VIII. STATICS.

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

I. In statics, forces are designated as pressure, tension, resistance, reaction: explain these terms. Shew how the direction of a tension may be changed without changing its magnitude.

2. It is assumed as a statical principle that when a force acts upon a body the effect of the force will be unchanged at whatever point of its direction it may be applied: illustrate this, and give some reasons for the assumption.

A string suspended from a ceiling supports three equal weights of four pounds, one at its lowest point and each of the others at equal distances from its extremities: find the tensions of the parts into which the string is divided by the weights.

3. If any number of forces acting upon a particle be represented in magnitude and way of action by the sides of a polygon taken in order, they will keep the particle at rest.

If the forces acting on a point be represented by the sides of a regular pentagon taken in order, shew at what angles the forces acting on the point are inclined to each other, and if the forces be so taken as acting on a point, prove independently of the polygon of forces that their resultant is

zero.

4. When three forces acting in one plane maintain a rigid body in equilibrium, shew that their lines of action either all meet in a point or are all parallel.

In the latter case, prove the relation that must subsist amongst the forces.

The horizontal roadway of a bridge is 30 feet long and weighs 6 tons, and it rests on similar supports on its ends; what pressure is borne by each of the supports when a carriage weighing 2 tons is one-third of the way across the bridge?

5. Shew that a system of forces not in equilibrium acting in one plane on a rigid body will be equivalent either to a single resultant or to a couple.

ABC is a triangle, AE, BF, CD lines drawn from the angles to the points of bisection of the opposite sides: shew that the forces represented by AE, BF, and CD are in equilibrium.

6. If three heavy particles are in one plane in given positions, shew how to find the distance of the common centre of gravity of the particles from either one of them.

If three heavy particles be placed in the angles A, B, C of a triangle, the weights of each being proportional to the opposite sides of the triangle a, b, c, prove that the distance of the centre of gravity of the particles from A is equal to

7. A common steelyard supposed uniform is 40 inches long, the weight of the beam is equal to the movable weight, and the greatest weight that can be weighed by it is four times the movable weight: find the place of the fulcrum.

8. State and prove the relation between the power and the weight when they are in equilibrium on a wheel and axle.

9. A uniform ladder 70 feet long is equally inclined to a vertical wall and the horizontal ground, both rough; the weight of a man with his burden ascending the ladder is 2 cwt., and the ladder weighs 4 cwt.: how far up the ladder can the man ascend before it slips, the tangent of the angle of resistance for the wall being and for the ground?

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10. When is an engine said to work with one horse-power?

How many cubic feet of water will an engine of 100 horse-power raise in one hour from a depth of 150 feet, if the modulus of the engine be '5; the weight of a cubic foot of water being 62.5 pounds?

IX. DYNAMICS.

Great importance will be attached to accuracy in results.

[N.B.-When needed, the acceleration due to gravity may be taken equal to 322 feet per second.]

I. Define the term acceleration, and explain how it is used in measuring the rate at which a variable velocity is changing.

Prove the formula 2s=ft2, connecting the space moved over, the acceleration, and the time.

2.

How will the measure of the acceleration due to gravity be changed, if the unit of time is altered from one second to one minute?

Find also what must be the unit of space, if gravity be represented by the number 14, when the unit of time is 5 seconds.

3.

Enunciate and prove the theorem known as the Parallelogram of

Velocities.

Two steamers, X and Y, are respectively at points A and B, 5 miles apart. X steams away with an uniform velocity of 10 miles an hour in a direction making an angle of 60° with AB. Find in what direction y must start at the same moment, if it steam with an uniform velocity of 103 miles per hour, in order that it may come into collision with X, and at what angle it will strike it.

4. Prove that the time of falling from rest down a chord of a vertical circle drawn from the highest point is constant.

If particles start from rest from a given point to run down a number of smooth inclined planes, shew that at the end of seconds they will all be at the same distance from a point feet below that from which they started.

5. Prove that a particle projected (in vacuo) in any direction not vertical, and acted on by gravity, will describe a parabola; and find the latus rectum of the parabola described.

6. A particle is projected from the foot of an inclined plane whose inclination is (B), and in a direction making an angle of 60° with the horizon. If its range on the inclined plane is equal to the distance through which another particle would fall from rest during the time which elapses before the first particle hits the plane, find ß.

7. Two balls of elasticity e impinge directly; find their subsequent velocities.

If e=, and their masses are as 2 to 1, and their respective velocities before impact as 1 to 2, and in opposite directions, shew that each ball will move back after impact with ths of its original velocity.

8. Two weights are connected by a string which passes over a smooth fixed pulley; determine the motion.

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If the weights, being each equal to 8 oz., are in equilibrium, and

oz. is then added to one of them, determine how long it will be in descending 10 feet, and what velocity it will acquire in so doing.

9. If a particle run down the arc of a smooth vertical circle, starting from rest at the highest point, prove that it will quit it on reaching a point whose perpendicular distance below the starting point is one-third of the radius.

IO. If a seconds pendulum be carried to the top of a mountain half a mile high, how many seconds will it lose in a day, if gravity vary as the inverse square of the distance from the earth's centre, which is supposed to be 4,000 miles from the foot of the mountain?

II. Prove that the principle of Virtual Velocities is true for a single movable pulley when the strings are not parallel.

Define the term horse-power, and find that of an engine which will travel at 25 miles per hour up an incline of 1 in 100, the weight of the engine and load being 50 tons, and the resistance 10 lbs. per ton.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1882.

PRELIMINARY EXAMINATION.

I. EUCLID (Books I.-IV. and VI.). (Obligatory.)
[Great importance will be attached to accuracy in working.]

I. Define a right angle. Draw a straight line at right angles to a given straight line from a given point in the same.

If the given point be at the extremity of the given line, shew how to draw the perpendicular without producing the given line.

2.

Prove that all the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

There are two regular polygons, the number of sides of one is double the number of sides of the other, and an angle of one polygon is to an angle of the other as 9 to 8; find the number of sides of each polygon.

3. Prove that the complements of a parallelogram which are about the diameter of any parallelogram are equal to each other.

If any point P be taken on the diagonal AC of a parallelogram ABCD, shew that the sum of the triangles APB, PCD is half the parallelogram.

4. If a given straight line be divided into any two parts, prove that the sum of the squares of the parts, together with twice the rectangle contained by the parts, is invariable, in whatever way the line may be so divided.