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Maths Grinds
20 Dec 2018

Question 1

A cyclist has a maximum acceleration of $2$ m/s$^2$, a maximum speed of $15$ m/s and a maximum deceleration of $4$ m/s$^2$. The cyclist wishes to travel a distance $s$ from rest to rest in the shortest time. Find the time taken in the two cases

(i) $s=105$ m and

(ii) $s=54$ m.

Draw a rough velocity-time graph for each case and explain why $84\frac{3}{8}$ m is a critical distance.

[Video Solution]

Question 2

A perfectly elastic particle falls vertically with speed $u$ on to a smooth plane inclined at an angle $\alpha$ to the horizontal, and rebounds, hopping down the plane.

i) Write down its displacement $\vec{r}$ from the point of contact after time $t$ in terms of $\vec{i}$ and $\vec{j}$, where $\vec{i}$ is drawn directly down the plane.

ii) Show that the length of the first hop is $\frac{4u^2 \sin \alpha}{g}$ and that

iii) the length of the second hop is double this.

[Video Solution]

Question 3

A ship $A$ is steaming with velocity $v$ m/s where $\vec{v}=5\sqrt{3}\vec{i}+5\vec{j}$ where $\vec{i}$ and $\vec{j}$ are point East and North, respectively. At midday a second ship $B$ has a position $10\vec{i}$ km with respect to $A$.

i) Find the minimum speed $B$ must have if it is to intercept $A$.

ii) If the maximum speed of $B$ is in fact $6$ m/s, show that it can steer in either of two directions to intercept $A$ and

iii) calculate the two times of interception.

[Video Solution]

Question 4

The diagram shows a light inelastic string with one end connected to a fixed point $a$ of a ceiling, passing under a heavy movable pulley $b$ of mass $10$ kg and then over a fixed pulley $c$ attached to the ceiling. To the other end $d$ of the string is attached a particle of mass $10$ kg hanging freely.

i) Show in separate diagrams the forces acting on the particle and on the pulley $b$ when they are released from rest.

ii) Show that the acceleration of $d$ is double that of $b$,

iii) and calculate the acceleration of $d$ and

iv) the tension in the string. (Neglect the inertia of both pulleys).

[Video Solution]

Question 5

A particle of mass $10$ kg hangs freely from the end $b$ of a light inextensible string of length $0.2$ m which is attached at the other end to a fixed point $o$. The particle is then projected horizontally with speed $1.4$ m/s.

i) Show in a diagram the forces acting on the particle when $ob$ is inclined at an angle $\theta$ to the downward vertical assuming that it has a speed $v$ m/s at that point.

ii) Use conservation of energy to determine $v$ and

iii) express the tension in the string in terms of $\theta$.

iv) Find where the particle comes to instantaneous rest and

v) show that the tension in the string is then $49$ N.

[Video Solution]

Question 6

A light string $rstu$ is attached to fixed points at its ends $r$ and $u$, so that $u$ is vertically below $r$. Particles of weights $50$ N and $100$ N are attached to the string at $s$ and $t$, respectively, and a horizontal force $x$ newtons is applied to the particle at $s$ so that the string is in equilibrium in a vertical plane through $ru$ with $\angle sru = \angle rut = 45^\circ$ and $\angle rst = 105^\circ$. Show in a separate diagrams the forces acting on the two particles and prove that $x=\left(250+100\sqrt{3}\right)$ N.

[Video Solution]

Question 7

An equilateral triangle is $pqr$ is formed from three uniform rods, each of length $2a$ and weight $W$, freely jointed at their ends. The triangle is freely suspended by a string attached to the midpoint $o$ of $pq$ so that it hangs symmetrically under gravity with $r$ vertically below $o$. Show in separate diagrams the forces acting on $pq$ and $pr$, and calculate the horizontal and vertical components of the reactions at $r$.

[Video Solution]

Question 8

i) Prove that the moments of inertia of a uniform circular disc of radius $0.4$ m and mass $5$ kg about an axis $oq$ through its centre $o$ and perpendicular to the disc is $0.4$ kg m$^2$.

[Video Solution]

ii) Such a disc can rotate freely about the axis $oq$ which is fixed horizontally. A light inextensible string is wound around the rim of the disc with one end attached to it, and to the other end is tied a particle $p$ of mass $2$ kg which hangs vertically. If the system is released from rest, show that the speed of $p$ is $2.8$ m/s after it has descended a distance $0.9$ m.

[Video Solution]

Question 9

An engine pulls a train along a level track against a resistance which at any time is $K$ times the momentum. The engine works at a constant power $25KMU_0^2$, where $M$ is the total mass of the train and engine and $K$, $U_0$ are constants. Show that the equation of motion of the train is

$$v\frac{dv}{dt}=K\left(25U_0^2-v^2\right),$$

and find the time taken to increase speed from $U_0$ to $4U_0$.

[Video Solution]

Question 10

a) State the principle of Archimedes.

[Video Solution]

b) A solid hemisphere of radius $a$ is help submerged in a liquid of density $\rho$ with its plane face horizontal and uppermost at a distance $2a$ below the free surface of the liquid. Calculate the magnitude, direction and line of action of

i) the force exerted by the liquid on the plane face,

ii) the total force exerted by the liquid on the surface of the solid.

[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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