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Maths Grinds
10 Dec 2018

## Question 1

A racing car covers a journey of $8.8$ km from rest to rest. It accelerates uniformly in the first minute to reach its maximum speed of $40$ m/s, it holds this speed for a certain time and then slows uniformly to rest with a retardation of magnitude three times that of the acceleration.

i) Draw a rough velocity-time graph and find the distance travelled in the three stages of the journey and the total time taken.

ii) If the maximum speed over the final kilometre of the journey had been restricted to $20$ m/s, show that the time taken from rest to rest would have been at least $22.5$ s longer than before, assuming the same rates of acceleration and deceleration as before.

[Video Solution]

## Question 2

From a point $p$ on horizontal ground an elastic particle is projected under gravity with a velocity of $\left(28\vec{i}+21\vec{j}\right)$ m/s, where $\vec{i}$ and $\vec{j}$ are unit vectors along the horizontal and upward drawn vertical, respectively.

i) Find the displacement $\vec{r}$ from $p$ at time $t$ seconds afterwards, and in particular when the particle is at its highest point.

ii) If at this point [the highest point] the particle strikes a fixed vertical wall, where the coefficient of restitution is $\frac{1}{2}$, find how far from $p$ the particle strikes the ground.

[Video Solution]

## Question 3

a) State the laws governing oblique, perfectly elastic collision between two spheres.

[Video Solution]

b) A small sphere collides obliquely with a similar sphere of equal mass at rest – both spheres being smooth and perfectly elastic.

i) Show that the paths of the two spheres after the collision are at right angles.

ii) Prove that there is no loss in kinetic energy.

[Video Solution]

## Question 4

A wedge of mass $2M$ rests on a smooth horizontal table with one of its plane faces inclined at $45^\circ$ to the horizontal. This plane face is smooth and on it is placed a particle of mass $M$ and the system is released from rest.

i) Draw separate diagrams showing the forces acting on the particle and on the wedge during the motion.

ii) By considering the acceleration of the particle in two components, one component down the plane and the other horizontal, show that the acceleration of the wedge is $\frac{g}{5}$ and that of the particle is $g\frac{\sqrt{13}}{5}$.

[Video Solution]

## Question 5

a) Using the usual notation, prove that

$$v^2=\omega^2\left(A^2-x^2\right)$$

represents simple harmonic motion.

[Video Solution]

b) A light flexible elastic string of natural length $2$ m and elastic constant $12$ N/m has one end $p$ tied to a point on a smooth horizontal table. To its other end is attached a particle of mass $3$ kg, which is placed on the table at a point $q$, distance $3.5$ m from $p$ and released from rest. Show that in the first stage of motion the particle moves with simple harmonic motion of period $\pi$ seconds and that the time taken to reach $p$ is $\left(\frac{\pi}{4}+\frac{2}{3}\right)$ s.

[Video Solution]

## Question 6

a) Show that a particle moving in a circle with constant speed is being accelerated towards the centre of the circle.

[Video Solution]

b) A small ring of mass $1$ kg is threaded on a smooth light flexible inelastic string of length $0.8$ m. The ends of the string are attached to two fixed points, distance $0.4$ m apart in the same vertical line, and the ring describes with constant speed a horizontal circle whose centre is the lower fixed point. Find the constant speed of rotation and show that the tension in the string is $12.25$ N.

[Video Solution]

## Question 7

A uniform rod of length $4$ m and weight $100$ N is smoothly hinged at one end to a rough horizontal floor. The rod rests on the smooth curved surface of a hemisphere whose plane face is on the floor. The rod is in equilibrium inclined at $45^\circ$ to the horizontal and the hemisphere, of weight $50$ N and radius $1$ m, is in limiting equilibrium.

i) Show in separate diagrams the forces acting on the rod and on the hemisphere.

ii) Find the reaction between the rod and the hemisphere,

iii) and prove that the coefficient of friction between the latter and the floor is $\frac{2}{3}$.

[Video Solution]

## Question 8

a) Prove that the moment of inertia of a uniform rod $ab$ of mass $M$ and length $2l$ about an axis through $a$, perpendicular to the rod, is $\frac{4}{3}Ml^2$.

[Video Solution]

b i) Such a rod [from part (a)] is free to rotate in a vertical plane about a fixed horizontal axis at $a$, with a particle of mass $2M$ attached to the rod at $b$. The system is released from rest with the rod vertical and the end $b$ above $a$. Show that the angular speed of the rod when it is next vertical is $\left(\frac{15g}{7l}\right)^\frac{1}{2}$.

ii) At this point the particle falls off. Find the height to which the end $b$ subsequently rises.

[Video Solution]

## Question 9

A particle of mass $M$ kg is projected vertically upwards from ground level with a speed of $70$ m/s. In addition to the weight of the particle, there is the resistance force of the air of magnitude $\frac{1}{20} M v^2$ newtons when the speed if $v$ metres per second. Show that the equation of motion during the upward journey is

$$-20\frac{dv}{dx} = \frac{v^2+196}{v}.$$

Prove that the maximum height is $\left(10 \ln 26 \right)$ m, and that the time taken to reach it is approximately $1.95$ s.

(Note: $\ln 26 = \log_e 26$)

[Video Solution]

## Question 10

a) State the condition for the equilibrium of a floating body.

[Video Solution]

b) A uniform rectangular board $abcd$ of weight $W$, floats with the diagonal $ac$ on the surface of the of the water, the lowest corner $b$ being attached to the bottom of the vessel by a light inelastic string.

i) Show in a diagram the forces acting on the board and prove that the specific gravity of the board is $\frac{1}{3}$.

ii) Find the tension in string.

[Video Solution]

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### 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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