Great im

[Full marks will be given for about two-thirds of this paper. portance is attached to accuracy. Gravitational acceleration may be taken = 32 feet per second per second, and π = 2]

I. Investigate the conditions that three parallel forces, whose lines of action are intersected by a straight line ABC in A, B, C respectively, may be in equilibrium.

A solid cone of weight W, height h, and radius of base r, rests with its slant side in contact with a smooth horizontal plane. Find (i.) the least vertical force which, applied at the vertex, will move the cone, and (ii.) the force at the vertex which will keep it at rest with its axis horizontal.

2. Find the magnitude and direction of the resultant of a system of forces acting in one plane on a particle, the magnitudes and the directions of the forces being given.

Five forces P, Q, X, R, Y act at O, the centre of a regular pentagon ABCDE, in the directions OA, OB, OC, OD, OE respectively. If the system be in equilibrium and P, Q, R be given, find the magnitudes of X and Y. [N.B.-Sin 18° = † (√5 − 1.)]

3. The sides AB and CD of a quadrilateral lamina ABCD are parallel and at a distance h apart. If AB=a, CD=b, find the distance of the centre of mass of the lamina from the straight line bisecting the opposite sides AD and BC.

A rectangle ABCD is bisected by a straight line cutting AB in E and CD in F. Show that the locus of the centre of mass G of either part is a parabola, and that the tangent at G to the locus is parallel to the corresponding line EF.

4. A rigid body is in equilibrium under the action of four forces in one plane, whose directions are not all parallel, and whose lines of action are not concurrent. If the magnitude of one of the forces be given, and the lines of action of all, show how the magnitudes of the other three may be found by the graphical method.

C and D are two small rough pegs, I foot apart in a straight line inclined to the horizon at an angle of 60°. A uniform rod ACDB of 10 pounds weight and 3 feet in length is placed in this line, under C and above D, so that AC=BD=1 foot. Find graphically the least weight which, suspended from the upper end B, will keep the rod in the above position, the angle of friction between the rod and the pegs being 45°.

5. Describe the common balance, and explain how its sensibility depends on the form of the beam.

6. Enunciate and explain the proposition known as the "Parallelogram of Velocities."

An insect crawls along the minute hand of a clock at the rate of 32 feet an hour. Find its velocity at the distance of one yard from the centre of the clock face.

7. A particle is projected up a rough inclined plane with velocity u. Find its position and velocity at the end of time t, the inclination of the plane to the horizon being i, and λ the angle of friction.

If the particle be projected with velocity due to falling from a point A above the point of projection, and through A a straight line be drawn at an angle X to the horizon, show that this line will meet the inclined plane at the point where the particle comes to rest.

8. If a particle be moving in a vertical plane under the action of gravity, prove that its path must be a parabola whose axis is vertical, and that the velocity at any point varies as the normal (terminated by the axis).

Find the two directions in which a particle may be projected, with a velocity of 60 feet per second, so as to hit an elevated object whose horizontal and vertical distances from the point of projection are 48 and 20 feet respectively; and prove that the times of describing the two paths are as 4 to 13.

9. Two spheres of masses m1 and m2 impinge obliquely with velocities u1 and u in directions making angles a1 and a respectively with the line of impact; find their velocities and directions after impact, e being the coefficient of restitution.

Three equal billiard balls A, B, C are lying on a billiard table; A and B are in contact, and equidistant from C. The ball C, projected very nearly in the direction of the point of contact, strikes A first and B immediately afterwards. Show that after impact the velocities of A and B are in the ratio of 4 to 3-e.

IO. Define work, potential energy and kinetic energy; and explain what is meant by the principle of the conservation of energy.

Three strings are knotted together at C. One string passes round a smooth peg A and supports a weight P at its free extremity; a second passes round a smooth peg B in the same horizontal line as A, and supports a weight Q; the third hangs vertically and supports a weight R. Prove that the work done in carrying C from its position of equilibrium to the peg A is

2 Qc sin2 10

where AB = c, and 0 is the angle which BC makes with the horizon in the position of equilibrium.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1897.

OBLIGATORY EXAMINATION.

I. EUCLID.

[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. In the absence of special directions to candidates, any of the propositions within the limits prescribed for examination may be used in the solution of problems and riders. Great importance will be attached to accuracy.]

I. Give definitions of parallel straight lines, rectangle, sector of a circle, reciprocal figures. State the enunciation of any proposition in the Sixth Book of Euclid in which duplicate ratio occurs, and give a numerical illustration.

2. Prove that if a straight line falls on two parallel straight lines it makes the alternate angles equal to one another.

3. To a given straight line, apply a parallelogram equal to a given triangle and having one of its angles equal to a given rectilineal angle.

4. Enunciate the two propositions which prove that, if P, Q, and R be three points in a straight line in any order, the sum of the squares on PR and QR differs from the square on PQ by twice a certain rectangle.

5. Divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part.

6.

Prove that if two circles meet they cannot have the same centre.

7. Prove that if, from the point of contact of a tangent to a circle, a chord be drawn, the angles which the chord makes with the tangent are equal to the angles which are in the alternate segments of the circle.

8. Prove that if two straight lines cut one another within a circle, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other.

9. Inscribe in a given circle an equilateral and equiangular hexagon.

IO.

Two obtuse-angled triangles have one acute angle of the one equal to an angle of the other, and the sides about the other acute angle in each proportionals; prove that the triangles are similar.

II. Prove that if four straight lines are proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means.

12. Prove that in equal circles, sectors have the same ratio which the arcs on which they stand have to one another.

13. If the vertical angle of an isosceles triangle is half the angle of an equilateral triangle, show that the base is greater than half of one of the equal sides.

14. ABCD is a quadrilateral in which ABC is a right angle; and the square on AD with twice the rectangle AB. CD is equal to the sum of the squares on AB, BC, and CD; show that the angle BCD is also a right angle.

15. A and B are fixed points on a circle whose centre is C, and P is a moving point on the circle; if AP revolves round A at a given rate, find the relative rates at which BP and CP revolve respectively round B and C.

16. ABCDEF is a regular hexagon inscribed in a circle whose centre is O, and P is any point on the smaller arc AB; show that the perpendicular from P on OC is equal to the sum of the perpendiculars from P on OA and OB.

17. If in a quadrilateral ABCD, CD touches the circle through A, B, and C; and the rectangle AB. AC is equal to the rectangle BC. CD; show that AB touches the circle through A, C, and D.