APPLIED MATHEMATICS: FIRST PAPER.

Examiner.-Prof. C. NIVEN.

Students in naval architecture and marine engineering of first year.

1. State the parallelogram of forces, and deduce from it that, if three forces acting on a particle be in equilibrium, each is proportional to the sine of the angle between the other two.

A small ring of given weight rests on the arc of a smooth circular hoop which is fixed in a vertical plane, being attached to the highest point by a string whose length is equal to the radius of the hoop; find the tension of the string and the pressure on the hoop.

2. Investigate the conditions of equilibrium of a particle acted on by any number of forces in one plane.

Three equal spheres rest in one vertical plane against each other, being suspended from a point by strings each equal to the radius of one of the spheres and attached to points in their surfaces. Find the tensions of the strings and the pressures between the spheres.

3. Define the center of gravity of a body, and prove that the center of gravity of a uniform triangular plate coincides with that of three equal particles placed at its angles.

Two uniform rods of the same substance and thickness, and of lengths 5 and 3 inches, are rigidly connected at one end A so as to be at right angles, and are suspended by a string so that the other ends are in the same horizontal plane. Show that the distance from A of the point on the longer rod at which the string is attached is 1.0225 inch.

4. Determine the resultant of a number of forces in one plane. Prove that the equation of the line of action of the resultant is

x'Z(Y) — y'Σ(X) = 2(xY — yX).

5. State the principal laws of friction which have been deduced from experiment. How can the coefficient of friction be found?

6. Determine the center of gravity

(i) of a sector of a circle.

(ii) of a segment of a circle.

7. Describe the different systems of pulleys, and find whether a weight of 15 lbs. will be able to support 2 cwt. in a system in which there are four movable pulleys, the strings around which are attached to a fixed beam, and each of which weighs one pound.

8. Explain the principle of the screw, and determine the mechanical advantage gained.

9. Explain the difference, in dynamics, between a ton and the weight of a ton. What are their numerical values on the foot-pound-second system? If the weight of a ton be the unit of force, and a minute and yard those of time and length, what will be the unit of mass?

10. Establish the equation v2 = V2+2gx for the motion of a falling body, and express it in the language of the science of energy.

A ball of 10 lbs. is dropped from a height of 289.8 feet, but, after falling half-way, it explodes into two equal parts, one of which is reduced by the explosion to rest. Find the subsequent motion of each part, and determine the kinetic energy developed by the explosion.

11. Find the range of a projectile on a horizontal plane passing through the point of projection.

A particle projected from a point, A, in the floor of a room returns to A after striking one of the opposite walls and the floor successively. Prove that if it strike the

wall at right angles and the floor once its elasticity, and that it strikes the floor half-way between the foot of the wall with half its original velocity and exactly opposite to its original direction.

12. Two balls, A, B, whose masses are as 2: 1, and which are moving in opposite directions, collide. If the first ball be brought to rest, and the coefficient of elasticity be, prove that their original velocities are as 7: 5.

APPLIED MATHEMATICS: SECOND PAPER.

Examiner.-T. S. ALDIS, Esq., M. A.

Students in naval architecture and marine engineering of first year.

1. Define fluid, vapor, gas.

Show that the pressure at any point in a fluid at rest is the same in every direction. 2. What do you mean by specific gravity? How would you compare the specific gravities of (a) two coins, (b) two samples of milk?

3. What is the center of pressure and the total resultant pressure? Find them in the case of an equilateral triangle one of whose sides (12 feet long) is on the surface and the opposite angular point 10 feet beneath it.

4. A hollow cylinder, a foot long, closed at the upper end, weighs as much as half the water it will hold. It is sunk, with the closed end uppermost, in a vertical position in water. How high will the water rise within it when the top is a foot below the surface? At what depth will it rest in equilibrium?

5. Explain how a ship can tack against the wind. A Chinese junk will run before the wind faster than an English ship. In what case, and why, will the ship outsail the junk?

6. In passing from the freezing to the boiling point, air expands .366 of its volume. A cubic foot of air at 60° F. (the barometer standing at 29 in.) weighs 527 grains. What will be the weight of a cubic foot of air when the thermometer is at 90° and the barometer at 28 in. ?

7. Determine the conditions of equilibrium of a floating body. Why is an ironclad with a low free-board specially unfitted to carry sail?

8. In Atwood's machine equal weights of 10 ozs. are suspended to the string which passes over the pulley and a bar of 1 oz. weight is placed across one. This, after falling through the space of a foot, passes through a ring which removes the 1 oz. weight. How far will the 10 oz. weight descend in the next minute?

9. Show how to calculate the space described in a given time under the action of a uniformly accelerating force, the motion being in a straight line.

A stone thrown down a rough board inclined at an angle of 30° neither gains nor loses velocity in its descent. What velocity will it gain by falling down the board (which is 20 feet long) when it is inclined at an angle of 60° ?

10. A particle revolves in an ellipse about a center of force in the focus. Calculate the law of attraction.

11. Explain the action of the governor of a steam-engine.

Show how to calculate

the position it will assume for a given number of revolutions per minute, neglecting all weights but those of the balls.

12. A smooth bead slides down the arc of a cycloid; determine the motion.

APPLIED MATHEMATICS.-FIRST PAPER.

Examiner.-T. S. ALDIS, Esq., M. A.

Students in naval architecture and marine engineering of second and third years.

1. State and prove, for direction, the principle of the parallelogram of forces, explaining clearly the assumptions you make.

ABCD is a quadrilateral figure. Forces act along BA, BC, DA, and DC proportional to them, show that their resultant is a single force represented by four times the straight line which joins the middle point of BD to the middle point of AC.

2. Show how to find the resultant of any number of forces acting on a rigid body. A cube has forces proportional to 1, 2, 3, 4 acting along the edges of one face taken in order. Forces proportional to 4, 1, 2, 3 act along the corresponding edges of the opposite face in the opposite direction. Find the resultant.

3. Show how to find the C. of G. of a solid of revolution, and find it in the case of a hemisphere.

4. State the laws of friction.

Show how to calculate the total friction in the case of a rope stretched round a rough cylinder.

5. Explain the principle of the arch, and show how you would calculate the curve required for a given arrangement of the load upon it.

6. What are the laws of motion? Give an experimental illustration of each. Investigate formulæ for the motion of a particle on an inclined plane under the action of gravity.

7. Calculate the motion of a body projected obliquely and acted on by gravity.

A building 20 feet high, 20 feet wide, and 30 feet long is surmounted by a gable roof rising 20 feet higher. A smooth stone is projected horizontally with a velocity of 2 feet per second just along one side of the ridge from one end of it. Find where it will strike the ground.

8. Calculate the motion of a particle acted on by a central force varying as the distance,

A weight hangs from a peg by an elastic string which it stretches to double its unstretched length. If the weight be slightly displaced, find the time of a small vertical oscillation.

9. A block of wood thrown on ice with a velocity of 10 feet per second is brought to rest after passing over 30 yards. A bullet of equal weight with the block is then shot into it with a velocity of 100 feet per second. Determine the subsequent motion.

10. A ball is dropped from a height of 10 feet on a plane inclined at an angle of 30°; the coefficient of elasticity is ; find the points where the ball will again strike the plane.

11. Two perfectly elastic particles are revolving in the same direction and in the same plane round a center of force varying inversely as the square of the distance. One is moving in a circular orbit, the other in a parabola whose latus rectum equals the diameter of the circle. They collide as the second particle is approaching the center of force. Determine the subsequent motions.

APPLIED MATHEMATICS: SECOND PAPER.

Examiner.-Prof. C. NIVEN.

Students in naval architecture and marine engineering of second and and third years

1. Determine the general equations of equilibrium of a fluid; and show that, when the external forces are such as arise from a potential, the surfaces of equal potential, of equal density, and of equal pressure coincide.

A heavy liquid is contained in a vessel and is also under the action of two centers of force which are in the same vertical line, and which exert equal forces at equal distances, but one of which is repulsive and the other attractive. The law of force being directly as the distance, prove that the free surface is a horizontal plane, and find the pressure at any point.

2. Define the whole pressure and resultant pressure on a surface immersed in a fluid; and show how to calculate them.

Prove that the total normal pressure on a spherical surface immersed to any depth

in water is the same as that on the circumscribed cylinder immersed to the same depth.

3. Find the center of pressure of a circle immersed in water to any depth.

4. Find the form of the free surface of a fluid which rotates uniformly, in relative equilibrium, round a vertical axis.

m

A cylindrical jar whose weight is th of the weight of water which it would contain, is filled (1—1)th full and is then placed, mouth downwards, on a horizontal table which is made to rotate uniformly round a vertical axis coinciding with the axis of the jar. Prove that the angular velocity necessary to cause the fluid to escape is the same as if the jar weighed 1th of the water it would hold and were (1—1)th full; and find this angular velocity.

n

m

5. Investigate the conditions of stability, for small displacements, of a body floating in water.

A pyramid on a square base, whose other faces are equilateral triangles, floats in water with its vertex immersed and base horizontal, find the condition of stability. How will the stability be affected by tilting it round different axes?

6. Investigate the law of density of a vertical column of still air of uniform temperature.

Find the law of density on the hypothesis that the temperature diminishes in harmonical progression as the height increases in arithmetical progression, the variation of gravity in ascending being disregarded.

7. State the hypotheses upon which the equation of fluid motion is founded; and prove the equation.

12

=

Ρ

8. Define the component velocities at any point of a fluid in motion; and, in the case of motion in one plane, find an expression for the quantity of fluid which flows, in given time, in through the boundary of a circle of radius a whose center is at the origin.

9. Given a plane figure of any form; find the line round which it has the least moment of inertia.

The diagonals of a square plate being drawn, the two opposite triangles are cut out; find the principal axes and moments of inertia of the remaining figure, and the moments of inertia about each of the edges of the figure.

10. State D'Alembert's principle, and investigate any conclusions which can be drawn from it for the motion of a rigid body under no forces.

11. State and prove the principle of the convertibility of the centers of suspension and oscillation of a pendulum.

A pendulum is formed of two uniform rods of equal lengths, but of different materials and thicknesses, connected at one end so as to be in the same straight line. Their masses are m, m', and the axis of suspension passes through the middle point of m; find the time of oscillation of the pendulum.

12. State and prove the equation of Vis Viva.

A rod AB is capable of turning round A in a vertical plane, the other end being attached to an elastic string BC which is fastened to a fixed peg vertically above 4, and such that AC=AB. The elasticity of the string is such that a weight equal to that of the rod would stretch it to three times its natural length AB. If the rod be

started from its position of stable equilibrium with an angular velocity find the subsequent motion until the string becomes slack.

NOTE H.

ADMIRALTY CIRCULAR IN REGARD TO PRIVATE STUDENTS IN NAVAL ARCHITECTURE AND MARINE ENGINEERING.

A limited number of students unconnected with the naval service will be permitted to receive instruction at the Royal Naval College, in the course laid down for acting second-class engineers and dockyard apprentices.

The full course will be for three sessions, of nine months each.

The fee (payable in advance before entry) is £30 for each session, or £75 for the full course. Students who have already paid one fee of £30 will be allowed to compound for the next two sessions by a payment of £50 at the commencement of the second session.

Proportionate fees will be paid by students attending special classes only.

Students not connected with the naval service will reside outside the precincts of the college.

Facilities for visiting the royal dockyards will be offered to all private students, being British subjects.

Applications for admission should be addressed to the secretary of the Admiralty, Whitehall.

My lords reserve entire discretion in the selection of the candidates to be admitted.

ENTRANCE EXAMINATIONS.

Private students will be examined before entrance, in accordance with the programme laid down in the general regulations established for the admission of students to the Royal Naval College, as follows, viz:

1. The ordinary rules of arithmetic.

2. Algebra up to quadratic equations, the three progressions, the binomial theorem, and the theory of logarithms.

3. The subjects of the first four books of Euclid's Elements; proportion and similar figures, or the definitions of the fifth book and the proportions of the sixth book of Euclid's Elements.

4. The definitions and fundamental formulæ of plane trigonometry, including the solution of plane triangles. De Moivre's formula and its principal applications.

5. Elements of statics, dynamics, and hydrostatics.

6. Co-ordinate geometry, up to the equations of the conic sections.

7. Geometrical drawing.

ANNUAL EXAMINATIONS.

All private students will be examined at the end of each session. Certificates of proficiency in the various subjects they may have studied will then be awarded.

NOTE I.

(Page -.)

EXAMINATION PAPERS, GUNNERY SHIP EXCELLENT.

SUB-LIEUTENANTS.

(July, 1876.)

1.-Explain fully how to divide an arc into degrees.

What is the angle between the axis of the gun and the keel line, when the pointer on the slide coincides with the zero mark?

S. Ex. 51-20