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ABCD is a square; forces of 1 lb., 6 lbs., and 9 lbs. act in the directions AB, AC, and AD respectively: find the magnitude of their resultant correct to two places of decimals.
3. A light triangular frame ABC stands in a vertical plane, C being uppermost, on two supports A, B, in the same horizontal line, and a weight of 18 lbs. is suspended from C. If AB=AC=6 feet, and BC=5 feet, find the pressures on the supports.
4. If four forces in equilibrium be parallel, two and two, prove that they form two unlike couples of equal moment.
A uniform ladder 13 feet long rests with one end against a smooth vertical wall, and the other on a rough horizontal plane at a point 6 feet from the wall. Find the friction between the ladder and the ground, the weight of the ladder being 56 lbs.
5. Find the centre of gravity of a pyramid, and show that it coincides with that of four equal particles placed at the angular points.
6. A triangular lamina ABC, obtuse-angled at C, stands with its side AC in contact with a table: show that the least weight which, suspended ·a2+362 - 2 from B, will overturn_it =} W where W=the weight of the triangle.
Interpret the above if c2 be > a2 +362.
7. A light rod rests wholly within a smooth hemispherical bowl of radius r, and a weight W is clamped on to the rod at a point whose distances from the ends are a, b. Show that, if @ be the inclination of the
rod to the horizon in the position of equilibrium, then sin 0 =
and find the pressures between the rod and the bowl.
a-b 2 r2-ab
8. Investigate the relation between the power and the weight in the case of a smooth screw.
What must be the length of the power arm of a screw having six threads to the inch in order that the mechanical efficiency may be 216?
9. State the laws of limiting friction.
The poles supporting a lawn-tennis net are kept in a vertical position by guy ropes, one to each pole, which pass round pegs 2 feet distant from the poles. If the coefficient of limiting friction between the ropes and pegs be, show that the inclination of the latter to the vertical must not be less than tan-1, the height of the poles being 4 feet.
10. Explain the terms virtual velocity and virtual moment of a force.
A uniform beam AB of weight W, movable in a vertical plane about a hinge at A, is kept in a given position by a force P applied at the end of a string PCB passing over C, a pulley vertically above A, and at a distance AC AB. Show that, if the system be slightly displaced owing to a small change in P
P× P's virtual velocity+ W× W's virtual velocity=0.
[The measure of the acceleration of gravity may be taken to be 32 when a foot and a second are the units of length and time.]
[Great importance will be attached to accuracy.]
1. Compare the velocities of the extremities of the hour, minute, and second-hands of a watch, their lengths being 48, 8, and 24 inches respectively.
Enunciate and prove the proposition known as the Parallelogram
Two points describe the same circle in such a manner that the line joining them always passes through a fixed point: show that at any moment their velocities are proportional to their distances from the fixed point.
3. Find the space described in time t by a particle projected with velocity u, subject to a constant acceleration f in the direction of its
If a, b, c, be the spaces described in the pth, qth, and rth seconds, prove that a (q-r)b(r− p)+c (p −q)=0.
4. If the acceleration due to gravity be taken as the unit acceleration, and the velocity generated in one minute as the unit of velocity, find the unit of length.
5. Two weights, P, Q, are connected by a weightless string which passes over a smooth pulley: find the acceleration and the tension of the string.
Two strings pass over a smooth pulley; on one side they are each attached to a weight of 5 lbs., and on the other to weights of 3 lbs. and lbs. respectively: find their tensions.
6. A body is projected in a given direction with a given velocity: find the range on a given inclined plane passing through the point of projection.
The angular elevation of an enemy's position on a hill & feet high is ß: show that, in order to shell it, the initial velocity of the projectile must not be less than √gh (1+cosec 8),
7. A ball of mass m moving with velocity u impinges directly upon a ball of mass m' moving with velocity u': find the velocities of the balls after impact, e being the coefficient of elasticity.
Two equal marbles A, B, lie in a horizontal circular groove at opposite ends of a diameter. A is projected along the groove, and after a time t impinges on B: show that a second impact will take place after a further interval 2.
8. A particle slides down a smooth curve in a vertical plane under the action of gravity: find the velocity of the particle in any position.
A bead slides on a wire bent into the form of a parabola whose axis is vertical and vertex upwards; if the bead be just displaced from its position of equilibrium, then at any subsequent time its velocity will vary as its distance from the axis.
9. A particle weighing oz. rests on a horizontal disc and is attached by two strings 4 feet long to the extremities of a diameter. If the disc be made to revolve 100 times a minute about a vertical axis through its centre, find the tension of each string.
10. A train weighing 200 tons is running at 40 miles an hour down an incline of 1 in 120: find the resistance necessary to stop the train in half a mile.
MATHEMATICAL EXAMINATION PAPERS
FOR ADMISSION INTO
Royal Military Academy, Woolwich,
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. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases, or third sides, equal; the angle which is contained by the two sides of the one is equal to the angle contained by the two sides, equal to them, of the other.
ACB, ADB are two triangles, on the same base AB and on the same side of it; AC is equal to BD, and AD to BC; and AD, BC intersect in 0: prove that the triangles OAB, OCD are each isosceles.
2. If a straight line falls upon two parallel straight lines, it makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle on the same side of the line, and the two interior angles on the same side of it together equal to two right angles.
AB and CD are any two diameters of a circle; prove that, if C and D be joined to B, the straight lines CB, DB will bisect the angles made with AB by a straight line through B parallel to CD.
3. If a parallelogram and a triangle are on the same base and between the same parallels, the parallelogram is double of the triangle.
Construct the greatest triangle which has two of its sides equal to two given straight lines each to each.
If a straight line is b'sected and produced to any point, the rectangle contained by the whole line thus produced and the part produced, together with the square on half the line bisected, is equal to the square on the line which is made up of the half and the part produced.
Find a point D in AB produced so that the rectangle AD, DB shall be equal to the square on a given straight line which is greater than half AB.
5. If a straight line is divided into two equal parts, and also into two unequal parts, the squares on the unequal parts are together double of the square on half the line, and of the square on the part between the points of section.
6. The angles in the same segment of a circle are equal to one another.
If two segments of circles have a common chord AB, and any points P and Q are taken on their arcs, the locus of the point of intersection of the bisectors of the angles PAQ, PBQ is another segment of a circle on the same chord AB.
7. Upon a given straight line describe a segment of a circle, which shall contain an angle equal to a given rectilineal angle.
8. Describe a circle about a given triangle.
Show that, if the centre of the circle described about a triangle coincides with the centre of the circle inscribed in it, the triangle must be equilateral.
9. The sides about the equal angles of equiangular triangles are proportionals, those sides being homologous which are opposite to the equal angles.
A and B are the points of intersection of two circles; CAD is a chord through A; and BC, BD are drawn: prove that BC is to BD as the diameter of the circle ABC is to that of the circle ABD.
IO. Similar triangles are to one another in the duplicate ratio of their homologous sides.
Of two equilateral and equiangular hexagons, one is inscribed in a circle and the other is described about the same circle: prove that the areas of the hexagons are to one another in the ratio of 3: 4.
II. In any right-angled triangle, a rectilineal figure described on the side subtending the right angle is equal to the similar and similarly described 'figures on the sides containing the right angle.
Explain the term "similarly described."