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Using squared paper, find the velocity in each position and the time taken by the body to get to each position counting from x = 0, the velocity then being 5 feet per second. (40.)

48. A structure has a hollow circular section 10 inches outside diameter and 8 inches inside. The resultant of all the loads and supporting forces acting on one side of the section has a component of 30 tons normal to the section and it acts at 2 inches from the centre; find the maximum and minimum stresses in the section. (30.) 49. Assume that a pipe of 10" diameter inside and 14′′ diameter outside, of homogeneous material can withstand no tensile stress except endwise. It is wound outside with hoop-iron in tension, which produces an external radial pressure Po on the material, the tangential compressive stress produced by this near the outside being 500 lb. per square inch when there is atmospheric pressure inside the pipe. Find po and Ро the tensile force in the hoop-iron per unit length of pipe. What is the tangential compressive stress every where ?

Find what is the greatest internal pressure that the pipe will withstand, and the stress everywhere when there is this pressure inside.

Prove the formulæ used by you.

(40.)

50. An inward flow turbine for a fall of 30 feet and 20 cubic feet of water per second; sketch the turbine; give its principal dimensions and speed and the angles of guide blades and What is the kinetic energy of a pound of water just entering the wheel? Neglect losses by friction.

vanes.

(40.) 51. Prove the formula used for finding the resultant pressure on a sloping plane interface, with any shape of boundary underneath the surface of a liquid. Also find the position of the resultant. Apply the formula to the case of a rectangle with its highest horizontal side 5 feet long at 20 feet below the surface, its other sides being 12 feet long and making 30° with the horizontal. (30.)

52. A symmetrically loaded beam of uniform section; given the diagram of bending moment when supported at its ends, what is the easy rule for obtaining the diagram when the beam is fixed at the ends? Prove the rule to be correct.

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HONOURS-PART II.

INSTRUCTIONS.

Read the General Instructions on page 1.

You may not attempt more than eight questions.

The value attached to each question is shown in brackets after the question.

NOTE.-No Candidate is eligible for examination in Part II. of Honours who has not already obtained a pass in Honours— Part I. in 1898, or a 1st or 2nd Class in Honours of the same subject in a previous year.

61. If a telegraph wire never reaches permanent set, find a formula approximately correct showing how the dip alters with the temperature.

(50.) 62. A uniform beam of rectangular section is fixed at the ends, is 20 feet long, has a load of 10 tons at its middle, and one of 7 tons at 5 feet from one end. Find the diagram of bending moment.

(50.)

63. A spiral spring of angle a, radius of coils r, length of wire 7, is fastened at its upper end; to the lower end we apply a small axial load F and a small couple L about the axis of the spiral; find the axial elongation and the rotation of the lower end.

(50.) 64. If the tractive force of a motor car on a level road of any kind is a function of the speed, show that for a given length of journey on a road of this kind with any gradients, if the duration of the journey is settled, to expend on the journey a minimum amount of propelling energy, the speed ought to be constant. Why is it not fair to assume from this result that the speed of a bicycle or animal-drawn carriage ought to be constant?

A very elementary knowledge of the calculus of variations seems to be needed.

(50.) 65. A round bar of steel, 1 inch diameter, 8 feet long, is subjected at the centres of its ends to equal and opposite loads of 1,500 lb. What is the stress if the rod is kept from bending? If the rod lies horizontally, only supported at the ends, what is the maximum stress due to its own weight? If now it is subjected to the endlong loads when lying horizontally and supported at the ends, what is the maximum stress? Take E = 3 × 107 lb. per square inch.

66. A body of weight Wis at the end of a horizontal lever OBW hinged at 0; it is supported by a vertical spiral spring attached to the point B. Given the stiffness of the spring, show how we find the time of vertical vibration, neglecting the inertias of lever and spring; the motion of the body being resisted by a friction of an amount which is proportional to the speed. Now let the support of the spring get a simple vertical vibratory motion, what is the motion of W after sufficient time has elapsed to still its natural vibrations? (50.) 67. Describe the reasoning and experimental work of Professor Reynolds which led to his law for friction of fluids in pipes. What is that law? How is it related to the well-known D'Arcy formula ?

(50.) 68. A steel tabe 7 inches internal and 10 inches external diameter has steel strip wound on it to the external diameter of 15 inches under a constant winding stress of 20 tons per square inch. Imagine no interstices in the layers of strip. Show by a curve the stress at various places in the solid metal and in the winding.

(50.) 69. A point x, y, z is displaced to x + u, y + v, z + w. Write out the six important kinds of strain; also what are called "the rotations." Express the connection between stresses and strains. Deduce the equations of equilibrium between the stresses and volumetric forces, and convert them into the general equations of strain.

(50.) 70. In what way may a piston ring be made, so that, without the help of auxiliary springs, it will produce a uniform pressure against the inside surface of the cylinder. Prove your statement to be correct.

(50.)

71. A link has motion parallel to a plane. Given the velocities and accelerations of two pins, how do we draw the velocity and acceleration diagrams which give at once the velocity and acceleration of any point in the link? Prove your two constructions to be correct. (50.) 72. The resultant of all the loads and supporting forces acting on one side of a cross section of a structure like a metal arch is given in position. Prove the rule by which we find the tensile or compressive stress at any point in the section. Such a cross section is a circular ring of 10 inches outside diameter and 8 inches inside; the resultant has a componeut of 30 tons normal to the section, and acts at 2 inches from the centre; find the maximum and minimum compressive stresses in the section.

(50.)

1899.

SUBJECT VIII. SOUND, LIGHT, AND HEAT.

EXAMINERS:

PROFESSOR A. W. RÜCKER, M.A., SEC.R.S.,

PROFESSOR W. GRYLLS ADAMS, M.A., D.Sc., F.R.S.,

AND

PRINCIPAL R. T. GLAZEBROOK, M.A., F.R.S.

If the rules are not attended to, your paper will be cancelled.

Elementary Stage.

INSTRUCTIONS.

You are permitted to answer only eight questions.

The marks allotted to each question are given in brackets.
Put the number of the question before your answer.

You are to confine your answers strictly to the questions Froposed.

Your name is not given to the Examiners, and you are forbidden to write to them about your answers.

The examination in this subject lasts for three hours.

SECTION 1.

1. Explain fully the motion of the particles of a vibrating body and the mechanism by which wave motion is transmitted when the vibrations are longitudinal.

(12.)

2. A sound is produced between two parallel obstacles from each of which several repetitions of an echo are produced. If the first and second echoes are heard threequarters and five-quarters of a second respectively after the sound was produced, when will the third echo be heard? If the velocity of sound is 1,100 feet per second, what is the distance between the obstacles? (12.)

3. What do you understand by the amplitude of vibration in a sound wave? Describe two experiments to show that the loudness of a sound depends on the amplitude of vibration of the waves. (13.)

4. A watch is moved away from an observer's ear until its ticking just ceases to be audible. On placing a glass rod between the watch and the ear the ticking is heard distinctly. Explain fully the cause of this.

SECTION II.

(12.)

5. A point of light is placed on the axis of a convex lens and between the lens and a plane mirror which is perpendicular to the axis of the lens. Draw a diagram showing the paths of some of the rays of light which fall first upon the mirror and afterwards upon the lens.

water.

(13.) 6. Describe any method of determining the refractive index of (12.) 7. Draw a careful figure to show the paths of the rays of light proceeding, from a point and passing through a thick plate of glass.

(12.) 8. Distinguish between the illuminating power of a source of light and the intensity of the light at a distance from the source, explaining how the two are connected.

SECTION III.

(12.)

9. What inferences do you draw from the observations that the readings of two thermometers containing different liquids agree at the freezing point and boiling point of water respectively, but not at other points of the scale? (12.) 10. What is meant by the specific heat of a substance? Fifty grammes of iron, whose specific heat is 11, are heated to a temperature of 100° C. and placed in 350 grammes of water at a temperature of 15° C. contained in a copper calorimeter whose mass is 110 grammes and specific heat 0.1; find the temperature of the water after the immersion.

(13.)

il. Water is a worse conductor of heat than iron. What experiments would you make to verify this?

(12.)

12. In the light of our present knowledge do you consider "latent heat a well-chosen name? Give reasons for your answer.

(13.)

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