whose axis is horizontal. The rods touch the cylinder at points distant Find the length of the rods, and prove

one-third of their length from B.

that the bending moment at the point of contact of either rod with the

4. A heavy circular wheel is suspended in a vertical plane from a point B in a rough vertical wall, by a smooth weightless wire through its centre O, and rests at right angles to the wall and touching it at the point A. A string is fixed to the rim of the wheel and passes over it on the side away from A. If the inclination (a) of the wire OB to the horizon is less than 28, where tan ẞ is the coefficient of friction between the wheel and the wall, show that a force, however great, applied along the string will not turn the wheel unless the direction of the string makes with BO produced an angle less than

sin-1

sin(a-B)
sin B

(Positive angles are measured from OA to OB.)

5. Find the position of the centre of gravity of a cylindrical bar with circular ends, whose density at any point is proportional to the square of the distance of the point from the end of the bar.

A cone of vertical angle 2a rests with its base on a rough plane inclined at an angle ẞ to the horizon, the coefficient of friction (μ) being greater than tan ß. A gradually increasing force is applied at the vertex of the cone parallel to the plane and downwards. Show that, when the force is large enough to disturb equilibrium, the cone will tilt over or slide down accord. ing as tan ẞ is less or greater than (μ- tan a).

6. A particle initially at rest is acted on by a force constant in direction and magnitude. Prove that the kinetic energy of the particle at any time is equal to the work done on it by the force.

A particle moves from rest down a rough plane inclined at an angle 20 to the horizon, tan being the coefficient of friction. Prove that in moving over a length s of the plane it acquires the same velocity as in falling freely through a distance s tan 0.

7. An engine draws a train whose weight (exclusive of the engine) is 100 tons. The power of the engine is such that when running on the level it exerts a pull of 2 tons weight on the front carriage, and the resistance due to friction, etc., is 11.2 lbs. per ton. Show that if the engine draws the same train from rest up an incline of 1 in 300 it will in one minute acquire a velocity slightly exceeding 15 miles per hour.

8. A bullet is fired with a velocity of 800 feet per second. Find (a) its greatest possible range on the horizontal plane through the point of projection; and (6) the height (approximately) to which the bullet ascends when the range is one-tenth of the greatest possible range.

9. A mass of weight W rests on the smooth surface of a horizontal table, also of weight W, and is connected by a weightless string, passing over a smooth pulley at the edge of the table, with a weight 2 W hanging freely, which is allowed to fall, the string being initially taut. If the table does not move, find the tension of the string, and show that the coefficient of friction between the table and the floor is not less than 1.

IO.

Two equal imperfectly elastic balls moving in the same straight line impinge directly upon one another. Find the change of kinetic energy produced by the impact.

A billiard ball A moving parallel to one side of the table, strikes another ball B (initially at rest) in the centre. After B has struck the cushion and then struck A again, the velocity of A is three-fourths of its initial velocity, but in the opposite direction. Show that, if e is the coefficient of elasticity between the balls and also between a ball and the cushion,

e3 + e2+3e = 4.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy,
Academy, Woolwich,

NOVEMBER, 1894.

OBLIGATORY EXAMINATION.

I. EUCLID (Books I.-IV. AND VI.).

[Ordinary abbreviations may be employed, but the method of proof must be geometrical. Proofs other than Euclid's must not violate Euclid's sequence of propositions. Great importance will be attached to accuracy.]

I. Define plane rectilineal angle, circle, gnomon, similar rectilineal figures.

2. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

3. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts are also themselves equal and parallel.

ABCD is a quadrilateral; show that, if the four parallelograms BCDP, CDAQ, DABR, ABCS be completed, the four straight lines AP, BQ, CR, DS will be equal and parallel.

4. If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the two parts.

Show also that if a straight line be divided into any three parts, the square on the whole line is equal to the squares on the three parts, together with twice the three rectangles whose sides are the three parts taken two and two together.

5. Describe a square equal to a given rectilineal figure.

6. If a straight line drawn through the centre of a circle bisect a straight line in it which does not pass through the centre, it cuts it at right angles; and if it cut it at right angles, it bisects it.

7. Straight lines in a circle which are equally distant from the centre are equal to one another.

8. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle are equal to the angles which are in the alternate segments of the circle.

Four circular coins, of different sizes, are placed upon a table so that each one touches two, and only two, of the remaining three; show that the four points of contact lie on a circle.

9. Inscribe a circle in a given triangle.

Prove that the centre of this circle lies inside each of the three circles described on the three sides of the triangle as diameters.

IO. Describe a circle about a given equilateral and equiangular pentagon.

II. Triangles which have one angle of the one equal to one angle of the other and the sides about the equal angles reciprocally proportional, are equal to one another.

Two straight lines AOC, BOD intersect in O and the lines AB, CD are drawn. From the greater of the two triangles AOB, COD cut off a part equal to the less by a straight line drawn through the point O.

12.

Parallelograms about the diameter of any parallelogram are similar to the whole parallelogram and to one another.

II. ARITHMETIC,

[N.B.-The working as well as the answers must be shown.]

3. A sum of £237. 185. 4d., lent at simple interest, amounts in three years to £270. os. 8d. What is the rate per cent. ?

4.

Find the value in cwts. qrs. and lbs. of '0234375 of 93 tons.

5. What fraction is 28 cubic inches of a cubic foot?

6. Find the value of 13 acres 2 roods 17 perches at £14. 14s. 8d. per acre.

7.

Find the greatest common measure of 10058, 4982, and 9823.

8. Justify, from first principles, each step of the process of addition of vulgar fractions and deduce the rule for the addition of decimals.

9. What was the cost of goods on which a man lost 20 per cent. by selling them for £64?

IO. A clock set right at noon on Tuesday loses at the rate of 192 seconds in 10 hours. What is the true time on the following Friday afternoon when the reading of this clock is 2 hours 36 minutes?

II.

Determine, without performing the divisions, the remainders that result from dividing 48909661 by 8, 16, 25, 9, and 11; and give a brief explanation of the reason from which you draw your conclusion in the first three cases.

12.

If it costs the same amount to keep 4 horses or 9 oxen, and 5 horses can do as much work as 8 oxen, which will it be more profitable to employ -20 horses and 32 oxen, or 12 horses and 48 oxen?