Leaving Certificate Examination 1931 Honours Applied Mathematics

Question 1

The motion of a particle along a straight line is given in the following table :-

Obtain approximately its velocity at intervals of $0.2$ seconds from $0.2$ seconds to $1.1$ seconds, and plot $V$ against $T$. Hence determine approximately when $\left(1\right)$ the velocity of the particle is zero and $\left(2\right)$ is acceleration is zero.

OR

If a particle moves according to the law $x=k\sin t$, explain how you can obtain, by calculus, expressions for its speed, and its acceleration at any instant. When, during the motion, are these magnitudes zero? How is this kind of motion usually described?

Question 2

A train acquires a speed of $45$ miles an hour in minutes. If the carriage wheels are $4$ feet in diameter, what is their angular velocity at this speed? What was their average angular acceleration?

Show, on a diagram, the magnitude and direction of the components of the acceleration of the highest point of the wheel during the accelerated motion of the train, and the acceleration of the same point when the train is moving with uniform speed.

Question 3

A stone is projected with velocity $v$ and elevation $\theta$ from a point $O$ in a horizontal plane, so as to hit a mark $P$ at a horizontal distance $h$ from $O$, and at a height $k$ above the plane. Show that $v$, $h$, $k$ and $\theta$ are connected by the relation:

$$ h\sin\theta\cos\theta-k\cos^2\theta=\frac{gh^2}{2v^2}$$.

Derive the condition that $\frac{h^2}{2v^2}$ should be a maximum when $h$ and $k$ are constants, and show that is is satisfied by $\theta=\frac{\pi}{4}+\frac{\alpha}{2}$ where $\alpha$ is the elevation of $P$ from $O$.

Question 4

In an Attwood’s machine, a mass of $2$ pounds is attached to each end of the cord. An additional mass of $0.6$ pound is placed on one side and is found to produce a velocity of $4.72$ feet per second at end of a descent from rest of $4$ feet. Compare this result with that given by the simple theory. Express in ft-lb. the kinetic energy of the masses and the work done by the weights, and account for the difference between these quantities.

Question 5

Two uniform rods $AB$ and $BC$ weighing $150$ grams each, and of lengths $20$ cm. and $40$ cm. respectively are freely jointed together at $B$. They are maintained in a horizontal straight line by three vertical strings, one attached to a point $D$ in $AB$, $6$ cm. from $A$, and the others at $B$ and $C$. Find the tensions in the strings.

Question 6

A uniform square lamina, $ABCD$ of 9 in. side, is divided into two parts by a line joining $A$ to a point $E$ in $DC$ where $DE=3$ in. State the distances of the centres of gravity of the triangles $ABE$ and $BCE$ from the sides $AB$ and $BC$. Find the distance of the centre of gravity of $ABCE$ from $AB$ and $BC$.

Question 7

A uniform rod, $106$ cm. long and weight $150$ grams, is suspended from a fixed point by strings $90$ cm. and $56$ cm. long attached to the ends of the rod. Find the tension in each string.

Question 8

Two particles $m_1$ and $m_2$ moving along the axes $OX$ and $OY$ respectively towards $O$ with velocities $u_1$ and $u_2$ collide at $O$. What are the components of the velocity of centre of gravity before the collision? Why do they remain unaltered by the collision? Show that the kinetic energy of the particles is equal to the kinetic energy of a mass $m_1 + m_2$ moving with velocity of the centre of gravity and a mass and a mass $\frac{m_1 m_2}{m_1 + m_2}$ moving with the relative velocity of either particle with respect to the other.

Question 9

Define the term Power. A motor-car weighing $32$ cwt. is travelling at uniform speed of $27$ miles per hour on a level road. On reaching a hill which descends with a uniform gradient of $1$ in $25$, it is allowed to run free, and the speed is observed to be the same as before. Calculate the resistance of the road and the horsepower expended on the level.

Question 10

A pendulum bob at the end of a string $60$ inches long, describes a horizontal circle with the string making an angle of $30$ degrees with the vertical. Find its angular velocity and the time of describing the circle. State clearly the principles applied in the solution of this problem.

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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