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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.]

What is meant by uniform velocity, and how is it measured? What by uniform acceleration, and how measured?

A body starting with a velocity of 10 feet describes 11 feet in one second. If the acceleration on it be uniform, find its value.

2. A steamer is going due N. with velocity v; the smoke from the chimney points 0 degrees South of E. If the wind is due W., find its velocity.

3. If the measure of an acceleration is f, what change in this measure is produced by taking double the unit of time as the new unit? Prove the result.

If the measure of gravity-acceleration be 1, what is necessary as to the units of space and time?

4. Two bodies start together from rest, from one point, in two diverging lines, each moving with a given acceleration. Show that their velocity of separation is uniformly accelerated during the motion.

5. How is angular velocity measured when uniform? A body revolves round an axis with a uniform angular velocity w; how many complete revolutions does it make per minute?

A point moves so that its angular velocities round two fixed points are always equal and in the same sense; find the path it describes.

6. A perfectly elastic ball strikes another equal ball moving in the same direction. Show from the principles of impact, and without assuming any formulae, that the balls exchange velocities.

7. Given the initial velocity and elevation of a projectile, construct geometrically its position, and the direction in which it is moving, at any given moment.

From your construction (or otherwise) show that if any number of projectiles are thrown together from one point, whatever be their initial motions, their directions of motion at any instant all pass through one point.

8. A vertical hollow cylinder, 3 feet in circumference, is revolving uniformly round its axis, making one turn per second. A heavy particle is let drop from a point at the mouth of the cylinder, so as just to touch the inside surface during its fall and leave a trace upon it. If the cylinder be afterwards unrolled, what form will the trace present?

9. A body moves uniformly with given velocity in a circle. Is there necessarily any force acting on the body? If so, state its magnitude and direction at any moment.

A smooth hollow sphere is revolving with uniform angular velocity w round a vertical diameter. Show that a heavy particle placed inside will only remain, resting against the side of the sphere, at one particular level.

If the angular velocity w be less than a certain limit, show that the particle will remain at the lowest point of the sphere.

10. State the principle of work with regard to the motion (1) of a particle acted on by force, (2) of a rigid body. If a body is at rest, and any forces are applied to it, and it moves under the action of those forces, until it arrives at another position where it is also at rest, what can be inferred as to the action of the forces?

A uniform rectangular block ABCD stands on its base AD on a rough floor. It is pulled at C by a horizontal force just great enough to begin to turn it round the corner D. If this same force continues to pull it horizontally at C till the block has turned through an angle 0, and then ceases, prove that the block will have acquired sufficient momentum to cause it just to overturn round D, provided sin 0=tana, where a<BDC.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1887.

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. The angles which one straight line makes with another straight line on one side of it, either are two right angles, or are together equal to two right angles.

The straight lines bisecting the angles at the base BC of an isosceles triangle ABC intersect in D. Show that AD produced bisects the angle BDC.

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.

Describe an isoceles triangle in which each of the angles at the base is one-fourth of the vertical angle.

3. Distinguish between a Problem and a Theorem, and explain what is meant by the converse of a Theorem.

Enunciate carefully, and prove, the converse of the following theorem : "Triangles on equal bases and between the same parallels are equal.”

W. P.

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4. If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the two parts.

Prove that, of all right-angled parallelograms having the same perimeter, the square has the shortest diagonals.

5. In every triangle the square on the side subtending an acute angle is less than the squares on the sides containing that angle by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle and the acute angle.

6. The diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote; and the greater is nearer to the centre than the less.

7. In a circle the angle in a semicircle is a right angle, but the angle in a segment greater than a semicircle is less than a right angle, and the angle in a segment less than a semicircle is greater than a right angle.

Two circles, whose centres are O, O', touch each other in the point A, and are met by a common tangent in the points B, C, respectively. BOD, CƠ'E, are diameters of the two circles. Show that BE, CD, intersect in A.

8. Inscribe a circle in a given square.

Two squares are described, one circumscribing and the other inscribed in a given circle. Prove that the diameter of the circle inscribed in the smaller square is equal to the radius of the circle described about the larger square.

9. How does Euclid test whether four magnitudes are proportionals? Find a fourth proportional to three given straight lines.

IO. If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means; and, if the rectangle contained by the means be equal to the rectangle contained by the extremes, the four straight lines are proportionals.

Describe a rectangle so that its sides are in a given ratio, and its area is equal to a given square.

11. Parallelograms about the diameter of any parallelogram are similar to the whole parallelogram and to one another.

Show that either of the complements is a mean proportional between the two parallelograms about the diameter.

II. ARITHMETIC.

(Including the Use of Common Logarithms.)

[N.B.-Great importance is attached to accuracy. Candidates are expected to use Arithmetical methods of solution.]

I. Multiply 909 by 990, and explain your method of work.

2.

What is the value of 6 oz. 12 dwts. of gold when I oz. 13 dwts. cost £4. 18s. 10d.?

3. State and explain the rule for division of fractions. Divide 33 by 79, and bring the result to its lowest terms.

4. If the sum of the digits of a number be divisible by 9, prove that the number itself is also divisible by 9.

5. Divide 0053 by 2'5, and 53 by 0025. Explain your rule for pointing in the quotient in each case.

6. Reduce £2. 15s. 6d. to the decimal of a pound: and find the difference between 625 and '0625 of a cwt.

7. Reduce 007648 to a vulgar fraction; and add together *125, 4°163, 1143, and 2.54 without reducing them to fractions.

8. If the carriage of 150 feet of wood, that weighs 3 stone per foot, costs £3 for 40 miles, how much will the carriage of 54 feet of marble, weighing 8 stone per foot, cost for 25 miles?

9. Find, by Practice, the cost of 92 tons 7 cwt. 3 qrs. 12 lbs. at £1. 125. 7d. per ton.

root.

IO. State and explain the rule of Pointing in the extraction of a square Extract the square root of 197.96492 to 7 places of decimals.

II.

Find the solid content of a block of marble 3 ft. 8 in. 4' long, I ft. 7 in. 6' broad, and 1 ft. 4 in. 9' thick (by duodecimals). Express your result in cubic feet and inches.

12. In how many years will £768. 175. 6d. amount to £1230. 45. at 5 per cent. simple interest?

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