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Maths Grinds' Blog

Blogging and Mathing
17 Aug 2018
Question 1

The following table gives the speed $v$ of a train $t$ seconds after leaving a station $A$ for portion of the journey between two stations $A$ and $B$, $4800$ ft. apart.

 Speed in feet per second. $0$ $18$ $34$ $40$ $44$ $45.5$ $45.5$ $42$ $35$ Time in seconds. $0$ $10$ $20$ $30$ $40$ $50$ $60$ $70$ $80$

Estimate the acceleration after $15$ seconds and the distance travelled in the $80$ seconds. If the train continues to move with the uniform speed of $35$ ft. per second until the brakes are applied, producing a uniform retardation of $2$ ft. per sec. per sec., find the total time taken in going from $A$ to $B$.

Question 2

(a) A particle describes simple harmonic motion of amplitude $a$, the period being $T$. Prove that at distance $x$ from its mean position its speed $v$ is given by $\frac{2\pi}{T}\sqrt{a^2-x^2}$.

(b) If the amplitude of the motion of bob of a simple pendulum is $1$ foot and the period $3$ seconds, find the time occupied by the bob in passing between points which are distance $8$” and $4$” from the mean position and are on the same side of it.

Question 3

Prove that two bodies moving in different straight lines with uniform velocities will meet if their relative velocity is in the straight line joining them.

A ship steaming in a direction $17^\circ$ North of East is seen by a submarine which is in a direction $10^\circ$ East of South from the ship. Find in what direction a torpedo travelling at $20$ miles per hour must be fired in order to strike ship.

Question 4

A bullet of mass $1$ oz. travelling horizontally with a speed of $1200$ ft. per second becomes embedded in a block of wood of mass $4$ lb. suspended by a light steel wire, the C.G. of the block being $4$ ft. below the point of suspension. Find the vertical distance through which the block rises.

Question 5

When the motors of an electric train of effective weight $100$ tons are working at the rate of $80$ H.P., a uniform speed of $30$ miles per hour is attained on the level. Assuming the frictional resistances to remain the same, what is the maximum speed attainable with $150$ H.P. when ascending an incline of $1$ in $250$?

Question 6

Show that the centre of gravity of $3$ equal particles placed at the three vertices of a triangle respectively coincide with the centre of gravity of the area of the triangle. The parallel side of a trapezium are of lengths $a$ and $b$ and the perpendicular distance between them is $h$. Find the distance of the C.G. from the side $a$.

Question 7

What are the laws of solid friction?

A body of weight $w$ is dragged along a horizontal plane, coefficient of friction $\mu$, by a cord inclined at an angle $\theta$ to the horizontal; find the tension in the cord. For what value of $\theta$ is the tension a minimum? A body is placed on a plane, coefficient of friction $0.15$ inclined at $50^\circ$ to the horizontal; find the velocity acquired in one second.

Question 8

A triangular roof-truss $ABC$ has a horizontal span $AC$ of $36$ feet and angle $ABC$ is $120^\circ$, $AB$ and $BC$ being of equal length. The roof-truss is hinged at $A$ and simply supported on rollers at $C$ (so that the supporting force at $C$ is vertical). It is loaded as follows: $1\frac{1}{2}$ tons perpendicular to $AB$ at its mid-point at $BC$. Find the reactions of the supporting forces at $A$ and $C$ and the tensions in $AC$.

Question 9

A body of weight $20$ lb. is moving due East with a velocity of $12$ ft. per second, and forty seconds later it is moving North-East with a velocity of $20$ ft. per second. What is the change in velocity? What constant force acting during the interval would produce the change? Show that in the latter case the hodograph is a straight line.

Question 10

Assuming that the wheels of a motor-car are $2\frac{1}{2}$ feet in diameter, what is the angular velocity of a wheel when the car s travelling at the rate of $50$ miles per hour? What is the velocity of a point on the surface of tyre (a) when vertical above the centre, (b) when the radius from the point to the centre is inclined at an angle of $60^\circ$ to the upward vertical? What would be the force acting on a stud of mass $\frac{1}{2}$ oz. embedded in the type when at its greatest height?

Citation:

Malone, D and Murray, H. (2016). Archive of Maths State Exams Papers. Accessed at: http://archive.maths.nuim.ie/staff/dmalone/StateExamPapers/

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