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IX. DYNAMICS.

[Great importance will be attached to accuracy of working.]

(N.B.-When needed the measure of the force of gravity may be taken as

32 feet.)

I. If a particle moves uniformly in a circle, distinguish between the angular and linear velocities. What is the relative velocity of two particles moving uniformly in a straight line (1) in the same, (2) in opposite directions? What is the relative angular velocity of the hour and minute hands of a clock, and their relative linear velocity, if the minute hand be 9 inches and the hour hand 3 inches in length?

Two trains whose lengths respectively were 130 and 110 feet, moving in opposite directions on parallel rails, were observed to be 4 seconds in completely passing each other, the velocity of the longest train being double that of the other; find at what rate per hour each train is moving.

2. Enunciate the first law of motion. State briefly the evidence on which we accept the truth of the law. How is the velocity of a body affected when acted on by a uniformly accelerating force? If a body projected upwards with a velocity (u), ascend through a space (s), obtain the equation v2=u2 – 64s.

A tower is 288 feet high; at the same instant one body is dropped from the top of the tower and another projected vertically upwards from the bottom, and they meet half way; find the initial velocity of the projected body, and its velocity when it meets the descending body.

3. Prove that the times of descent down all chords in a vertical circle, whether drawn from the highest or the lowest points of the circle, are

constant.

Two vertical circles whose radii are 10 and 6 feet touch each other at the highest point; a straight line is drawn from the point of contact to meet the outer circle; find the time of describing from rest the portion of this line intercepted between the two circles.

4. What is understood by the parallelogram of velocities? Find the resultant velocity of two uniform component velocities.

A particle moves in a straight line along a horizontal smooth plane with a velocity of 3 feet per second; after 2 seconds a velocity of 8 feet per second is imparted to it in a direction at right angles to its original motion; find the distance of the particle from its starting point after it has been in motion for 4 seconds.

5. Find the time of flight and greatest height of a projectile with reference to the horizontal plane passing through the point of projection.

If (t) be the time in which the projectile reaches a point P, and (ť) the time from P until it strikes the horizontal plane through the point of projection, prove that the height of P above that plane is tť. verify the expression for the greatest height.

Hence

6. State the third law of motion. If (W) be the weight of a body in pounds, what assumptions are made and what units are referred to in

obtaining the expression ( for the mass of the body?

A heavy body is placed on a smooth horizontal table, a pressure of 6 lbs. acts continuously upon it; at the end of three seconds the body is moving with a velocity of 48 feet in a second; find the weight of the body.

7. A weight (W) is drawn up a smooth inclined plane by means of a string, to the other end of which a weight (P) is attached that hangs freely over the top of the plane; find the accelerating force and the tension of the string.

If both (P) and (W) be 8 lbs., the inclination of the plane 30°, and the string be just on the point of breaking, find the greatest weight which the string would support if it were suspended from a fixed point vertically.

8. Define an impulsive force. How is such a force estimated? When one elastic ball impinges directly on another, describe briefly the action supposed to take place during their impact.

An elastic ball (m) moving with a given velocity impinges in direct impact on (m') at rest; find the velocity of (m') after impact, and determine the ratio of the relative velocity of the balls after impact to the original velocity of (m).

9. Point out briefly how a simple pendulum may be used to determine the force of gravity at the place where it swings.

A pendulum, whose length is L, makes (m) oscillations in a day its length is changed so that it makes (m+n) oscillations in a day; shew that

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IO. In the theory of work what is meant by a foot-pound? Shew that the kinetic energy of a body in motion is equal to half its "vis viva."

A train is moving on a horizontal rail at the rate of 15 miles an hour; if the steam be suddenly turned off, how far will it run before it stops, the resistances being taken at 8 pounds per ton?

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

JUNE, 1883.

PRELIMINARY EXAMINATION.

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

[Great importance will be attached to accuracy.]

I. If two straight lines cut one another, the vertical, or opposite, angles shall be equal.

ABC is a triangle, BD, CE lines drawn making equal angles with BC, and meeting the opposite sides in D and E and each other in F: prove that if the angle AFE is equal to the angle AFD the triangle is isosceles.

2. Triangles on equal bases, and between the same parallels, are equal to one another.

ACB is a triangle, CD, BE parallel lines meeting AB and AC produced respectively in D and E : prove that if the triangles BCE, ACB are equal D is the middle point of AB.

3. In any right-angled triangle, the square which is described on the side subtending the right angle is equal to the squares described on the sides which contain the right angle.

If the squares on the first and third sides of a quadrilateral are together equal to those on the second and fourth, the diagonals intersect at right angles.

4. If a straight line be bisected, and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced.

W. P.

I

7

5. If a straight line drawn through the centre of a circle, cut a straight line in it, which does not pass through the centre, at right angles, it shall bisect it.

Two equal circles have a common chord AB. If a chord AC of one of them, equal to AB, produced backwards pass through the centre of the other, AB is equal to the radius of either circle.

6. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same part of the circumference.

ABC is an isosceles triangle; on BC as base construct a triangle whose vertical angle shall be equal to either of the equal angles B or C, and which shall be similar to the given triangle.

7. In equal circles, equal angles stand on equal circumferences, whether they be at the centres or circumferences.

AB is a fixed chord in a circle APQB, PQ another chord of given length; shew that if AP, BQ meet in R, R will be on the circumference of the same circle for all positions of PQ.

8. Describe a circle about a given triangle.

If ABCD is a parallelogram, and BE makes with AB the angle ABE equal to the angle BAD, and meets DC produced in E, the circles described about the triangles BCD, BED will be equal.

9. If the radius AB of a circle is divided in C so that the rectangle AB, BC is equal to the square on AC, and the chord BD is equal to AC, the circle of which AD is a chord and which touches BD will pass through C, and the triangle ACD will be isosceles.

10. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those sides produced, proportionally.

Prove the following construction for trisecting a line AB in G and H: On AB as diagonal construct a parallelogram ACBD; bisect AC, BD, in E and F. Join DE, FC, cutting AB in G and H.

II. Triangles which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.

If ABC is a triangle right-angled at B, and BD the perpendicular on AC is produced to E so that DE is a third proportional to BD and DC, the triangle ADE will be equal to the triangle BDC.

I.

II. ARITHMETIC.

(Including the use of Common Logarithms.)

[N.B.-Great importance will be attached to accuracy.]

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divided by 161??

4. What fraction of 2 is the quotient of 11

5. Add together 5125 of a yard, '62734 of a pole, and 018325 of a furlong; subtract the result from 0049 of a mile, and express the answer as the decimal of a yard.

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9. Reduce of half a rood to the decimal of ¦ of an acre.

10. Express 9 hours 19 minutes and 3 seconds as the decimal of

9 days.

II.

12.

Divide 110 tons II cwt. 2 qrs. 16 lbs. 8 oz. by 79.

Find the dividend on £9,648. 8s. at 16s. old. in the £.

13. At what rate per cent. per annum will £1,885. 15s. amount to £2,569. 6s. 8d. in 7 years and 3 months, simple interest?

14. The French unit of volume is the stère, which is a cube whose side is a mètre.

Supposing a linear yard to be of a mètre, find to two places of decimals the difference, in cubic inches, between the volume of a cubic yard and that of a stère.

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