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Maths Grinds
28 Sep 2018
##### Question 1

Two ships $A$ and $B$ are $10$ miles apart, $A$ being due west of $B$ and steaming due East at $12$ miles per hour. $B$ is steaming due North at $18$ miles per hour. When will they be nearest one another and what will be their distance apart at that time?

##### Question 2

A particle slides down a smooth inclined plane $12$ feet long in $4$ seconds : find its acceleration and the inclination of the plane to the horizontal.

How far from the bottom should a second particle be started on the plane one second after the first had started from the top so that the two particles will reach the bottom together?

##### Question 3

On a rectangular sheet of iron, $12$ inches by $9$ inches, marks are made on two adjacent edges at a distance of $3$ inches from one corner. These marks are joined by a straight line and the sheet folded down along this line. Find the centre of gravity of the sheet so folded.

##### Question 4

Describe any method by which the acceleration due to gravity may be determined experimentally. Mention the causes of error likely to affect the result.

##### Question 5

An object of $18$ lb. wt. is placed upon a sheet of glass resting on a horizontal table and a string attached to the object passes horizontally over a pulley at the end of the table. When a weight of $2$ lb. is hung on the free end of the string, the object is drawn from rest through a distance of $6$ feet in $3$ seconds. Find the coefficient of friction between the object and the glass and also the tension in the string.

##### Question 6

A uniform iron bar $AB$, $5$ ft. long, weighs $10$ lb., has a $5$ lb. weight fixed to it at a point $1$ ft. from $A$, and is supported by means of cords attached to its ends from a peg $C$. The cord $CA$ is $3\frac{1}{2}$ ft. long and the cord $CB$ is $4$ ft. long. Find the inclination of the bar to the horizontal in the position of equilibrium and find the tensions in the cords.

##### Question 7

The height of a lift from the ground $t$ seconds after starting is $h$ feet, given by the following table :-

$t$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$
$h$ $1.6$ $4.5$ $8.5$ $14.4$ $22.7$ $33.3$ $44$ $54.8$ $63$ $69$ $72.5$ $74$

Draw the distance-time graph for the lift. Find the greatest velocity of the lift.

Draw a rough velocity-time graph, and use it to give a short account of the variation in the acceleration throughout the journey.

##### Question 8

An inverted cone, of angle $60^\circ$, has a smooth inside surface and rests with its axis vertical. A smooth particle moves in a horizontal circle on the inside of the cone with a uniform speed of $8$ feet per second. Find the distance of the particle from the vertex of the cone.

##### Question 9

A particle is moving in a straight line with Simple Harmonic Motion. When it is $5$ ft. and $12$ ft. from the centre of its path, its speed are $24$ ft./sec. and $10$ ft./sec. respectively. Find the period and the amplitude of the motion. Find also the maximum speed and the maximum acceleration of the particle.

##### 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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