, and $(-1,4)$ respectively. Locate the centre of gravity (or the centre of mass) of the system.

Two discs of radii $r$ and $\frac{r}{2}$ are placed flat on a horizontal plane with a distance $d>2r$ between their centres. If $X$ is their centre of gravity and $O$ the centre of gravity of the large disc calculate the length of $OX$.

A square hole is punched out of a circular disc (radius $\alpha$), a diagonal of the square being a radius of the circle. Show that the centre of gravity of the remainder of the disc is at a distance $\frac{\alpha}{4\pi-2}$ from the centre of the circle.

##### Question 7

Explain “normal reaction” and “angle of friction”.

Deduce $\theta \leq \lambda$ where $\lambda$ is the angle of friction and $\theta$ the angle between the normal reaction and the reaction (resultant reaction).

A uniform rod $AB$ $4$ feet long leans against the smooth edge of a table three feet in height with one end of the rod on a rough horizontal floor (as in diagram). If the rod is one the point of slipping when inclined at an angle of $60^\circ$ to the horizontal, find the coefficient of friction.

##### Question 8

Two particles $P$ and $Q$ move with velocities $\vec{v_1}$ and $\vec{v_2}$ respectively. Use diagrams to illustrate clearly the velocity of $P$ relative to $Q$ when $\vec{v_1}$ and $\vec{v_2}$ are

(i) parallel and in the same direction,

(ii) parallel and in opposite directions,

(iii) non-parallel.

A steamer going north-east at $14$ knots observes a cruiser which is $10$ miles south-east from the steamer and which is travelling in a direction $30^\circ$ east of north at $25$ knots.

Illustrate by diagram the cruiser’s course as i appears to the steamer. Find the velocity of the cruiser relative to the steamer, in magnitude and direction, and find also the shortest distance between the two ships.

##### Question 9

Establish an expression for the pressure at a depth $d$ in a liquid of density $\rho$ at rest under gravity. Hence show that the resultant thrust on a plane surface immersed vertically in the liquid is the product of the area of the plane surface and the depth of its centre of gravity.

Find the resultant horizontal thrust on a rectangular lockgate $20$ feet wide when the water stands at a depth of $8$ feet on one side and $4$ feet on the other.

(1 cu. ft. of water weighs $62\frac{1}{2}$ lbs.).

##### 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/

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