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6. In the use of the simple machines called "mechanical powers,' what is meant by a mechanical advantage? In any two of these machines state numerically the mechanical advantage.

Describe the construction of a pair of toothed wheels used as mechanical power, and find the equilibrium ratio between the power and the weight; express also that ratio when the teeth are very small compared with the radii of the wheels.

7. Shew how the two following laws of statical friction may be established by experiment:

(1) Limiting friction is proportional to the pressure:

(2) Limiting friction is independent of the surfaces in contact.

A weight of 30 lbs. is just supported on a rough inclined plane (coefficient of friction 2) whose height is ths of its length. Shew that it will require a force of 36 lbs. acting parallel to the plane just to be on the point of moving the weight up the plane.

8. A uniform ladder rests between a vertical wall and the horizontal ground, both rough; if the coefficient of friction for the ladder and wall be and for the ladder and ground; find the angle which the ladder makes with the ground when it just begins to slide.

9. State the principle of virtual velocities as applicable to the mechanical powers, and prove that it holds good on a smooth inclined plane when the power acts at any angle to the plane.

Explain the result if the expression for the virtual velocities of the power and weight in this case are equal.

IO.

cally?

How is work measured? How is horse-power estimated numeri

An engine is required to raise in 3 minutes a weight of 13 hundredweight from a pit whose depth is 840 feet; find the horse-power of the engine.

IX. DYNAMICS.

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

[N.B.-When needed the measure of the accelerating force of gravity may be taken as 32.]

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Find to three decimal places the velocity of the extremity of the minute hand of a watch an inch long, if π= the units of space and time being inches and minutes respectively.

2.

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If the velocity of the point A in fixed space is known, and also the velocity of the point B relative to A, how is B's velocity in fixed space determined?

If a watch be laid face upwards on a table and moved without rotation parallel to the line from 9 to 3 o'clock with the velocity with which the extremity P of the minute hand moves when the watch is at rest, prove that the velocity of P in fixed space is perpendicular and proportional to the line joining P at the instant with the position of P at any half-hour.

3. Find the space described by a moving particle in any time when there is uniform acceleration in the line of motion, and apply the result to determine the motion of a projectile.

A wet open umbrella is held with the handle upright and made to rotate round that handle at the rate of 14 revolutions in 33 seconds. If the rim of the umbrella be a circle of one yard diameter and its height above the ground be 4 feet, prove that the drops shaken off from the rim meet the ground in a circle of 5 feet diameter,

the air being neglected.

being 22; the effect of

4. Three bodies are projected simultaneously from the same point and in the same vertical plane, one vertically upwards, another at the angle of elevation 30°, and the third horizontally. If their velocities be in the ratio of 1: 1:√3, prove that they will always be in a straight line.

How does this line move in space?

5.

If APB be a vertical circle whose highest and lowest points are A and B, prove that the time down either AP or PB from rest is equal to the time down AB.

If the point be taken in AB such that AQ=AP, and if AP produced meet the tangent at B in R and a body slide down APR from rest, prove that the times of the body being within and without the circle are in the ratio of AQ to BQ.

6. Define the terms Energy (Kinetic and Potential), Work, and Power, and state what is meant by the "Conservation of Energy."

If the unit of Energy be that required to raise 1 lb. through foot (without gain of velocity), find the number of units of Kinetic Energy in a mass of 1 oz. moving 10 feet per second.

7. Find the velocities of two equal perfectly elastic spheres after oblique impact.

Prove that the directions of the relative velocities of the spheres before and after impact are equally inclined to the line of centres at the instant of impact.

8. Find the acceleration of a particle describing a circle with uniform velocity.

In question 3 prove that the force required to keep a drop of water attached to the umbrella's rim makes an angle whose tangent is with the vertical. If the drop weighs or of an oz., find to four decimal places the magnitude of this force.

9. State the Third Law of Motion.

A train weighing 50 tons is moving on a level at 30 miles an hour when the steam is shut off, and the brake being applied to the brake-van the train is stopped in a quarter of a mile. Find the weight of the brakevan, taking the coefficient of friction between its wheels and the rails to be one-sixth, and supposing the unlocked wheels of the train to roll without any sliding.

10. Explain the principle of the pendulum.

Two pendulums oscillating at two different places lose and 7 seconds a day respectively, and if the places at which they oscillate be interchanged they lose and 7' seconds; prove that t+T=t'+' nearly.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1883.

PRELIMINARY EXAMINATION.

I.

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

[Great importance will be attached to accuracy.]

Draw a straight line perpendicular to a given straight line of unlimited length, from a given point without it.

2. If a side of a triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles are equal to two right angles.

If ABC be any triangle, and through D the middle point of AB, DE is drawn parallel to BC, and BE be drawn to bisect the angle ABC and meet DE in E, AEB will be a right angle.

3. The complements of the parallelograms which are about the diameter of any parallelogram are equal to one another.

If ABCD be a parallelogram, and from K, a point on the diagonal AC, EKF be drawn parallel to AD to meet AB in E and DC in F, and HKG be drawn parallel to AB to meet AD in H and BC in G; the triangles AGF, AEH are together equal to the triangle ABC.

4. If a straight line be divided into any two parts, the squares on the whole line, and on one of the parts, are equal to twice the rectangle con

W. P.

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tained by the whole and that part, together with the square on the other part.

If AB be divided in C so that the square on AC is double the square on CB, the sum of AB and CB will be equal to the diameter of the square on AB.

5. If two circles touch each other internally, the straight line which joins their centres, being produced, shall pass through the point of contact. 6. The angles in the same segment of a circle are equal to one another.

A and B are the extremities of an arc of a circle, and Q any point on the arc is joined to A and B. If P be a point on AQ or AQ produced such that QP=QB, prove that P will lie on an arc of a fixed circle passing through A and B.

7. If from any point without a circle two straight lines be drawn, one of which cuts the circle but does not pass through the centre, and the other touches it: the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square on the line which touches it.

8. About a given circle describe a triangle equiangular to a given triangle.

9. If the angle ACB of the triangle ACB is bisected by CE, cutting AB in E, and another point D be taken in AB produced, such that the angle ECD is equal to CED, CD will touch the circle described about ACB.

IO. If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular to one another, and shall have those angles equal which are opposite to the homologous sides.

If APB is a semicircle of which AB is the diameter, and C the centre, Na point on CB, and AB is produced to 7 so that

AT: AC=AN: CN,

and PT is the tangent drawn from T, CNP will be a right angle.

II. If four straight lines be proportionals, the similar rectilineal figures similarly described upon them shall be proportionals.

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