A tank 24 feet long, 12 feet broad, 16 feet deep, is to be filled by water in a well, the surface of which is always at a depth of 80 feet below the bottom of the tank; determine the work done in filling the tank, and find the horse-power of a steam-engine whose modulus is 5, that will fill the tank in 4 hours, the weight of a cubic foot of water being 62.5 pounds.

I.

IX. DYNAMICS.

Candidates are not expected to answer the whole of this paper.

Define linear and angular velocity.

If a railway carriage be moving at the rate of 30 miles an hour, and the radius of one of its wheels be 2 feet, what is the angular velocity of the wheel if there be no sliding? What also is the relative velocity of the centre and highest point of the wheel?

Why are garden rollers generally divided into two parts?

2. Two particles are started simultaneously from the points A and B, 5 feet apart, one from A towards B with a velocity which would cause it to reach B in 3 seconds, and the other at right angles to the former and with ths of its velocity.

Find their relative velocity in magnitude and direction, the shortest distance between them, and the time at which they are nearest to one another. 3. When a body is projected vertically upwards prove that the velocity is the same at the same points in the ascent and descent.

If the time of a body's fall from a certain height at one place on the earth's surface be m seconds less than that at another place, and the velocity

acquired in the fall be a feet per second greater, prove that

geometric mean of the accelerations of gravity at the two places.

a

m

is the

If a gun be fired at the elevation 0, where 0 is very small, so as to hit a mark at a distance a, and b be the range when fired at an elevation of 45°, prove that is equal to

5.

a

26

approximately.

Find the acceleration of a particle sliding down a smooth inclined plane. What is the vertical component of this acceleration?

If two particles P and Q start simultaneously from A, one sliding down the plane AB at the angle a to the horizon, and the other falling freely, prove that their relative vertical acceleration is (gcos2 a).

Hence prove that the line PQ is always perpendicular to AB.

6. Two unequal weights W and W' are connected by a string passing

over a smooth pulley. Find the acceleration.

Hence explain the principle and use of Attwood's machine.

7. If the weights in the last question be contained in scale-pans attached to the ends of the string, prove that if the weights of the pans be neglected, the pressure between each pan and the contained weight is equal to the tension of the string.

Find the pressures when the weight of each pan is w, and prove that they are to each other as W: W'.

8. A ball A impinges directly upon an exactly equal and similar ball B lying upon a smooth horizontal plane; ife be the coefficient of elasticity prove that after the impact A's velocity will be to B's velocity as 1+e: 1 - e.

9. If, after the impact described in the foregoing question, A moves on and impinges directly upon a perfectly elastic vertical wall from which it rebounds and meets B a second time, prove that after the second impact A will be reduced to rest.

Find the total loss of vis viva. What becomes of the kinetic energy which has thus disappeared?

IO. Find the acceleration of a particle moving in a circle with uniform velocity. Two weights W and W' are placed on a smooth table, and connected together by a string passing through a small fixed ring on the table. If they are projected with the velocities v and v' at right angles to the string, find the ratio in which the string must be divided by the ring in order that both weights may describe circles round the ring as centre.

II. A string AP of length a is fastened to a point A and carries a weight P. If P be projected vertically upwards from the position in which AP is horizontal, find the least velocity of projection with which it may describe a circle round A, and state what happens if the velocity be less than this.

If P be projected vertically upwards when AP is below the horizontal line, and at an angle a to it, find the least value of v that P may ultimately describe a circle in this case.

12. If a particle move under the action of gravity in a smooth cycloidal groove whose axis is vertical and vertex downwards, prove that the acceleration varies as the distance from the vertex.

13. If the particle in the last question start from rest from an extremity of the base of the cycloid, and if T be the time in which it reaches the vertex, prove that at any time t, the height of the particle above the vertex is equal to its depth below the base at the time T-t.

14. A pendulum oscillating seconds at one place is carried to another place at which it loses 2 minutes a day. Compare the accelerations of gravity at the two places.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1880.

PRELIMINARY EXAMINATION.

I. EUCLID (Books I.-IV. and VI.). (Obligatory.)

I. Distinguish between a postulate and an axiom. In which propositions are the 2nd postulate and the 12th axiom of Euclid respectively used for the first time? What axioms are used in Prop. IV., Book I.?

2. The angles at the base of an isosceles triangle are equal to one another; and if the equal sides be produced, the angles on the other side of the base shall be equal to one another.

State fully the converse of this proposition.

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

Complete the figure which is necessary for the proof of the second part of this proposition.

How many of the exterior angles of any triangle must be obtuse?

4. Parallelograms upon the same base, and between the same parallels, are equal to one another.

Of all triangles that can be drawn upon a given base and between the same parallels, shew that an isosceles triangle has the least perimeter.

W. P.

I

2

5. Describe a square that shall be equal to a given rectilineal figure. Divide a given straight line into two parts, such that the rectangle contained by them shall be equal to the square of their difference.

6. If two circles touch one another internally, the straight line which joins their centres, being produced, shall pass through the point of con

tact.

If two circles touch externally at E, and AB, CD be any two parallel diameters of the circles, shew that the straight lines AD, BC will pass through E.

7. If a straight line touch a circle, the straight line drawn from the centre to the point of contact shall be perpendicular to the line touching the circle.

8. The angles in the same segment of a circle are equal to one another.

If PQ, RS, two chords of a circle, intersect within the circle, shew that their inclination to one another is equal to one half of an angle at the centre of the circle standing upon an arc equal to the sum of the arcs PR and QS.

9. Inscribe an equilateral and equiangular pentagon in a given circle. Give a construction (without proof) for the inscription of a regular twenty-sided figure in a given circle.

IO. If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another.

Find a point C in ACB, a given segment of a circle, such that the straight line AC is double of the straight line CB.

II. Describe a rectilineal figure which shall be similar to one given rectilineal figure, and equal to another given rectilineal figure.

II. ARITHMETIC. (Obligatory.)

(Including the use of Common Logarithms.)

[N.B.-Great importance will be attached to accuracy in numerical results.]

I. Add together 2 of 2, 3 of 3, 1 of 3%, and 1 of 51.

9. Express I furlong 4 poles as the decimal of a mile.

IO. What is the difference between 0057 of a lb. troy and of a dwt.? Express the answer as the fraction (vulgar) of a dwt.

11. Find, by Practice, the dividend on £3,245. 155. at 13s. 9d. in the £.

12. Find the amount of £2,060 in 3 years at 4 per cent. simple interest, neglecting fractions of a penny.

13. Multiply by duodecimals 3 ft. 5 in. by 4 ft. 9 in., and the product by 8 ft. 7 in. What does the answer become when expressed in cubic feet, cubic inches, and the fraction of a cubic inch?

14. A, B, and C working together can do a piece of work in 6 days. A could do it alone in 24 days. After working together for 2 days A is taken ill. How long will B and C take to finish it?