9. Find the following integrals—

IO.

dx

x

x x

I fin2 ads, felogede.

Evaluate the definite integrals

12. Determine the length of any arc of a parabola measured from its vertex.

VIII. STATICS.

1. Assuming that the resultant of two equal forces bisects the angle between them, prove that the resultant of two forces, one of which is double the other, acts in the diagonal of the parallelogram of which they are sides.

2.

A straight uniform lever, weighing 10 lbs., rests on a fulcrum onethird of its length from one end; it is loaded with a weight of 4 lbs. at that end; find what vertical force must act at the other end to keep the lever at rest.

3. Four forces act along the sides of the rectangle ABCD, and are measured by those sides; the first three, AB, BC, CD, act in a contrary sense to the fourth, AD. Find their resultant and its line of action.

4. Three forces, PA, PB, PC, diverge from the point P; and three others, AQ, BQ, CQ, converge to the point Q. Show that the resultant of the six is represented in magnitude and direction by 3PQ, and passes through the centre of gravity of the triangle ABC.

5. It is required to decompose a force whose magnitude and line of action are given into two equal forces passing through two given points. Give a geometrical construction for solving the problem (1) when the two points are on the same side of the line; (2) on opposite sides.

6. Two uniform rods, AB, BC, jointed together at B, are placed in a horizontal line, ABC, and supported by two props, one at A, the other between B and C. Determine the position of the latter, given the lengths and weights of the two rods.

7. A weight is supported on a smooth inclined plane by a force acting along the plane. Find the force, also the pressure on the plane.

If the plane be rough, prove that this pressure is unaltered; and explain how far the force may be determined.

8. Three given weights are rigidly connected together; prove that the resultant weight always passes through one point, fixed as regards the weights, however the system be displaced in the plane of the weights. Does this hold if the system be displaced out of its original plane?

9. Two weights, joined by a string passing over a pulley, rest on two smooth inclined planes of the same height, and whose highest points coincide at the pulley. Apply the principle of virtual velocities to show that the weights are as the lengths of the planes, and that, however they are displaced, their centre of gravity moves horizontally.

Determine by the same principle whether these theorems hold when the pulley is not at the vertex of the planes.

IO.

Four weightless rods are freely jointed together, forming a quadrilateral ABCD. If two equal and opposite forces are applied at two points, one on the rod AB, one on the opposite rod CD, determine the conditions that they shall balance each other.

If, instead of forces, two equal and opposite couples be applied to the rods AB, CD, what is necessary for equilibrium?

State generally in each case what will happen if the conditions are not satisfied.

IX. DYNAMICS.

[The measure of the acceleration of gravity may be taken to be 32 when a foot and a second are units of length and time.]

I.

[Great importance will be attached to accuracy.]

Define uniform velocity and uniform acceleration, and state how each is measured.

2. A body whose weight is W lies on a smooth table. What horizontal force applied to it will cause it to move with unit acceleration, a foot and a second being the units of space and time?

3. From a train moving with velocity V, a carriage on a road parallel to the line, at a distance d from it, is observed to move so as to appear always in a line with a more distant fixed object, whose least distance from the railway is D. Find the velocity of the carriage.

4. A particle moving from rest with (unknown) uniform acceleration acquires a velocity v in the time t; find the space described.

5. A particle is projected vertically upwards with a velocity u; find the height it will reach.

Show that the velocities with which it passes through any assigned point, in its ascent and fall, are equal.

6. Find the conditions that two given non-elastic balls, on striking each other, shall both be reduced to rest. If they are perfectly elastic, what will happen after the collision?

Show this directly, without employing general formulas.

7. State (and prove) with what acceleration a particle will fall down a smooth inclined plane. State also (without proof) the acceleration, if the plane be rough: given μ the coefficient of friction.

A heavy slab whose under surface is rough, but the upper smooth, slides down a given inclined plane. Find the acceleration with which a small particle laid on its upper surface will move along the slab.

8. If a train ascends a gradient of 1 in 40 by its own momentum for a distance of 1 mile, the resistance from friction, &c., being 10 lbs. per ton, find its initial velocity.

9. Find the greatest range which a projectile with an initial velocity of 1,600 ft. per second can attain on a horizontal plane.

Show also that for a small difference of elevation, not exceeding 10, whether of excess or defect, the range attained will fall short of the maximum by less than 1 feet.

IO. A body of mass m is placed at a distance R from a centre of force O, which attracts it with a force P. Find the velocity and direction with which it must be started so as to describe a circle round 0.

If the force of attraction, for different distances, varies inversely as the squares of the distances, and for different bodies, directly as their masses, prove that if several bodies move round O in concentric circles, the squares of the times of revolution are as the cubes of the radii.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1888.

PRELIMINARY EXAMINATION.

1. 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.]

1. Define a right angle, and a rhombus.

If, at a point in a straight line, two straight lines on opposite sides of it make the adjacent angles together equal to two right angles, they are in the same straight line.

AOB, COD are two intersecting straight lines, and each of the figures AOCE, BODF is a rhombus. Show that the straight line EF passes through O, and that AC, BD are parallel.

2. The opposite sides and angles of a parallelogram are equal, and either diagonal bisects it.

Straight lines drawn through A, C, the extremities of one diagonal of a parallelogram ABCD, respectively perpendicular and parallel to the other diagonal, intersect in E. Prove that BE CD.

W. P.

I

19