VIII. STATICS.

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

Full marks may be obtained by doing three-fourths of this paper.

I. If the parallelogram of forces be true for the direction of the resultant of any force and each of two other forces taken separately, then it will be true for the direction of the resultant of that force and the two other forces taken together.

2. Deduce the triangle of forces and the polygon of forces from the parallelogram of forces.

If the resultant R of the two forces P and Q inclined to each other at any given angle make the angle with P, prove that the resultant of the 0

forces (P+R) and Q at the same angle will make the angle - with P+R.

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3. If any number of forces in one plane act upon a rigid body, prove that they may be always replaced by a single force acting at any point and a couple.

4. Find the centre of gravity of a pyramid.

If ABCD be a tetrahedron, and if the plane CDE passing through the edge CD cuts AB in E, prove that the line joining the centres of gravity of the tetrahedrons ABCD and AECD is parallel to AB.

5. Find the conditions of equilibrium when any number of forces in one plane act on a rigid body.

Four heavy rods, equal in, all respects, are freely jointed together at their extremities so as to form the rhombus ABCD. If this rhombus be suspended by two strings attached to the middle points of AB and AD, each string being inclined at the angle ℗ to the vertical, prove that in the position of equilibrium the angles of the rhombus will be 20 and π- 20.

6. Find the power necessary to support the weight W in a system of movable weightless pulleys in which each string is attached to the weight.

If in such a system each pulley have the weight w, and the sum of the weights of the pulleys be W', and P and W be the power and weight in this case, prove that the power P+w would support the weight W+W' in the same system, if the pulleys had no weight.

7. Find the relation of the power to the weight in the inclined plane, the power acting at any angle to the plane.

A wedge with angle 60o is placed upon a smooth table, and a weight of 20 lbs. on the slant face is supported by a string lying on that face passing through a smooth ring at the top and supporting a weight W hanging vertically. Find the magnitude of W. Find also the force necessary to keep the wedge at rest (1) when the ring is not attached to the wedge, (2) when it is so attached.

8. State the principal laws of statical friction, and find the answers in the last case, supposing the slant face of the wedge to be rough, the co

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efficient of friction being and the 20 lbs. weight on the point of moving √3 down.

9. Distinguish between stable and unstable equilibrium. Two particles A and B are connected by a rod AB and laid upon a smooth table. The particle A is acted on by a force F parallel to a line Ox in the table and in the direction from 0 to x, and by an equal force F parallel to another line Oy from O towards y. The particle B is acted on by two exactly equal and opposite forces to those on A. Prove that wherever AB is placed on the table there are two positions of equilibrium for AB, stable when A is further from O than B, and unstable when A is nearer to 0.

IX. DYNAMICS.

[N.B.-When needed the force of gravity may be taken as 32 feet. Great importance will be attached to accuracy in results.]

I.

Full marks may be gained by doing eight-ninths of this paper.

When a body moves with a uniform velocity, establish the relation

s to which connects the time, space, and velocity.

The velocity of the extremity of the minute hand of a clock is 48 times the velocity of the extremity of the hour hand, which is 3 inches long; find the length of the minute hand.

How does it

2. Define acceleration, and state how it is measured. appear that the accelerating force of gravity is independent of the weight of falling bodies? Give instances in which the force accelerating the motion of a body may be half the force of gravity. Find the space described in a given time by a body starting with a given velocity, and moving with uniform acceleration.

A body projected perpendicularly downwards describes 720 feet in (^) seconds, and 2,240 feet in (27) seconds; find (t) and the velocity of projection.

3. If a body be projected down a smooth inclined plane with a velocity V, prove that the velocity at the foot of the plane will be independent of the length of the plane if the height of the point of projection above the horizontal plane be given.

A body begins to slide down a smooth inclined plane from the top, and at the same instant another body is projected upwards from the foot of the plane with such a velocity that the bodies meet in the middle of the plane ; find that velocity of projection, and determine the velocities of each body when they meet.

Enunciate the second law of motion, and refer briefly to any experimental facts which lead to its adoption.

Three velocities, whose ratios are as √3+ :√6: 2, are simultaneously impressed on a particle and the particle does not move; find the angles at which the directions of the velocities are inclined to each other.

5. Find the range on a horizontal plane of a projectile in vacuo. Determine the angle of projection when the range is equal to the height due to the velocity of projection.

Find also the direction in which the projectile is moving at any point of its curve.

6. A projectile is projected from the foot of an inclined plane whose inclination is (B) with velocity (v); if (0) be the angle between the direction of projection and the inclined plane, prove that the time of flight when the 27 sin 0

projectile strikes the plane is

Shew that the projectile will strike the plane at right angles if cot ẞ=2 tan 0.

7. Define moving force, and shew how the equation w=mg is obtained, where (w) is the weight and (m) the mass of a body.

Describe Attwood's machine, and shew how the force of gravity (g) might be ascertained by means of this machine.

8. A weight (P) hanging freely descends, raising a weight (W) by means of a string passing over a smooth peg; find the accelerating force and the tension of the string.

When (W) has been in motion from rest for 3 seconds the string is suddenly cut; find (P) so that (W) may ascend through 16ths of a foot before it begins to descend, the weight of (W) being four ounces.

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9. A ball (m) impinges directly on a ball (m') at rest, with a given velocity (v); find the velocity of each after impact, the modulus of elasticity being (e).

10.

Describe the mathematical assumptions by means of which the time of an oscillation of a simple pendulum describing a small circular arc is found to be π

If a pendulum that oscillates seconds be lengthened by its hundredth part, find the number of oscillations it will lose in 24 hours.

II. How is the kinetic energy of a moving body expressed in terms of its "vis viva"? A train of 20 tons is moving at the rate of 30 miles an hour, what is the measure of its "accumulated work"? Explain the term foot-pounds.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy,
Academy, Woolwich,

JUNE, 1882.

PRELIMINARY EXAMINATION.

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

[Great importance will be attached to accuracy.]

i. What is Euclid's definition of a straight line? Is this definition available for demonstration? What axiom concerning straight lines furnishes the test required? Enunciate and prove the proposition in Euclid in the proof of which this axiom is first referred to.

2. Equal triangles upon equal bases in the same straight line and towards the same parts are between the same parallels.

ABC is a triangle; join D and E, the middle points of AB and AC: prove, by the use of propositions of the first book only, that DE is parallel to BC.

3. Describe a parallelogram equal to a given triangle and having one of its angles equal to a given rectilineal angle.

Describe a parallelogram, the area and the perimeter of which shall be each equal to the area and perimeter of a given triangle.

4.

Enunciate and prove the proposition from which the corollary is inferred that "the difference of the squares of two unequal straight lines is equal to the rectangle contained by their sum and difference."

Find the straight line the square of which shall be equal to the rectangle contained by the sum and difference of two given straight lines.

5. Divide a straight line into two parts so that the rectangle contained by the whole line and one of the parts shall be equal to the square of the other part.

If AB be divided in C so that the rectangle AB, BC is equal to the square on AC, and CD be taken equal to BC, shew that AC is divided in D so that the rectangle AC, AD is equal to the square on CD.

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

Two circles cut one another, through a point of intersection draw a straight line terminated by the circumferences, so that the chords so intercepted in each circle may be equal.

7. The angle in a semicircle is a right angle, 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.

When are segments of circles said to be similar? If two circles touch each other externally, any straight line drawn through the point of contact will cut off similar segments; when will all the four segments cut off be similar? .

8. Describe a circle about a given triangle.

Hence shew that the perpendiculars drawn from the middle points of the sides of a triangle meet in the same point.

9. Describe a circle about an equilateral and equiangular pentagon; assuming that the straight lines bisecting each of the angles of the pentagon intersect in the same point.

If A, B, C, D, E be the angles of the pentagon taken in order, prove that the line CE is parallel to AB.

10.

Define similar rectilineal figures; if the figures be triangles, is there anything superfluous in the definition?

Similar triangles are to each other in the duplicate ratio of their homologous sides.

ABC is a triangle, AE and BF intersecting in G are drawn to bisect the sides BC, AC in E and F; compare the areas of the triangles AGB, FGE.

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II. If an angle of a triangle be bisected by a straight line which cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight line which bisects the angle.

If ABC be a right-angled triangle, whose right angle B is bisected by BF, cutting the base in F and meeting the circumference described about ABC in D, prove that the rectangle contained by BD and BF is equal to twice the area of ABC.