VII. MECHANICS.

[Full marks may be obtained for about two-thirds of this paper. Great importance is attached to accuracy. N.B.-g may be taken = 32 feet per second per second.]

I. One end, B, of a light cord is fixed; the cord passes over a fixed peg, A, in the horizontal line through B, and the other end, C, of the cord hangs down vertically below A. If a mass of weight P is suspended from C, find the magnitude and direction of the pressure on the peg.

2. Give the definitions of the coefficient of friction and the angle of friction between two bodies. Describe also any method by which these magnitudes have been measured.

A mass of 190 lbs. is placed on a rough inclined plane the tangent of whose inclination to the horizon is, the coefficient of friction being; find the magnitude of the horizontal force which will just suffice to drag the body up.

What is the magnitude of the horizontal force which will just prevent the body from sliding down?

3. Define the moment of a force about an axis (or a point, explaining when this latter expression may be used).

A and B are two fixed points in a given plane, 10 inches apart; a force Pacts through A, but has any direction whatever in the plane; if P has always a moment of 60 inch-pounds' weight about B, give a graphic representation of the various magnitudes and directions of P.

What is the least, and what the greatest, value of P?

4. A ladder, AB, 15 feet long, rests against the ground at A and against a rough vertical wall at B, the coefficients of friction at A and B being and respectively; the centre of gravity, G, is 6 feet from A; find the inclination to the horizon at which the ladder will be just about to slip.

If the ladder is placed at an inclination tan-12, and a boy whose weight is of that of the ladder ascends it in this position, how far will he be able to go before the ladder begins to slip?

5. Which of the accelerations, 15 miles per hour per 2 minutes and 2 inches per second per second, is the greater ?

If a point moves from rest with the first of these, through how many feet will it move in to seconds?

6. If a particle slides down a rough inclined plane of inclination i, the coefficient of friction being μ, find its acceleration.

If tan i = 1, and u = 1, in what time will the particle move from rest over 125 feet of the plane?

7. A mass of 8 ounces hangs from a spring balance in a balloon : what tension will be indicated by the balance,

(a) if the balloon is moving upwards with constant velocity;

(b) if it is moving upwards with an acceleration of 2 feet per second per second ?

8. Enunciate the two principles on which the solution of the problem of the collision of two spheres depends.

Two spheres are approaching one another, their centres moving in one and the same straight line; their masses are 10 and 6 ounces, their respective velocities 40 and 60 feet per second, and their coefficient of restitution is; find their velocities after collision.

If they are in contact for 4 of a second, find, in ounces' weight, the magnitude of the mean pressure between them.

9. If a particle moves in a circle of radius r with a velocity v, prove that it has an inward normal acceleration equal to

22

r

.

If the mass of the particle is 2 ounces, the radius of the circle 8 inches, and the velocity at all points 16 feet per second, what is the magnitude of the resultant force acting on the particle in each position, and what is its precise direction?

Point out the erroneous conception involved in the term "centrifugal force."

10. State, in a general way, the kind of effect produced on the trajectory of a projectile by the resistance of the air.

If the resistance of the air can be neglected when a projectile is fired at an elevation tan-112 with a velocity of 520 feet per second, when and where will the projectile strike an inclined plane passing through the point of projection, the inclination of this plane being tan -11%?

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

JUNE, 1896.

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. Great importance will be attached to accuracy.]

I. ABC and DEF are two triangles. If the sides BC, CA, AB are equal to the sides EF, FD, DE respectively, prove that the angle BAC is equal to the angle EDF.

2.

ABC is any triangle, D any point inside the triangle. Prove that BD and DC are together less than BA and AC, and that the angle BDC is greater than the angle BAC.

The angle BAC is bisected by a straight line meeting BC in E, and P is any point on this straight line within the triangle. Prove that

(i.) BA and AC are respectively greater than BE and EC;
(ii.) BA and AC are respectively greater than BP and PC.

3. Write down Euclid's axiom with regard to parallel straight lines.

OA, OB are two finite straight lines meeting at O. C and D are any two points in OA and OB respectively. Through C and D straight lines are drawn at right angles to OC and OD respectively. Prove that these straight lines, if produced indefinitely in both directions, must meet.

4.

C is a point in a finite straight line AB. Prove that the square on AB is equal to the squares on AC and CB, together with twice the rectangle contained by AC and CB.

Express this result algebraically.

5. ABC is an acute-angled triangle. AD is drawn perpendicular to BC. Prove that the squares on AB and BC are together equal to the square on AC and twice the rectangle contained by BC and BD.

If M is the middle point of BC, prove that the difference of the squares on AB and AC is equal to twice the rectangle contained by BC and MD.

6. If a chord and a diameter of a circle intersect at right angles, prove that the diameter bisects the chord.

:

AB is a diameter of a circle. PQ is a chord not at right angles to AB. AM and BN are drawn perpendicular to PQ. Prove that PM = NQ.

7. If two circles touch one another externally, prove that the straight line which joins their centres passes through the point of contact.

8. Show how to draw tangents to a circle from a given point outside it.

Pand Q are points on a diameter of a circle produced, and are at equal distances from the centre. PR and QT are tangents from P and Q, the points Rand 7 lying on opposite sides of PQ. Prove that PRQT is a parallelogram.

9. Prove that angles in the same segment of a circle are equal.

AB and CD are two intersecting chords of a circle. If the arcs AD and BC are together equal to the arcs DB and CA, prove that AB and CD are at right angles to one another.

10.

Show how to describe a triangle whose sides shall touch a given circle and whose angles shall be equal to the angles of a given triangle.

II. Show how to find a mean proportional between two given straight lines.

12. AB is a diameter of a circle, CD is a chord at right angles to it. If any chord AP drawn from A cuts CD in Q, prove that the rectangle contained by AP and AQ is constant for different positions of P.

3. In how many years will £1500 amount to £1781. 5s. od. at 21 per cent. per annum simple interest ?

4. Find the value of 30875 of a mile in furlongs, poles, and yards.

5.

of £53.

Express the sum of 71⁄2 guineas 31⁄2 crowns and 54 florins as a fraction

6. What is the rental of an estate of 645 acres 3 roods 25 poles at £2. 11s. 4d. per acre?

7. Find the least common multiple of 555, 1221, and 2035.

8. Find the cost of a carpet, for a room 24 feet long and 171 feet wide,

at 3s. 3d. per square yard, if a margin 2 feet wide be left uncovered.

9. A watch is offered for sale for £5. 15s. od.; and, if that price is reduced by 5 per cent., the dealer who is selling it will still make 94 per cent. profit: how much did the watch cost him ?

10.

State the rule for determining the remainder in a division sum when it is worked by dividing by two factors of the divisor successively. instead of by their product.

As an example, reduce 107 lbs. to quarters by dividing by 4 and by 7 ; and explain why you do what you do to find how many pounds there are

over.

II. A man goes bankrupt with £1160 assets. His liabilities are the present value of three loans, at simple interest, amounting in all to £1800; one of them obtained 8 years ago at 4 per cent.; another, double of the first, 4 years ago at 5 per cent.; and the third, equal to the sum of the other two, a year ago at 8 per cent. How much does each of the three creditors lose?