5. In a system of three movable pulleys in equilibrium, in which the strings are parallel and each is attached to the weight (usually called the 3rd system), find the tension of the string which passes over the fixed pulley, the weights of the pulleys, which are all equal, being taken into

account.

If the weights of the pulleys are neglected and each string is attached to a bar which keeps horizontal, find the point in the bar from which the weight is suspended.

6. Give examples of stable and unstable equilibrium. A cone and a hemisphere of the same material are cemented together at their common circular base: if they are placed on a horizontal plane, the hemisphere being in contact with the plane, find the height of the cone that the equilibrium may be neutral, it being given that the centre of gravity of a hemisphere divides a radius in the ratio of 3 to 5.

7. If any number of forces act in one plane on a rigid body, state the three conditions of equilibrium.

A beam whose weight is (W) rests with its ends on two inclined planes whose angles of inclination are (a) and (B); prove that the sum of the pressures on the planes is

COS

α-β

8. On what hypotheses is the relation between (P) and (W) usually obtained on a screw considered as a mechanical power? If the screw be rough, and (R) the sum of the normal resistances on the thread of the screw, (W) the weight supported, (ø) the angle of resistance, and (a) the W cos where (P) is just cos (a + )'

pitch of the screw, obtain the equation R= on the point of prevailing over (W).

9. The same force acting parallel to two inclined planes of 30° and 60° inclination can just move a given weight up the plane of less inclination, and can just prevent a weight twice as great from moving down the plane of greater inclination; prove that the coefficient of friction which is the same for both planes is nearly 13.

IO. State the principle of "virtual velocities." If a weight (W) be supported in equilibrium on a smooth inclined plane by a weight (P) hanging vertically and passing over a fixed pulley placed in the prolongation of the height of the plane, prove that the principle of virtual velocities holds good, and show that for a small displacement the centre of gravity of (P) and (W) will neither ascend nor descend.

II. A heavy ladder is placed in a given position, between a vertical wall and the horizontal ground, both being considered equally rough; a workman of given weight ascends the ladder with a given load, show how to determine by a geometrical construction whether the ladder will slip.

IX. DYNAMICS.

[Great importance will be attached to accuracy.]

I. The measure of a certain velocity in feet per second is v; what is it in miles per hour?

2. Two trains, each 200 feet long, are moving towards each other with velocities of 20 and 30 miles per hour respectively. Find the time which elapses from the instant when they first meet till they have completely cleared each other.

3. State exactly what is meant by saying that the accelerating force of gravity is 32.

In the equation F=mf, if a foot and a second are the units of space and time, and a weight of one pound the unit of force, what is the unit of mass?

4. Two bodies which weigh 9 lbs. and 16 lbs. respectively at the earth's surface, are placed in space at a distance asunder of 100 feet, no force acting on either. If they were now to attract each other with a constant force equal to 1 lb. at all distances, find after what time they would meet.

5.

A body slides down a rough inclined plane 100 feet long, the sine of whose inclination = 0.6, and coefficient of friction : find its velocity at the bottom.

If projected up the plane with a velocity which just carries it to the top, find what height it would reach if thrown vertically upwards with that velocity.

6. Two perfectly elastic balls whose masses are M, M', moving in the same direction, strike each other. If the hindmost ball is reduced to rest by the blow, show that its velocity must have been more than double that of the other.

7. A train runs from rest down an incline of 1 in 100 for a distance of 1 mile (no engine attached): it then runs up an equal gradient with its acquired velocity for a distance of 500 yards before stopping. Assuming the principle of work, find the total resistance, frictional or other, in pounds per ton, which has been opposing its motion.

8. A body is projected horizontally with a given velocity. Prove that it describes a parabola, and determine the position of its focus.

9. Two bodies are projected from the same point, one later than the other by T seconds, so as to describe the same parabola. Show that they are nearest to each other when in the same horizontal line (if that is V sin a T possible) and that this occurs at the interval of time after the g second body was projected. Explain what circumstances as to the data are required, if this be possible.

2

IO. Two weights, P, Q, are connected by a light string passing over a smooth fixed peg. Find the acceleration of the system. Mention any experimental use which has been made of this contrivance.

If two equal weights, P, P, are in equilibrium, connected in this way, and a third weight, P, is laid on one of them, find by how much the pressure on the peg is increased.

II. A particle starting with a velocity u, falls down a smooth vertical curve of any form. State what its velocity is when it has arrived at any given point of the curve.

A particle falls down a vertical circle, starting from rest at the highest point. If, when at any point, its velocity be resolved into two components, one passing through the centre, the other through the lowest point of the circle, prove that the latter is of constant magnitude.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1885.

PRELIMINARY EXAMINATION.

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

[Ordinary abbreviations may be employed; but the method of proof must be geometrical. Great importance will be attached to accuracy.]

I. Define a right angle, and show how to draw a straight line at right angles to a given straight line, from a given point in the same.

If one diagonal of a quadrilateral figure bisect the two angles at its extremities, it will bisect the other diagonal at right angles.

2. If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are equal to two right angles.

The longer sides of a parallelogram are twice as long as the shorter sides. Show that the straight lines joining the middle point of one of the longer sides with the ends of the opposite side, are perpendicular to each other.

3. If a parallelogram and a triangle be on the same base and between the same parallels, the parallelogram shall be double of the triangle.

4. If a straight line be divided into two equal and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line.

W. P.

I

13

If any point D be taken in the base BC of an isosceles triangle ABC, the rectangle under BC, CD will be equal to the difference between the squares on AB, AD.

5. If any point be taken without a circle, and straight lines be drawn from it to the circumference, one of which passes through the centre, of those which fall on the concave circumference, the greatest is that which passes through the centre, and of the rest, that which is nearer to the one passing through the centre is always greater than one more remote; but of those which fall on the convex circumference, the least is that between the point without the circle and the diameter; and of the rest, that which is nearer to the least is always less than one more remote.

Prove that the two lines which are equally inclined to the shortest line are equal.

6. Define a segment of a circle, and shew how to bisect its circumference.

Prove that the straight lines joining the ends of the base of a segment of a circle with any point on its circumference, are equally inclined to the straight line passing through that point and the point of bisection of the circumference.

7. About a given circle describe a triangle equiangular to a given triangle.

Prove that the equilateral triangle inscribed in a circle is one-fourth of the equilateral triangle described about the same circle.

8. Inscribe an equilateral and equiangular pentagon in a given circle. 9. The sides about the equal angles of triangles which are equiangular to one another are proportionals.

Prove that the diameters of the circles circumscribing the two triangles formed by joining the vertex with any point in the base of a given triangle are proportional to the sides of that triangle.

IO. If four straight lines be proportionals the similar rectilineal figures similarly described on them shall also be proportionals; and if the similar rectilineal figures similarly described on four straight lines be proportionals, those straight lines shall be proportionals.

II. If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.