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Maths Grinds
1 Oct 2018
##### Question 1

$ABC$ is a steel rod, the portion $AB$ being straight and of length $4$ feet and the portion $BC$ being curved. The whole rod lies in one plane. The end $A$ is hinged to the floor and a load of $2$ cwt. is suspended from $C$ by a rope which is $3$ feet from $AB$ which is kept vertical by applying a horizontal force $3\frac{1}{2}$ feet from $A$. The eight of the whole rod may be taken to be $\frac{1}{2}$ cwt. acting at a distance of one foot from $AB$ when the latter is in the vertical position. Find the horizontal force and the reaction at the hinge in magnitude and direction.

##### Question 2

$ABCD$ is a trapezium, $AB$ being parallel to $DC$. $AB=a$, $DC=b$. Show that the centroid of the area is one the line joining the mid-points $E$, $F$ of $AB$, $DC$ and that it divides $EF$ in the ratio $(a+2b):(2x+b)$.

##### Question 3

It is found that when a plane is inclined at $30^\circ$ to the horizontal a block of iron weighing $8$ lb. slides uniformly down the plane. Show that the angle of friction is $30^\circ$.

When the plane is inclined at $45^\circ$ to the horizontal the block is dragged slowly up the plane by a cord inclined at an angle $\theta$ to the plane. By means of the triangle of forces, or otherwise, find the tension in the cord and show that the tension is least when $\theta=30^\circ$.

##### Question 4

A wheel of radius $3$ feet is rolling, without slipping, on a horizontal plane ; the centre moving a uniform velocity of $18$ feet per second. What is the angular velocity of the wheel?

What is the velocity of the extremity of the radius which is inclined at $30^\circ$ to the upward vertical through the centre of the wheel, the angle being measured in the direction of rotation of the wheel?

##### Question 5

A $4$ lb. weight is drawn up a rough plane, coefficient of friction $\frac{1}{4}$, inclined at $30^\circ$ to the horizontal by a string, passing over a light smooth pulley at the top of the plane, to which is attached an $8$ lb. weight. Find the acceleration.

Calculate the kinetic energy when the system has moved a distance of $8$ ft. and show that it is equal to the loss in potential energy minus the work done against friction.

##### Question 6

A car weighing $2000$ pounds starts under the action of a force which decreases uniformly from $240$ lb. wt. at time $t=0$ to $80$ lb. wt. at $t=20$ seconds when it remains constant. The resistances to motion remain constant throughout and are equal to $80$ lb. wt. Show that the accelerating force at time $t$ $(0 \leq t \leq 20)$ is $160-8t$ lb. wt.

Find (a) the velocity at time $t=20$ sec.; (b) the distance travelled at time $t=20$ sec.; (c) the horsepower required when $t=10$ sec. and also when $t>20$ secs.

##### Question 7

A gun fires a projectile with initial velocity $u$ and elevation $\alpha$. Show that (a) the range $r=\frac{u^2}{g}\sin 2 \alpha$; (b) the greatest height $h=\frac{u^2}{2g}\sin^2\alpha$, neglecting air resistance.

Show that the maximum range which can be obtained with the same initial velocity is $2h+\frac{r^2}{8h}$.

##### Question 8

A cyclist is travelling round a curve $100$ feet radius at $15$ miles per hour. At what angle must the track be inclined radially so that there may be no tendency to slip?

If $\mu=0.2$ what is the maximum speed on a level track of the same radius so that skidding may not occur? ($\mu$ is the coefficient of friction).

##### Question 9

If the period of a simple harmonic vibration is $T$ seconds and the amplitude $a$ feet, show that the maximum acceleration is $\frac{4 \pi^2 a}{T^2}$.

A boy weighing $8$ stone standing on a plank oscillates vertically in simple harmonic motion of amplitude of $0.5$ foot and period $1$ second. Find the greatest and least reactions of the plank.

##### 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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