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### Maths Grinds' Blog

Blogging and Mathing
17 Oct 2018
##### Question 1

If the velocities of two bodies $A$ and $B$ are known, indicate by means of a diagram, or otherwise, how the velocity of $A$ relative to $B$ may be found.

When a man is travelling due east at $4$ miles per hour, the wind appears to blow from the north, but when he is travelling north east at $16$ miles per hour, it appears to blow from a point $30^\circ$ east of north. Find the velocity of the wind in magnitude and direction.

##### Question 2

Two masses, $A$ and $B$, weighing $2$ lb. and $1$ lb. respectively, are connected by means of a light string passing over a smooth pulley. $A$ is at rest on the floor and $B$ is handing freely. The mass $B$ is raised a vertical distance of $8$ feet and is then allowed to fall freely. Find how logn after it has been jerked into motion it will take the mass $A$ to return to the floor. Find, also, the kinetic energy of the system immediately before, and immediately after, the mass $A$ is jerked into motion.

##### Question 3

An engine is travelling round a curve of radius $480$ feet on a horizontal railway track, the rails of which are at the same level and $5$ feet apart. If the centre of gravity of the engine is $6$ feet above the level of the rails and is in vertical plane which bisects the distance between them, find the greatest speed wt which the engine can travel found the curve without losing contact with one of the rails.

Find, also, the distance which the outside rail must be raised so that there may be no lateral thrust on the rails when the engine is travelling at $30$ milers per hour.

##### Question 4

One end of a uniform ladder, weighing $60$ lb., rests on a rough plane inclined at angle of $20^\circ$ to the horizontal, and the other end rests against a smooth vertical wall at the top of the plane. The coefficient of friction between the ladder and the plane is $\frac{1}{\sqrt{3}}$. If the ladder is one the point of slipping, find its inclination to the wall, also, its total reaction on the plane.

##### Question 5

A train, starting from rest at the bottom of an incline of $1$ in $32$ and moving with uniform acceleration, attains a speed of $30$ miles per hour when it has travelled one mile up the incline. If the engine weighs $40$ tons and the rest of the train $120$ tons, and if the resistances to motion are equivalent to $12$ lb. wt. per ton, find the tension in the coupling between the engine and the rest of the train. Find, also, the H.P. at which the engine is working when the train attains the speed of $30$ miles per hour up the incline.

##### Question 6

A piece of metal of uniform thickness is in the shape of a square of side $4$ inches together with an isosceles triangle whose base is one od sides of the square. The centre of gravity of the piece of metal is in the base of the isosceles triangle. The top of the triangular portion is removed by cutting along a line which joins the middle points of the equal sides of the isosceles triangle. Find how far the centre of gravity of the remaining piece is from the intersection of the diagonals of the square.

##### Question 7

The side-wall of a house is $48$ feet high, and the roof which is smooth makes an angle of $30^\circ$ with the horizontal. An object on the roof slides from rest at a point $16$ feet from the bottom edge down along the line of greatest slope and then falls freely to the ground. With what velocity, in magnitude and direction, and how far from the bottom of the side-wall, will the object strike the ground?

##### Question 8

What do you understand by “simple harmonic motion”?

When a particle moving with simple harmonic motion is travelling towards its mean position and is at a distance of $5$ feet from it, its velocity and acceleration are $15$ feet per sec. and $15$ feet per sec.$^2$, respectively. Find (a) the amplitude, (b) the periodic time, (c) the least time it will take the particle to reach its mean position.

##### Question 9

The walls of a rectangular swimming bath are vertical and the floor slopes uniformly so that the water is $6$ feet deep at one end and $18$ feet deep at the other end. The bath is $150$ feet long and $60$ feet wide. Find the total thrust of the water (a) on the wall at the deep end, (b) on one of the side-walls.

A line drawn parallel to the surface of the water on the wall at the deep end divides the wall into two portions, the thrusts on which are equal. Find how far the line is below the surface of the water.

[$1$ cu. ft. of water weighs $62\frac{1}{4}$ lb.]

##### 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/

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