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Maths Grinds
2 Apr 2018
##### Question 1

State the conditions of equilibrium of three forces acting on an extended body in one plane.

$C$ is a point vertically above $A$, the pivot of light rod $AB$, and $AC=AB$. $B$ is attached to $C$ to $C$ by a cord and a mass $m$ is attached to $B$. Show that the thrust on the rod is Independent of the length of the cord and find the tension of the cord when $AB$ makes an angle $\theta$ with the horizontal.

##### Question 2

State the laws of friction.

A ladder with its centre of gravity at its mid-point rests with one end of the ground and the other against a vertical wall. Show that the greatest inclination to the wall consistent with equilibrium is $\tan^{-1} \frac{2\mu}{1-\mu^2}$ where $\mu$ is the coefficient of friction both with ground and wall.

##### Question 3

A body revolves with initial angular velocity $\omega_0$ and uniform angular acceleration $\alpha$: write down equations giving the angular velocity and the angle described in time $t$ and derive and equation not involving $t$.

A wheel is making $n$ revolutions per minute and $t$ seconds later it is found to be making $n’$ revolutions per minute: what is its acceleration (supposed uniform)? How many revolutions has the wheel made in the interval.

##### Question 4

Th resistance to a train weighing $W$ tons and travelling at $v$ miles per hour is $R$ lbs. wt. per tone: find the rate of working of the engine.

The train consumes $w$ tons of coal per hour and the burning of $1$ lb. of coal produces $s$ ft.-lbs. of energy: what proportion of the energy is usefully employed by the engine?

Evaluate when $W=100$, $v=60$, $R=10$, $w=\frac{1}{2}$, $s=10^7$.

##### Question 5

The equation of the path of a projectile referred to horizontal and vertical axes is $y=x-\frac{x^2}{64}$; find the angle at which it was projected and the initial velocity. Find also the direction of motion after $t$ seconds. (Note.-$g$ may be taken as $32$).

##### Question 6

A number of unequal particles are distributed in a straight line: find a formula for the position of the centre of gravity. Two masses $m_1$ and $m_2$ are attached to the ends of a light string passing over a smooth peg: show that the acceleration of the centre of gravity is $\left(\frac{m_1-m_2}{m_1+m_2}\right)^2g$.

##### Question 7

A mass $m$ hangs from a light spiral spring. Show that, if $m$ is pulled down slightly and released, it will move with simple harmonic motion. Find the greatest velocity and the periodic time of $m$, showing clearly on what they depend.

##### Question 8

In a pulley system a weight $W$ is raised with uniform acceleration by means of a load $Q$: show that the ratio of the accelerations of $P$ and $W$ is equal to the velocity ratio of the machine, friction and the weight of the pulleys being neglected.

If in any system of pulleys there is equilibrium when a weight $Q$ is supported by a load $P$, show that if $P$ be increased to $Q$, $W$ will ascend with acceleration $\frac{gP\left(Q-P\right)}{P^2+QW}$.

##### 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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