### My Journal

Maths Grinds
30 Nov 2018
##### Question 1

A bullet of mass $m$ is fired with speed $v$ into a fixed block of wood and is brought to rest in a distance $d$. Find the resistance to motion assuming it to be constant.

Another bullet also of mass $m$ is then fired with speed $2v$ into another fixed block of thickness $2d$, which offers the same resistance as the first block. Find the speed with which the bullet emerges, and the time it takes to pass through the block.

##### Question 2

Prove that the formula $x=ut+\frac{1}{2}at^2$ represents the distance $x$ travelled in time $t$ by a body moving in a straight line with constant acceleration $a$.

A train takes $4\frac{1}{2}$ minutes to travel between two stations $S_1$ and $S_2$ which are $4500$ meters apart. It starts from rest at $S_1$ and finishes at rest at $S_2$, by travelling with uniform acceleration for the first minute and with uniform deceleration for the last $\frac{1}{2}$ minute. Find the train’s constant speed during the remainder of the journey.

If a second train, travelling with a constant speed of $1000$ m/min, in the same direction passes $S_1$ as the first train leaves this station, find when overtaking occurs. (The lengths of the trains may be neglected).

##### Question 3

A particle is projected under gravity with an initial velocity $v_0$ at an angle $\theta$ to the horizontal. Find its position and the direction of motion after time $T$ in terms of $T$ in terms of $v_0$, $\theta$, $g$ and $T$.

A particle is projected from the top of a cliff which is $425$ ft. above sea level and the angle of projection is $45^\circ$ to the horizontal. If the greatest height reached above the point of projection is $200$ ft, find the speed of projection and the time taken to reach this greatest height.

Find when and where the particles strikes the sea.

(Take $g$ to be $32$ ft/sec$^2$).

##### Question 4

Prove that the bob of a simple pendulum moves in simple harmonic motion – stating any assumptions made.

The string of such a pendulum is $2$ ft. long and the bob is released from rest when at a distance $\frac{1}{4}$ ft from the equilibrium position. Calculate the time taken to travel halfway to the equilibrium position and the speed of the bob then.

(Take $g$ to be $32$ ft/sec$^2$).

##### Question 5

By deriving an expression for the necessary acceleration, prove that a particle of mass $m$ moving in a circle of radius $r$ with speed $v$ must have a force of magnitude $\frac{mv^2}{r}$ pointing towards the centre acting on it.

A particle of mass $4$ lbs. moving on the inside smooth surface of a fixed spherical bowl of radius $2$ ft is describing a horizontal circle of radius $\sqrt{3}$ ft. Find the constant speed of rotation and the reaction of the sphere on the particle.

(Take $g$ to be $32$ ft/sec$^2$).

##### Question 6

Show that the centre of gravity of a uniform triangular lamina coincides with the centre of gravity of three equal particles placed at the vertices of the triangle. Hence find the centre of gravity of a uniform trapezium $ABCD$, of weight $W$, in which $2 AB = CD$ and $AD = BC$.

A particle of weight $w$ is attached at $D$ and the system is suspended by a string attached to the midpoint $F$ of $CD$. If in the position of equilibrium $F$ is vertically above $A$ show that $q=\frac{2W}{9}$.

##### Question 7

A particle of mass $10$ lbs. is placed on a rough inclined plane. The least force acting up the plane which will prevent the particle slipping down the plane is $2$ lbs. weight. The least force acting up the plane which will make the particle slip upwards is $10$ lbs wt. Show that the coefficient of friction is $\frac{1}{2}$ and that the inclination of the plane is $\alpha$ where $\sin\alpha=\frac{3}{5}$.

Find the least force required to move the particle up the plane.

##### Question 8

Two equal uniform rods $AB$ and $BC$ each of weight $W$ are freely jointed at $B$. The system is suspended freely from $A$ and a horizontal force $\frac{W\sqrt{3}}{2}$ is applied at the lowest point $C$. If in the equilibrium position the inclination of $AB$ to the downward vertical is $30^\circ$, find the corresponding inclination of $BC$ and the supporting force at $A$.

##### Question 9

A small uniform cylinder of density $\rho$, mass $m$, total length $l$ and uniform cross section floats in a liquid of density $2\rho$ with its axis vertical. Find the thrust on the cylinder when it is displaced vertically in the liquid, without being completely immersed, through a distance $x$ from the equilibrium position. Show that if it is released in this position, it will oscillate with simple harmonic motion of period $2\pi\sqrt{\frac{l}{2g}}$.

##### 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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