UC-NRLF 
 
LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFORNIA 
 
 GIFT OK 
 
 Received <c&&&r: , i8 
 A ccessions No . 
 
QUESTIONS 
 
 GENERAL PHYSICS 
 
 IN FOUR PARTS 
 
 Based on the tenth edition of Everett's Translation of 
 DESCHANEL'S NATURAL PHILOSOPHY 
 
 PART I 
 
 STATICS 
 
 INCLUDING 
 
 MECHANICS, HYDROSTATICS AND PNEUMATICS 
 
 BY HAROLD WHITING, PH.D. 
 
 Associate Professor of Physics in the University of California 
 
 BERKELEY, CALIFORNIA 
 
 i93 
 
COPYRIGHT, 1893 
 BY HAROLD WHITING 
 
TABLE OF CONTENTS. 
 
 PART I. STATICS. 
 
 (A) MECHANICS, OR THE STATICS OF SOLID BODIES. 
 
 PAGE 
 
 Introductory - i 
 
 I. Units of Measurement 3 
 
 Forces and Displacement. 
 
 II. Crude Ideas of Force 4 
 
 III. Force Related to Displacement 5 
 
 IV. Gravitation Units of Force - 6 
 V. Absolute Units of Force 7 
 
 VI. Specification of a Force 8 
 
 VII. Composition and Resolution of Forces Meeting at a Point 9 
 
 VIII. Equilibrium of Forces Meeting at a Point 12 
 
 IX. Action and Reaction 13 
 
 Forces Without Definite Centres or Points of Application. 
 
 X. Transmissibility of Force 15 
 XI. Equilibrium of Forces Acting at Two Points (Struts 
 
 and Ties) - 16 
 
 XII. Translation and Rotation 16 
 
 XIII. Composition and Resolution of Couples in the same Plane 1 7 
 
 XIV. Composition of a Force and a Couple in the same Plane 19 
 XV. Composition of Parallel Forces - 20 
 
 XVI. Equilibrium of Three Parallel Forces 21 
 
 XVII. The Lever 22 
 XVIII. Composition and Equilibrium of Couples in Parallel 
 
 Planes 23 
 XIX. Composition and Resolution of Couples in Non-Parallel 
 
 Planes 24 
 
 XX. Wrenches 25 
 
 XXI. Elasticity of Solid Bodies - - 27 
 
Forces having Definite Centres or Points of Application. 
 
 XXII. Centre of Gravity 
 
 XXIII. Centre of Gravity of Geometrical Figures 
 
 ' XXIV. Centre of Gravity of Suspended Bodies 
 
 XXV. Stability and its Relations to Centre of Gravity - 
 
 XXVI. Sensitiveness of a Balance, as related to Centre of Gravity 
 
 Work. 
 
 XXVII. Work, and its Relations to Centre of Gravity 
 
 XXVIII. Work as a Criterion of Stability 
 
 XXIX. Principle of Work applied to Mechanical Powers 
 
 XXX. Conservation of Work 
 
 XXXI. Loss of Work by Friction - 
 
 XXXII. Efficiency 
 
 XXXIII. Power 
 
 PAGE 
 2 9 
 
 31 
 
 33 
 35 
 
 37 
 
 39 
 4i 
 
 42 
 
 45 
 46 
 48 
 49 
 
 () HYDROSTATICS, OR THE STATICS OF FLUID BODIES. 
 
 Properties of Fluids. 
 
 XXXIV. Solids and Fluids Distinguished 
 
 XXXV. Liquids and Gases Distinguished - 
 
 XXXVI. Fundamental Assumptions in Hydrostatics 
 
 XXXVII. Conditions of Equilibrium in Fluids 
 
 Hydrostatic Pressure. 
 
 XXXVIII. Pressure 
 XXXIX. Balancing Columns - 
 XL. Atmospheric Pressure 
 
 XLI. Flow of Liquids Affected by Air Pressure 
 XLII. Pumps 
 
 Buoyancy. 
 
 XLHI. Centres of Pressure and Buoyancy 
 
 XLIV. Flotation 
 
 XLV. Principle of Archimedes 
 
 XLVI. Atmospheric Buoyancy 
 
 XLVII. Apparent Specific Gravity - 
 
 Pneumatics. 
 
 XLVIII. Law of Avogadro 
 
 XLIX. Molecular Theory of the Pressure of Gases 
 L. Law of Boyle and Mariotte 
 LI. Unequal Compressibility of Gases 
 LII. Vapors and Gases Distinguished - 
 
 50 
 5i 
 54 
 55 
 
 57 
 60 
 62 
 64 
 65 
 
 67 
 
 68 
 70 
 72 
 74 
 
 76 
 
 77 
 78 
 
 79 
 80 
 
 END OF PART I. 
 
[UHIVBRSITtf 
 
 QUESTIONS ON GENERAL PHYSICS. 
 
 INTRODUCTORY. 
 
 The following questions have been prepared for the use of 
 college students in connection with a lecture and laboratory 
 course in General Physics. The asterisk (*) indicates topics with 
 which the student should be already familiar in his preparatory 
 work. The dagger (f) indicates topics to be treated or supple- 
 mented in the course of lectures. The double dagger (J) indicates 
 topics illustrated by experiments either in the lecture-room or in 
 the laboratory. Questions containing problems to be solved by 
 the student are marked " Prob." References are as follows : 
 D., to sections in DESCHANEL'S Natural Philosophy, translated 
 by EVERETT, tenth edition ; L,. , to laboratory exercises ; Q., to 
 questions in the list immediately preceding the reference, unless 
 otherwise indicated by section or number. References which 
 are incomplete, and explanatory remarks, which may be omitted 
 in recitation, are enclosed in parentheses. 
 
 i.f State certain rules for expression to be observed in note- 
 books and in recitation. 
 
 2 . State the distinction between ' 4 Natural " (or ' ' Physical ' ' ) 
 phenomena and other phenomena. In which does observation 
 play the more important part? D. i. 
 
 3.f Show that "Science," even in its most general sense, 
 necessarily involves classification or arrangement of some sort. 
 
2 INTRODUCTORY. 
 
 4. Explain the principles upon which Natural (or Physical) 
 Science has been divided broadly into Natural History and 
 Natural Philosophy. In which of these divisions does classi- 
 fication, and in which does the study of cause and effect receive 
 the relatively greater amount of attention ? D. i. 
 
 5. Upon what farther principles has Natural Philosophy 
 been divided into Astronomy, Biology, Chemistry, and Physics, 
 (or Natural Philosophy in its more restricted sense) ? D. 2. 
 
 6.f To which of the above divisions is the Experimental 
 Method virtually restricted, and in what does the Experimental 
 Method consist ? 
 
 7. Should the descriptive portions of Astronomy, Biology, 
 and Chemistry (e.g., Mineralogy) be included under Natural 
 Philosophy or under Natural History ? Q. I. 4. 
 
 8.f Distinguish ordinary descriptive physics, whether studied 
 in the class-room or in the laboratory, from general physics 
 (whether mathematical or simply deductive), on the one hand 
 and from truly experimental (inductive) physics, including 
 physical investigation and measurement, on the other hand. 
 
 9.f Show that, from the nature of the subjects allotted to 
 Physics, this science is able to carry both inductive and deductive 
 methods farther than any other science. 
 
 io.f What faculty is particularly trained by quantitative 
 experiments or measurements ? 
 
 n.f State some of the advantages of a course in general 
 and experimental physics. Show that such a course trains the 
 faculties of expression, observation, classification, experimenta- 
 tion, inference, explanation, and judgment (the last three being 
 the most important branches of reasoning, viz., inductive, 
 deductive, and quantitative). 
 
 i2.*tName the principal branches of Physics. Explain the 
 following terms and their mutual relations : Dynamics, Kinetics, 
 Statics, Mechanics, Hydrostatics, Pneumatics, Thermics, Optics, 
 Acoustics, Electricity and Magnetism. D. 3 and 9. 
 
3 UNITS OF MEASUREMENT. [I. 
 
 i.* Define the foot and the metre. 
 
 2.* Explain the use of a graduated scale for measuring 
 length. 
 
 3.* Define the pound and the gram as units of mass. D. 160. 
 
 4.* Explain the construction and use of a set of weights 
 (with the method of substitution). 
 
 5.* Define the (mean solar) second. 
 
 6.* Explain the use and construction of a clock. 
 
 7.* Define velocity, and state in what units it is expressed. 
 
 8.* Define the area of a surface, and name different units of 
 area. 
 
 9.* Find the area of a plane rectangular surface of length, 
 /, and breadth, b. 
 
 10.* Define volume, and state in what units it is expressed. 
 
 ii.* Find the volume of a block of length, /, breadth, 
 , and thickness, /. 
 
 12.* Define density, and give an example. D. 160. 
 
 13.* Distinguish absolute and relative density. D. 160. 
 
 14.* Find the density, d, of a body of mass, m, and volume, 
 v. (D. 1 60.) 
 
 15.* Find the mass, m, of a body of density, d, and volume, 
 v. D. 1 60. 
 
 1 6.* Find the volume, v, of a body of mass, m, and 
 density, d. (D. 160.) 
 
 1 7. *| Discuss the experimental determination of the density 
 (i) of a solid, (2) of a liquid, and (3) of a gas, by direct 
 measurements of mass and linear dimensions, and show that 
 the same fundamental principles apply to each case. (D. 162.) 
 
 1 8.* Define the pound weight and the gram weight as units 
 of force (in latitude 45). D. 161. 
 
 19.* Explain the signification and use of the prefixes, mega-, 
 kilo-, hekto-, deka-, deci-, centi-, milli-, and micro-. 
 
 20.* Explain the signification of the hyphen in such words 
 as foot-pound, kilogram-metre, etc. 
 
 21.* Explain the signification of "per" in such expressions 
 as pounds per square inch, or centimetres per second. 
 
4 II. 
 
 CRUDE IDEAS OF FORCE. 
 
 i.f Show that the sense of touch is closely connected with 
 our ideas of force. 
 
 2. What connection exists, in general, between force and 
 the muscular effort necessary to produce or to resist it? D. 7. 
 
 3.f Name certain cases in which the maintenance of a force, 
 like continued muscular effort, requires the continuous expendi- 
 ture of energy of some sort ; and where, if the source of energy 
 be cut off, exhaustion ensues, and the force ceases. 
 
 4.f What is meant by the "fatigue" of metals, etc., under 
 continued strain ? Would a perfect solid exhibit such effects ? 
 
 5.f Are we justified in supposing that the maintenance of 
 inanimate forces in general, like the maintenance of muscular 
 tension, in the absence of a fresh supply of energy, necessarily 
 involves fatigue or exhaustion to any considerable extent? 
 Illustrate by the force of gravity upon the moon. 
 
 6.f What misconception is likely to arise from the use of 
 muscular tension as the type of a force ? 
 
 y.f Does the modern idea of force contain any reference to 
 the supply of energy by which it may happen to be maintained ? 
 
 8.f What restriction, not observed by earlier writers, exists 
 in the modern use of the word force ? 
 
 9.f Criticise the expressions, force of a waterfall, force of the 
 wind, force of gravity. 
 
 ro.f Consider a "push," a "pull," or a "shove" as instances 
 of a force. What does each imply as to the direction of the 
 force with respect to the agent ; as to the point of application 
 of the force, and as to the state of tension or compression 
 produced ? 
 
 n.f Criticise the definition of a force as a "push or a pull." 
 (HALL'S Elementary Ideas.) To what extent is it supposed 
 that molecular forces can be so classified ? Is it thought that 
 electromagnetic forces fall into either category ? 
 
III. 
 
 FORCE RELATED TO DISPLACEMENT. 
 
 i.f What in general is the effect of a force upon a point in 
 an elastic body ? 
 
 2.f What relation exists by definition between the direction 
 of the force and the direction of the displacement which it 
 produces in a homogeneous elastic medium ? 
 
 3.f What obvious quantitative relation exists by definition 
 "between the magnitudes of several forces acting in a given 
 direction and the magnitude of the resulting force ? 
 
 4.J Describe one or more experiments showing that the 
 displacement produced by two or more forces acting at a given 
 point in the same direction is the sum of the displacements 
 produced by the separate forces when acting alone. 
 
 5.f State how forces are measured (relatively) in Statics, and 
 upon what experimental evidence the accuracy of this measure- 
 ment depends. 
 
 6.f Explain HOOKE'S law, ut tensio, sic vis. 
 
 7. What is meant by a "dynamometer?" D. 7. 
 
 8. Name some form of dynamometer in common use. D. 7. 
 9'* Describe an ordinary "spring balance." D. 7. 
 
 10. % Describe the calibration of a " spring balance." 
 
 ii.* Explain the use of a " spring balance" for the measure- 
 ment of forces. 
 
IV. 
 GRAVITATION UNITS OF FORCE. 
 
 1. Distinguish between the uses of the word "weight" 
 in the sense of "mass" and in the sense of "force." Which 
 use is adopted by most modern writers? D. 161. 
 
 2. What name is given to such units of force as the pound- 
 weight, or the weight of a kilogram ? D. 8. 
 
 3. What are meant by gravitation units of Torce ? D. 8. 
 
 4.f Describe an experiment with an ordinary balance showing 
 that the weight of a bod}* at the top of a high tower is less than 
 at the foot of the tower. 
 
 5.f What inference concerning the constancy of gravity is 
 drawn from the behavior of clocks controlled by gravity 
 pendulums when transported from one latitude to another ? 
 
 6.f What inference concerning the constancy of the action 
 of springs is drawn from the behavior of chronometers with 
 balance-wheels controlled by hair-springs when transported from 
 one latitude to another ? 
 
 7-t State the result of testing the weight of a given body in 
 different latitudes by means of a spring balance. 
 
 8. Explain the result of testing the weight (?) of a given 
 body in different latitudes by means of an ordinary balance. 
 D. 161. 
 
 9. In an unknown latitude, what kind of balance would you 
 prefer for measuring "weight" in each of its two senses? Q. 
 
 10. In what sense of the word weight may it be said to be 
 constant, and in what sense to vary with the latitude ? Q. 
 
 11. Is the variation in gravitation units of force with latitude 
 or altitude sufficient to impair their commercial value? D. 8. 
 
V. 
 ABSOLUTE UNITS OF FORCE. 
 
 i.f What is meant by an " absolute unit of force ?" Explain 
 by the action of such a unit upon a spring balance in different 
 latitudes. 
 
 2.f Discuss the possibility of graduating a spring balance so 
 as to indicate the magnitudes of forces in absolute units. 
 
 3.f Name the absolute units of force adopted in the English 
 and in the Metric Systems. 
 
 4.f Define the " poundal " in terms of the weight of a pound 
 in latitude 45 degrees, where the velocity acquired by a falling 
 body in one second is 32. 172 feet per second. 
 
 5.f Define the " dyne " in terms of the weight of a " gram " 
 in latitude 45 degrees, where the velocity acquired by a falling 
 body in one second is 980.61 centimetres per second : the 
 equivalent of 32.172 feet per second. 
 
 6.f Give instances of familiar forces measured in poundals 
 and in dynes. 
 
 7. In latitude 45 degrees, how would you reduce grams to 
 dynes ? pounds to poundals ? and the reverse ? Q. 
 
 8.f What branch of Physics has been greatly simplified 
 by the use of absolute units of force ? 
 
VI. 
 SPECIFICATION OF A FORCE. 
 
 i. Can a force be applied practically to a point? Give 
 reasons for your answer. D. 12. 
 
 2.* What is meant by the point of application of a force? 
 D. n. 
 
 3. What is meant by the line of action of a force ? Is the 
 line of action of a force synonymous with the direction of the 
 force? D. 13. 
 
 4. * What is meant by the magnitude of a force ? 
 
 5. What elements are necessary for the specification of a 
 force ? Q. 
 
 6. Correct DKSCHANEI/S statement that "a force is com- 
 pletely specified when its magnitude, its point of application 
 and its line of action are all given. ' ' State what interpretation 
 must be given to the words " line of action " in this statement 
 in order that it may be true, and whether this interpretation is 
 or is not in keeping with his use of the expression ? D. 1 1 and 13. 
 
VII. 
 
 COMPOSITION AND RESOLUTION OF FORCES 
 MEETING AT A POINT. 
 
 1. 1 Describe an experiment showing that the direction and 
 magnitude of the displacement produced by a given force acting 
 on a point in a homogeneous elastic medium is independent of 
 the forces already existing at that point, provided that these 
 forces are constant in magnitude and in direction. 
 
 2. Show that all geometrical propositions relating to 
 displacements apply also to the forces which these displacements 
 represent. Q. 
 
 3. Two forces acting on a point produce independently the 
 displacements, AB and BC. What is the resulting displace- 
 ment ? Q. 
 
 4. What name is given to a single force which will, acting 
 alone, produce the same displacement as two or more than 
 two other forces when acting together? D. 15. 
 
 5. Define the resultant of two or more forces. D. 15. 
 
 6.f Show that, if forces are measured by the displacements 
 which they produce, the resultant of two forces is represented in 
 direction and magnitude by the diagonal of a parallelogram, the 
 sides of which represent the two forces in question. Q. 
 
 7.* Describe the ''parallelogram offerees," and its use in 
 finding resultants. D. 16. 
 
 8.* Show that two forces, represented by the lines AB and BC, 
 have a resultant represented by a line, AC, in the same plane as 
 AB and BC. D. 14, 15. 
 
 9.* Explain the "triangle of forces," its relation to the 
 " parallelogram of forces," and its use in finding resultants. 
 D. 16. 
 
 10. Show that any number of forces, AB, BC, CD, . . . YZ, 
 have a resultant, AZ. D. 18. 
 
10 COMPOSITION OF FORCES. [vil. 
 
 n.f Consider whether the statement in the last question 
 applies to forces in one plane only, or to forces in any plane. 
 
 12. Show that the resultant of three forces represented by 
 the lines, AB, BC, and CD, is represented by the diagonal of the 
 parallelepiped constructed on these three lines. D. 18. 
 
 13. Show that in the parallelepiped of forces, the order of 
 combination is indifferent. D. 18. 
 
 14. Describe the use of the parallelepiped of forces for finding 
 resultants of three forces. D. 18. 
 
 15.* What name is given to two or more forces which together 
 produce the same displacement as a single force of given 
 magnitude and direction? D. 15, 21. 
 
 1 6.* Define the word component, D. 15, 31. 
 
 17. Show that if the magnitude and direction of a force 
 and two of its three components are given, the magnitude and 
 direction of the remaining component is determined. Q. 
 
 1 8. Show that, if the magnitude and direction of a force and 
 the direction of its three components are given, the magnitude 
 of each component is determined. Q. 
 
 19.* What name is given to the process of finding the 
 components of a force in two or three different directions ? D. 31. 
 
 20. Show that a force may in general be resolved into 
 components, if these components are unlimited either in number 
 or in direction, in an infinite number of ways. Q. 
 
 21. Name certain circumstances under which the resolution 
 of a force into components leads to definite results. D. 31, Q. 
 
 22.* State certain necessary limitations between the magnitude 
 of the resultant of two forces and (the sum and difference of) 
 these forces ; also between the sum and the resultant of any 
 number of forces. Q. 
 
 23. What name is given to components of a force when these 
 are at right angles to each other ? and to the process of resolution 
 in such cases ? D. 32. 
 

II COMPOSITION OF FORCES. [VII. 
 
 24. What kind of composition and what kind of resolution 
 is understood unless otherwise stated ? D. 32. 
 
 25. Show that a force can have three and only three 
 rectangular components, and that it is completely specified when, 
 and only when, these three components are explicitly or 
 implicitly given. Q. 
 
 26. Find by the Pythagorean proposition the relation between 
 a force and its three rectangular components. Q. 
 
 27. A point, A, is displaced from A' to A", find (by 
 rectangular projection) how much it has approached an infinitely 
 distant point, B. Prob. 
 
 28. What is meant by the (rectangular) component of a 
 displacement or a force in or along a given direction, line, or 
 plane? D. 32. 
 
 29. State the trigonometric relation between a force AB, and 
 its (rectangular) component along the line, AC. D. 32. 
 
 30. What in general is the trigonometric relation between a 
 force and its component in a line (or plane) making the angle A 
 with a line parallel to the direction of the force? D. 32. 
 
 31. Show that a force can have no (rectangular) component 
 in a direction at right-angles with itself. Q. 
 
 32. Show that the component of a displacement in a given 
 direction is determined solely by the component of the force in 
 that direction, and is not affected by either of the other two 
 components. Q. 
 
 33. Show that the component displacement in any direction 
 due to one of the components of a force acting alone in this 
 direction is the same as that due to the whole force. Q. 
 
12 VIII. 
 
 EQUILIBRIUM OF FORCES MEETING AT A POINT. 
 
 1 . When is a point in an elastic body said to be in equilibrium 
 under the action of forces estimated by the displacements which 
 they produce ? D. 9. 
 
 2. Show that a point in an elastic body is in equilibrium if 
 the resultant of all the forces acting upon it is zero. Q. 
 
 3.f State the result of plotting lines, AB, BC, CD, DE, etc., 
 so as to represent in magnitude and direction forces producing 
 equilibrium at a point. 
 
 4-f What is meant by the "polygon of forces?" 
 
 5.* When are three forces acting at a point in equilibrium ? 
 D. 14. 
 
 6.* Show that three forces acting at a point cannot be in 
 equilibrium unless they are contained in the same plane. Q. 
 
 7.* Explain the "triangle of forces" as applied to forces in 
 equilibrium. D. 14. 
 
 8.J Explain the determination of the weight of a body by 
 means of two spring balances with cords meeting in a knot from 
 which the body is suspended, and illustrate by examples. L. 
 
 9. 1 Explain the determination of the weight of a body 
 suspended by a cord by means of a spring balance acting upon a 
 knot in this cord, so as to produce a known angular deflection. L. 
 
 10. How do you find the relative magnitudes of three forces 
 of known direction, producing equilibrium at a point? 
 
 n.t What is meant by the " equilibrant " of a number of 
 forces ? 
 
 i2.f Find the equilibrant of the forces, AB, BC, . . YZ. Q. 
 
 13. Find the equilibrant of two furces, AB and BC, and show 
 that it lies in the plane of these forces. D. 14. 
 
 14. Show that in the triangle of forces the forces and their 
 equilibrant represent rotatiori in the same direction about any 
 point in the interior of the triangle. D. 14. 
 
 15. Show that the equilibrant of any number of forces is in 
 all cases equal and opposite to their resultant. Q. 
 
i 3 IX. 
 
 ACTION AND REACTION. 
 
 1. A small light spring of the normal length, A' B' , is 
 stretched to the new length, A" B" . Find the displacement of 
 B relatively to A, and the displacement of A relatively to B, 
 (ist) taking AB as the positive direction, and (2d) taking BA as 
 the positive direction. What obvious relations exist between the 
 magnitudes and between the signs of these four displacements? 
 Prob. 
 
 2. Show that a force, if represented by a relative displace- 
 ment, necessarily exists in four different aspects. Q. i. 
 
 3.f When a small light spring is stretched from the normal 
 length, A B' , to a new length, A" B" , what displacement is taken 
 as a measure of the force exerted upon the spring by B ? by the 
 spring at B ? upon the spring at A? by the spring at A ? 
 
 4.f Consider the modifications (if any) which would be 
 introduced into your answers to the questions above by the 
 substitution of compression for extension. 
 
 5. What relation exists by definition between the forces 
 exerted at a given point upon and by a given thing (or agent)? Q. 
 
 6.* Considering the force exerted by a spring upon a fixed 
 support as the " action " of the spring, what name is given to the 
 force exerted by the support upon the spring ? 
 
 7.* Considering the force exerted by any agent (the hand, for 
 instance), upon a spring as the "action" of that agent, what 
 name is given to the force exerted by the spring upon the 
 agent ? 
 
 8.* If by the action of a spring, AB, we mean the force 
 exerted by it at A, what name do we give to the force exerted by 
 it at B ? 
 
 9.* If by the action of any agent upon a spring, AB, we 
 mean the force exerted upon the spring at A, what name do we 
 give to the force exerted upon the spring at B ? 
 
 io.f Distinguish action and reaction exerted at a point from 
 action and reaction exerted through some medium (for instance, 
 a spring) at two different points. 
 
 n.f Show that action and reaction are (in each case) names 
 for two different aspects of one and the same stress. 
 
14 ACTION AND REACTION. [iX. 
 
 i2.f Show that action and reaction in an elastic body, being 
 measured by relative displacement, are equal and opposite 
 whether that body be at rest, in uniform motion, or under 
 acceleration. 
 
 i3-f Show that action and reaction are necessarily simulta- 
 neous. 
 
 i4.*fState the principle of action and reaction. 
 15.* Give instances of action and reaction. D. 10. 
 
 16.* Correct the following statement: "every action is 
 followed (!) by an equal and opposite reaction." Q. 
 
 17.* Contrast the scientific use of the word "reaction " with 
 its popular use, exemplified in the phrase, "reaction of public 
 opinion." Q. 
 
 1 8.* Criticise the illustration of action and reaction by the 
 successive swings of a pendulum. Q. 
 
 iQ.f In applying the principle of action and reaction to astro- 
 nomical phenomena, what assumption is made with respect to 
 the velocity of propagation of gravity ? Q. 
 
 20. Why cannot the principle of action and reaction be 
 applied indiscriminately (as in D. 10) to forces which, like 
 mechanical stresses (e.g. the tension of a cord) or like magnetic 
 attractions and repulsions, require (in some cases) a perceptible 
 amount of time for their propagation ? Q. 
 
 2i.f Show that certain objections to the principle of action 
 and reaction disappear if the forces in question are applied at (or 
 indefinitely near) the same place. 
 
 22.* Apply the principle of action and reaction to the pressure 
 between a body and the table upon which it rests ; to the force 
 exerted by the hand in pulling a rope ; to the tension between 
 the two halves of a rope during the l ' tug of war " ; to the pull 
 of traces upon a whifBetree in starting a wagon, etc., etc. State 
 accurately what constitutes the pair of eqral rnd opposite forces 
 in each case. 
 
 23. f Show that the action and reaction between two bodies, 
 though equal and opposite, being applied to different bodies, do 
 not tend to produce equilibrium in either, considered by itself. 
 
15 
 
 X. 
 TRANSMISSIBILITY OF FORCE. 
 
 i.f Describe an experiment with a chain of light spring 
 balances illustrating the transmission of force. 
 
 2.f Prove, by the principle of action and reaction, that force 
 is transmitted along a chain of light spring balances without 
 loss. 
 
 3.f Show, by considering a medium transmitting force as a 
 sort of chain, that the equality of action and reaction can be 
 predicated for every point in the medium. 
 
 4-f Describe one or more cases of the transmission of force 
 between two points in a light rigid body. 
 
 5.J Describe one or more experiments illustrating the trans- 
 mission of force by a cord, or other flexible substance. 
 
 6.* Criticise the reason usually assigned (loss of force) for 
 hitching a horse as close as possible to his load. Q. 
 
 7. What relation exists between the forces at various points 
 of a line, AB, any one of which would be neutralized by a given 
 force at A, in the line, AB ? (Axiom). 
 
 8.f Show that a force of given magnitude in the direction, 
 AB, may be transferred to any point, B, in its line of action 
 without modifying the result. D. 13. 
 
 9. \ Describe one or more experiments with a rigid body held 
 by elastic forces, or otherwise, so as to be sensitive to any change 
 in the magnitude, as well as in the direction or line of action of 
 a force, and state the effect of changing the point of application 
 of a given force along its line of action. 
 
 10.* What is meant by the transmissibility of a force along 
 its line of action ? D. 13, Q. 
 
 n.f Show that the line of action of a force may be substituted 
 for its point of application in all cases involving only a single 
 position of a body, and not concerning the displacement of the 
 point itself. Q. 
 
i6 XI. [xn. 
 
 EQUILIBRIUM OF FORCES ACTING AT TWO POINTS. 
 
 i. What is meant by a "rigid body" in mechanics ? D. 12. 
 
 2.f What is meant by a " strut " or a " tie " ? and what kind 
 of force is each most suitable for transmitting ? 
 
 3.f What relation is found (experimentally) to exist between 
 the forces transmitted by a strut or a tie, and the line joining the 
 points of application of these forces ? 
 
 4.* Show, by general considerations of symmetry, that two 
 equal and opposite forces applied at two points, A and B, of a 
 rigid body, parallel to AB, should produce no displacement 
 (either of translation or of rotation) in the body as a whole. 
 
 5.f When is a rigid body said to be in equilibrium ? D. 9. 
 
 6. State the (three) conditions of equilibrium of two forces 
 applied at two points in a rigid body. D. 13. 
 
 7. State the conditions of equilibrium of any number of 
 forces with a common line of action. D. 13. 
 
 XII. TRANSLATION AND ROTATION. 
 
 i.* Distinguish displacements, (whether produced by forces or 
 not) into two classes, according to whether translation or rotation 
 is produced. D. 5. 
 
 2.f Describe an experiment with a globe illustrating the fact 
 that all displacements are resolvable into translation combined 
 with rotation about one or more axes. 
 
 3. Show that the displacement of a body is completely 
 specified by that of the centre and two points on the surface of a 
 sphere in which it is included. Prob. 
 
 4-t Show (by drawing great circles through any two points, 
 A and B, on the surface of a sphere, before and after rotation) 
 that there must ba two antipodal points, (the intersections of the 
 great circles) which are not affected by any given displacement 
 of a spherical surface about its centre. 
 
 5. Show that every displacement of a body about a point in 
 it must have a definite axis. Q. 4. 
 
 6. Show that every displacement must be resolvable into 
 one of translation, combined with one of rotation about a given 
 axis. Q. 4, 5. 
 
17 XIII. 
 
 COMPOSITION AND RESOLUTION OF COUPLES IN 
 THE SAME PLANE. 
 
 i.* Distinguish a push or a pull from a twist, torsion from 
 tension or compression, according to the displacement produced. 
 D. 5 and 6. 
 
 2.* State the effect of two equal and opposite forces applied 
 at the points, A and B, at right angles with the line, AB, in 
 an elastic medium. 
 
 3.* What combination of forces is necessary to produce a 
 twist ? 
 
 4. What name is given to a pair of equal and opposite forces 
 having different lines of action ? to any combination of forces 
 producing a twist ? D. 26, 27. 
 
 5. Define the word " couple," and tell what is meant by the 
 "arm," the "moment," the "plane," and the "axis" of the 
 couple. D. 26, 27. 
 
 6.f Distinguish right and left-handed couples : also similar 
 and opposite couples. 
 
 7-t Show that two equal and opposite couples in the same 
 plane, having equal forces and equal arms not parallel, give rise 
 to a pair of equal and opposite resultants, having for their 
 common line of action a diagonal of the parallelogram formed by 
 the prolongation of the component forces, and, therefore, neutralize 
 each other, as far as rotatation or twist is concerned. 
 
 8. Show that two similar couples in the same plane, with 
 equal forces and arms, being capable of neutralizing the effect of 
 a given couple, are equivalent to each other. (Axiom). 
 
 9.J Describe one or more experiments showing that the 
 translation or rotation of a couple in its own plane does not 
 modify the effect which it produces. 
 
 10. Find the resultant of two similar couples with equal forces 
 and with arms, AB and BC, in the same straight line. Prob. 
 
 n. What assumption is made in the last question ?.s to the 
 disappearance of two equal and opposite forces at a point ? and 
 what postulate (as to the introduction of equal and opposite 
 forces at a point) will enable you to resolve the resultant couple 
 back into its components ? Prob. 
 
1 8 COMPOSITION AND RESOLUTION OF COUPLES. [xill. 
 
 i2.f Show that a couple with given forces and arm, A, may 
 be resolved into A couples with the same forces, but with unit 
 arms. 
 
 I3.f Show that a couple with given arm and forces, F, can be 
 resolved into ^couples with the same arm, but with unit forces. 
 
 i4.f Show that a couple with forces, /% and arm, A, can be 
 resolved into F X A couples with unit forces and arms. 
 
 Q- 12, 13. 
 
 15. What name is given to the product of the force and arm 
 of a couple ? D. 26. 
 
 i6.f Describe one or more experiments with a torsion 
 apparatus showing that the effect of a couple is proportional to 
 the force, if the arm is constant ; to the arm, if the force is 
 constant ; and to the moment, if the force and arm both vary. 
 State the (empirical) law established by these experiments. 
 
 17. State the necessary relation existing between the moment 
 of a couple and the twist produced in an elastic medium, in which 
 a given couple always produces a given additional twist. Prob. 
 
 1 8. Show (by resolution into unit couples) that the resultant 
 of any number of couples in the same plane is a single couple 
 in that plane, with moment equal to the algebraic sum of the 
 moments of the components. Q. 
 
 ig.f Show that a couple can be balanced only by an equal and 
 opposite couple. D. 27. 
 
 20. J Describe one or more experiments (with E. H. HALL'S 
 apparatus) illustrating the conditions of equilibrium of two 
 couples in the same plane. 
 
 2i.f State the conditions of equilibrium of two or more couples 
 in the same plane. D. 27. 
 
 22. What is meant by the moment of a force about a point? 
 D. 24. 
 
 23.")" Show that the moment of a couple is equal to the 
 difference (or algebraic sum) of the moments of its two forces 
 about any point in its plane. 
 
 24. f Apply the principle of moments about a point to the case 
 of equilibrium as to rotation about this point. 
 
XIV. 
 
 COMPOSITION OF A FORCE AND A COUPLE IN THE 
 SAME PLANE. 
 
 1. Show that a change in the line of action of a force is 
 equivalent to the introduction of a couple whose moment is the 
 product of the force and the distance between its old and new 
 lines of action, and whose plane contains these lines. D. 28. 
 
 2. Find the direction of rotation introduced by the displace- 
 ment of the points of application of forces in special cases, (e.g., 
 a right-handed displacement of a downward force). Prob. 
 
 3.f The point of application of a force of magnitude and 
 direction, AB, is moved through a distance, AC, at right-angles 
 to AB ; find the nature of the action which brings this about, 
 and the complete specifications for this action. 
 
 4.f Show, conversely, that the resultant of any force and any 
 couple in the same plane is a single force in this plane, equal in 
 magnitude to the component force, parallel to this force, and at a 
 (perpendicular) distance from it, equal to the quotient of the 
 moment of the couple by the magnitude of the force. 
 
 5. How would you construct a couple equivalent to a given 
 couple, so as to show that its effect upon a given force in its own 
 plane is simply to displace the line of action of this force ? 
 D. 28. 
 
 6. Find the direction of the displacement produced in the 
 points of application of forces by couples in special cases, (e.g., 
 a left-handed couple and an upward force). Prob. 
 
20 
 
 XV. 
 COMPOSITION OF PARALLEL FORCES. 
 
 1. Show that the transfer of two equal forces, having the 
 same direction, to a line midway between them is equivalent to 
 the introduction of two equal and opposite couples, and hence 
 does not affect the result. Q. XIV. 
 
 2. Show that the resultant of two equal forces in the same 
 direction is equal to the sum of these forces, acting always in a 
 line midway between the lines of action of the components. Q. 
 
 3.f Show that the transfer of any two forces, AX and BY, 
 having the same direction, to a common point, P, dividing the 
 line, AB, into parts such that AP X AX is equal to BP X BY, 
 being equivalent to the introduction of equal and opposite couples, 
 does not modify the result. 
 
 4f Extend the demonstration called for in the last question 
 to the case of any two unequal forces in opposite directions. 
 
 5.f Show that the resultant of any two forces having the 
 same direction is a single force in the same direction, equal in 
 magnitude to the sum of the component forces, with its line of 
 action in the same plane with, and between the lines of action 
 of these forces, and distant from the line of action of each in the 
 inverse proportion of the magnitudes of these forces. D. 20. 
 
 6. Specify the resultant of two unequal parallel forces in 
 opposite directions. D. 20, Q. 
 
 7. Compare the moments of two parallel forces about any 
 point in their plane with the moment of their resultant about 
 this point. D, 22, 24. 
 
 8. Show (by successive composition) that any number of 
 parallel forces, whose algebraic sum is not zero, must have a 
 single force for their resultant. Prob. 
 
21 
 
 XVI. 
 EQUILIBRIUM OF THREE PARALLEL FORCES. 
 
 i.* Show that three parallel forces cannot be in equilibrium 
 unless one is opposite to the other two. Q. XV. 
 
 2.* Show that the equilibrant of two parallel forces must 
 have the same line of action as their resultant. Q. XI. 
 
 3.* Show that one of three parallel forces in equilibrium must 
 be equal in magnitude to the resultant of the other two, and 
 hence equal to the algebraic sum of the other two. (Axiom). 
 
 4.* Show that the equilibrant of two forces in the same 
 direction must lie between them, and in the same plane with 
 them. Q. 
 
 5.f Show that the line of action of the equilibrant of two 
 forces in the same direction must divide the distance between 
 their lines of action into parts inversely proportional to the 
 magnitudes of the two forces, in order that equal and opposite 
 couples may be produced, and that this condition is consistent 
 with those above. 
 
 6. Compare the moment of two parallel forces about any 
 point in their plane with the moment of their equilibrant about 
 this point. Q. 
 
 7. Where is the equilibrant of two parallel forces in opposite 
 directions ? and what becomes of it as the forces become more and 
 more nearly equal ? D. 26. 
 
 8.* State the (three) conditions of equilibrium of three 
 parallel forces. D. 19. 
 
22 
 
 XVII. 
 
 THE LEVER. 
 
 i.* Describe an ordinary (straight) lever, stating what is 
 meant by the "arms" of the lever, the "fulcrum," (F) 
 "power," (/>) and "weight," (W). D. 56, 57. 
 
 2.* Distinguish three classes of levers, stating which of the 
 three forces is in the middle in each case. D. 56, 57. 
 
 3.* In which class of levers is the weight always greater 
 than the power ? always less ? sometimes greater and sometimes 
 less ? 
 
 4.* Which force in each class of levers is equal to the sum of 
 the other two ? 
 
 5.* State the law of levers. Q. XVI. 
 
 6.f Show that each of the three forces in equilibrium in the 
 lever is proportional to the distance between the lines of action 
 of the other two. 
 
 7.^ Describe one or more experiments with the arithmetical 
 lever, and state what points each illustrates. D. 25. 
 
 8.* What is meant by a bent lever? D. 57. 
 
 9. Distinguish the "arms " of a bent lever from the " arms " 
 of the couples which are in equilibrium. D. 57. 
 
 10* When are the forces acting on a bent lever in equili- 
 brium ? Q. XVI. 
 
 n.J Describe one or more experiments (with E. H. HALL'S 
 apparatus) illustrating the equality of moments in the bent lever. 
 
23 
 
 XVIII. 
 
 COMPOSITION AND EQUILIBRIUM OF COUPLES IN 
 PARALLEL PLANES. 
 
 i.f Describe the condition of the rod of a torsion apparatus 
 during torsion. 
 
 2.f Show that a rod subject to torsion can offset the action of 
 a couple. 
 
 3. When a rod is twisted by a couple acting at one end of it, 
 what is the nature of its reaction against the agent through 
 which the couple is produced ? (Axiom). 
 
 4. Show (by considerations of symmetry) that a uniform 
 free rod, subject to torsion, must react equally at both ends 
 against agents producing the torsion. (Axiom). 
 
 5.t What is meant by a " shaft " or " axle," and when is it 
 in equilibrium ? 
 
 6.f Show that couples, like forces, exist under four aspects, 
 and obey the same laws of action and reaction. 
 
 7.f Prove that a couple is transmitted without loss at right- 
 angles to its plane, from one section of a body to another, and 
 hence from one end of the body to the other. 
 
 8. What is meant by the transmissibility of couples along 
 their axes ? Prob. 
 
 9.f Show that the plane or axis of a couple has no definite 
 position in space, but is determined only by parallelism to a given 
 plane or axis. Prob. 
 
 io.f Show that the resultant of any number of couples with 
 parallel planes is a single couple, with plane parallel to those of 
 the components, and with a moment equal to the algebraic sum 
 of the moments of the components. D. 27, Q. XIII. and XVIII. 
 
 1 1. 1 Describe one or more experiments (due to E. H. HALL) 
 illustrating the conditions of equilibrium between couples in 
 parallel planes. 
 
 i2.f Show that a couple is completely specified when its 
 moment and general direction of rotation are known. Prob. 
 
24 
 
 XIX. 
 
 COMPOSITION AND RESOLUTION OF COUPLES IN 
 NON-PARALLEL PLANES. 
 
 i.f Find the locus of all antipodal points in a given spherical 
 surface at which tangential forces can be applied so as to produce 
 a couple equivalent to a given couple. 
 
 2.f Find two antipodal points in a given spherical surface 
 where tangential forces can be applied so as to produce two 
 couples equivalent to any two given couples in non-parallel 
 planes. 
 
 3.f Show that the parallelogram of forces can be applied to 
 the composition of couples in non-parallel planes. 
 
 4-f Prove that the resultant of two couples in non-parallel 
 planes is a single couple in some intermediate plane. 
 
 5. Apply the Pythagorean proposition to the relation between 
 the moments of two component couples in planes at right-angles, 
 and the moment of their resultant. Prob. 
 
 6.f Show that a couple can be resolved into components, the 
 first parallel to any plane, and the second making a right-angle 
 (or any other angle) with the first. 
 
 7. Find the trigonometric relation between a couple and its 
 (rectangular) component in any plane, making the angle A with 
 the plane of the couple. Prob. 
 
 8,f Find the component of a couple in any plane whose 
 normal makes an angle, A, with the axis of the couple. Prob. 
 
 9-t What is meant by the moment of a force about a given 
 axis? and what distinction is made between positive and negative 
 moments ? 
 
 io.f Show that the algebraic sum of the moments of two equal 
 and opposite forces about any axis is equal to the component of 
 the couple in a plane at right- angles to this axis. 
 
 n.f Show that, in a state of equilibrium, the algebraic sum of 
 the components of all the forces along any axis, and also the 
 algebraic sum of their moments about this axis must be zero. 
 
XX. 
 
 WRENCHES. 
 
 1. Show that every force acting upon a body (whether part 
 of a couple or not) can be resolved into a force with line of action 
 passing through any given point, P, and a couple. D. 28, 29 ; 
 Q. XIV. i. 
 
 2. What is the nature of the resultant in the last question, 
 of all the forces passing through a single point, /*? Of all the 
 couples ? D. 29 ; Q. VII., XVIII., XIX. 
 
 3t Show that all possible forces acting on a body can be 
 resolved and recompounded into a single force with line of 
 action passing through a given point and a single couple. 
 D. 29; Q. 2. 
 
 4.f Show that the couple in the last question can be resolved 
 into two components, one at right-angles to the force, the other 
 parallel to it, and state the result of combining one of these 
 components with the force. D. 29, Q. XIX. 6, XIV. 4. 
 
 5.f Show that all possible forces acting on a body (whether 
 parts of couples or not), are equivalent to a single force with 
 definite line of application, and a single couple at right-angles 
 with this force. D. 29, Q. 
 
 6. What name is given to a force combined with a couple at 
 right-angles to it ? D. 29, 30. 
 
 7. Define the word "wrench" as used in (theoretical) 
 mechanics. D. 29, 30. 
 
 8. Show that all possible combinations of forces are reducible 
 to a wrench. D. 29, 30 ; Q. 
 
 9-f Distinguish right and left-handed wrenches, according to 
 the direction of rotation, as looked at in the direction of the 
 force. 
 
26 WRENCHES. [XX. 
 
 io.f Show that the reaction against a right or left-handed 
 wrench, being equal and opposite in respect to both force and 
 couple, is a wrench of the same kind, as concerns the relation 
 between linear and rotatory displacement. 
 
 n.f Distinguish wrenches farther into two classes, according 
 to whether extension or compression is produced. 
 
 I2.f Discuss the wrenches exerted in certain familiar mechan- 
 ical processes (such as driving a screw). 
 
 13-f Show that extension in one part of an elastic system 
 implies compression in some other part. 
 
 14. In what respect are the wrenches exerted by two different 
 parts of an elastic system on each other similar, and in what 
 respect dissimilar ? Q. 
 
 i5.f Discuss the longitudinal and torsional effects of mutual 
 wrenches in the ordinary spring curtain-roller. 
 
 1 6. State the law of action and reaction as applied to 
 wrenches. D. 30. 
 
2 7 
 
 XXI. 
 
 ELASTICITY OF SOLID BODIES. 
 
 i.f Name three effects of a wrench upon an elastic body. 
 
 2.* Distinguish stretching, bending, twisting, and com- 
 pression. 
 
 3. What is meant by elasticity ? by limits of (perfect) 
 elasticity? by a .set ? or permanent strain ? D. 126. 
 
 4. Define a stress, a strain, and a coefficient or modulus of 
 elasticity. D. 128. 
 
 5. 1 Describe one or more experiments illustrating effects of 
 longitudinal forces upon wires or rods. 
 
 6.f State the law connecting the stretching of a rod with the 
 force applied, with the length of the rod, and with its area of 
 cross-section. 
 
 7. What is meant by YOUNG'S modulus of elasticity? 
 D. 128. 
 
 8.f Give some idea of the relative magnitudes of the forces 
 necessary to produce a given stretch in rods of different materials, 
 but of the same dimensions. 
 
 g.f State the usual effect of stretching on the diameter of a 
 rod, and what is meant by " POISSON'S ratio." 
 
 io.f What connection (if any) exists between the relative 
 magnitudes of the forces necessary to produce a given amount of 
 stretching, and the relative magnitudes of those required to 
 produce a given amount of bending in bodies of given 
 dimensions ? 
 
 1 1. 1 Describe one or more experiments relating to the laws 
 of bending. 
 
28 ELASTICITY OF SOLID BODIES. [XXI. 
 
 12. f State the laws connecting the bending of a. beam with 
 the load, and with the length, breadth, and thickness of a beam. 
 
 134 Describe one or more experiments relating to the laws of 
 torsion. 
 
 i4-t State the laws connecting the twisting of a rod with its 
 length, with its breadth, and with the magnitude of the couple 
 applied. 
 
 15. f Give some idea of the relative magnitudes of the couples 
 necessary to produce a given amount of twisting in bodies of the 
 same dimensions, but different materials. 
 
 i6.f What is meant by a modulus or coefficient of torsion ? 
 
 17. What is meant by elasticity of volume? by resistance to 
 compression? by compressibility ? D. 129. 
 
 i8.f Explain how the compressibility of solids (as well as 
 liquids) can be demonstrated by an (OERSTED'S) piezometer. 
 
 19. f What connection (if any) exists between YOUNG'S 
 modulus, the modulus of torsion, and the resistance to compres- 
 sion for a given material ? 
 
 20. Show that HOOKE'S law applies to cases of stretching, 
 bending, twisting, and compression. 
 
XXII. 
 CENTRE OF GRAVITY. 
 
 i.f When is a force applied to a rigid body said to have a 
 definite point of application ? (D. 34). 
 
 2.f What is meant by a centre of force? by the centre of two 
 or more forces ? 
 
 3-t Distinguish between a true centre of force, as for instance 
 an ideal atom, and a virtual centre of force, as for instance the 
 centre of Saturn's rings. 
 
 4. Show that the centre of two parallel forces (not equal and 
 opposite) having definite points of application lies in the line 
 joining these points of application. D. 34. 
 
 5. Where is the point of application of the resultant of two 
 parallel forces (not equal and opposite) ? D. 34. 
 
 6. Show that if any number of parallel forces (whose 
 algebraic sum is not zero) have definite points of application, 
 they give rise to a resultant with a definite centre or point of 
 application. D. 34. 
 
 7. Show that the forces which the earth's gravity exerts 
 upon the molecules of a small body at the earth's surface, having 
 definite points of application, and being (practically) parallel, 
 must have a definite centre or point of application. D. 33, 34. 
 
 8. What name is given to the centie or ] cint of i f pl.'catic n 
 of the (parallel) forces which gia\ ity exerts i yon the different 
 elementary particles of a body ? D. 34. 
 
 9. Define the term " centre of gravity," and state whether 
 it is applicable to solids only, or also to liquids and gases. 
 D- 34. 35- 
 
 io.f Show that if masses are measured by the forces which 
 gravity exerts upon them, the centre of gravity coincides (by 
 definition) with the "centre of mass." 
 
30 CENTRE OF GRAVITY. [XXII. 
 
 n.f Would the statement in the last question hold if the 
 masses were not measured by the forces which gravity exerts 
 upon them ? or if the forces in question were not parallel ? 
 
 12. Under what circumstances does the centre of (the forces 
 due to) attraction coincide with the centre of mass ? Q. 
 
 13. Under what conditions does the resultant of two or more 
 forces have a definite centre or point of application ? Q. 
 
 14. Where is the centre of gravity of two equal masses? of 
 two unequal masses ? D. 34. 
 
 15. f Show that the centre of gravity of two, three, or four equal 
 masses is at the geometrical centre of this line, triangle, or 
 pyramid which they define. (D. 37-39). 
 
 i6.f Show that three or more unequal masses must have a 
 fixed centre of gravity, independent of the order in which they 
 are considered. (D. 37). 
 
 17. Show (by geometry) that in a system of mass, M, 
 consisting of two masses, m^ and m 2 , at the heights, /i, and // 2 , 
 the height, H, of the centre of the gravity is such that 
 MH=m^ h^ -j- m 2 h 2 . D. 22, 45. 
 
 1 8. Extend the principle of the last question to the case of 
 any number of masses. D. 23, 45. 
 
 19. Show that the principle of the last two questions applies 
 not only to the heights, H, h^ h z , etc., of the centre of gravity 
 and separate masses composing a system, but also to their 
 distances, D, d^ d 2 , etc., from any plane. D. 22, 23. 
 
XXIII. 
 CENTRE OF GRAVITY OF GEOMETRICAL FIGURES. 
 
 1. What assumption (as to the distribution of weight) is 
 made in calculating the position of the centre of gravity of bodies 
 approximating to geometrical figures ? D. 35. 
 
 2. What is meant by the centre of gravity of a geometrical 
 solid? surface? or line ? D. 35. 
 
 3.* Show that the centre of gravity of a unifoim thin rod (or 
 line) is at its centre, and that it will balance about any axis 
 passing through this centre. D. 36. 
 
 4.f Show that the centre of gravity of two equal triangles or 
 pyramids, symmetrically situated with respect to their common 
 apex, is at this apex. 
 
 5. Prove that if a figure can be cut up into strips, slices, 
 pairs of triangles, pairs of pyramids, etc., each having its centre 
 of gravity at or in a given point or axis, the centre of gravity of 
 the whole figure must be at or in this point or axis. Prob. 
 
 6. Show that if any figure is symmetrical with respect to a 
 centre (so that it can be cut up into equal and opposite pairs of 
 pyramids, with common apex at this centre) its centre of gravity 
 must lie at this centre. Prob. D. 36. Q. XVIII. 
 
 7. Show that if any figure is symmetrical with respect to -an 
 axis, (so that it can be cut up into equal and opposite pairs of 
 triangles with common apex on this axis) its centre of gravity 
 must lie on this axis. Prob. (Q. XXIII.) 
 
 8. Show that if any figure is symmetrical with respect to a 
 plane (so that it can be cut up into uniform thin strips, each with 
 its centre of gravity in this plane), the centre of gravity of the 
 figure must lie in this plane. Prob. (Q. XXIII.) 
 
 9. Where is the centre of gravity of a figure symmetrical 
 with respect to two axes ? an axis and a plane ? two planes ? 
 three planes ? Prob. 
 
32 CENTRE OF GRAVITY OF GEOMETRICAL FIGURES. [XXIII. 
 
 10. Find the centre of gravity of a circle, square, ellipse, 
 parallelogram, cube, block, parallelepiped, prism, cylinder, 
 sphere, ellipsoid, etc. Prob. D. 36. 
 
 11. Show (by cutting up a triangle into strips parallel to the 
 base, and considering the centre of gravity of each strip) that 
 the centre of gravity of a triangle must lie in its medial 
 line. D. 37. 
 
 12. Show that the centre of gravity of a triangle must lie at 
 the point of intersection of its medial lines, and find (by 
 geometry) the relative altitudes of the apex and centre of gravity. 
 D. 37, Q. ii. 
 
 13. Show (by cutting up a pyramid into sections parallel to 
 the base) that if a line be drawn from the apex to the centre of 
 gravity of the base (so as to pass through the centre of gravity 
 of each section), this line will contain the centre of gravity of 
 the pyramid. D. 38. 
 
 14. Show that the centre of gravity of a pyramid must lie at 
 the point of intersection of two or more lines drawn as in the 
 last question, and find (by geometry) the relative altitudes of 
 the centre of gravity and apex of a triangular (or any other) 
 pyramid. D. 38. 
 
 15. Find the centre of gravity of a cone by the same method 
 as in the last question. D. 38. 
 
33 
 
 XXIV. 
 
 CENTRE OF GRAVITY OF SUSPENDED BODIES. 
 
 i.f Find the moment of the force exerted by gravity on a 
 body of weight, W, about a horizontal axis in a vertical plane at 
 the distance, D, from the centre of gravity. (D. 58). 
 
 2.f A body of weight, W, and centre of gravity at the 
 horizontal distance, D, from a fulcrum (or axis) is balanced by a 
 force, P, acting on a horizontal arm, A. Find the equal and 
 opposite moments. (D. 58). 
 
 3. In the last question, express each of the four quantities, 
 W, Z>, P, and A, in terms of the other three. Prob. Q. 3. 
 
 4.J How can you find the weight of a body by balancing it 
 on an axis by means of a known weight ? Q. 
 
 5-J How can you locate the centre of gravity of a body of 
 known weight when balanced on an axis with another body of 
 known weight ? Q. 
 
 6.J Show how the weight of any object, great or small, can 
 be found by a graduated lever of known weight. Q. 
 
 7-1 Show how the position of a bullet of known weight in a 
 gun-barrel of known weight can be found from the displacement 
 of the centre of gravity. Q. 
 
 8. * Describe one or more experiments with levers showing 
 that their weight acts always as if concentrated at their centre of 
 gravity. 
 
 9.* Explain and describe one cr mure experiments illustrating 
 the behavior of bodies suspended at their centre of gravity. 
 
 10. Show that a body is in equilibrium if suspended at any 
 point in a vertical line passing through the centre of gravity. 
 D. 41. 
 
 IT.* What is the result of suspending a body at a point not 
 contained in the vertical line passing through the centre of 
 gravity ? 
 
34 CENTRE OF GRAVITY OF SUSPENDED BODIES. [XXIV. 
 
 12. State the condition of equilibrium in a body suspended 
 at a point. D. 41. 
 
 13.* Describe the experimental location of the centre of 
 gravity of a body by the method of suspension. D. 43, 44. 
 
 14. J Describe and explain one or more experiments in which 
 a body is suspended upon a horizontal axis in a vertical plane 
 containining the centre of gravity, and state how the centre of 
 gravity may be located by such experiments. 
 
 15.* What is the result of suspending a body upon a horizontal 
 axis not containing the centre of gravity ? 
 
 1 6. What is the result of suspending a body upon an oblique 
 axis containing the centre of gravity ? not containing it? Prob. 
 
 17.* Describe the condition of a body suspended about a 
 vertical axis. 
 
 1 8. State the conditions of equilibrium of a body suspended 
 upon an axis, in terms of the position of the centre of gravity 
 with respect to this axis. D. 41. 
 
 19. Show that in all the cases above, where a body is not in 
 equilibrium, the action of gravity is such as to cause the centre 
 of gravity to descend. D. 16. 
 
 20. Show that a heavy body is in equilibrium when, and only 
 when, the nature of its restraints is such as to prevent the centre 
 of gravity from descending. Prob. 
 
 2 1. 1 Explain the action of a double-pointed cone in rolling up 
 two inclined, but diverging rails. 
 
 22. Explain the action of a ball in rolling into the deepest 
 portion of a hollow. D. 46. 
 
 23. Show that, in general, the centre of gravity tends to 
 descend. D. 46. 
 
35 
 
 XXV. 
 
 STABILITY, AND ITS RELATIONS TO CENTRE OF 
 
 GRAVITY. 
 
 1. Show that a body, in which two points are fixed, is 
 equivalent to one suspended upon an axis passing through these 
 points. Prob. 
 
 2. How many points in a body must be fixed to prevent 
 movement of any kind ? Prob. 
 
 3.f Three points in the same horizontal plane are fixed. Find 
 the direction of the force on each, according to the position of 
 the centre of gravity. 
 
 4. Describe the forces by which a tripod is usually held in 
 place, stating the direction of these forces, and whether they 
 vary uniformly or " per saltum " under displacement. D. 40, 52. 
 
 5. State the conditions under which a body may be fixed by 
 the upward reaction of three forces. Q. 
 
 6.f What name is given to the area of a horizontal section 
 included between the forces in the last question ? 
 
 7.* Define, in general, the "area of support" of a bod) T . Q. 
 
 8.f Show that a body is stable if its centre of gravity lies 
 above its area of support. (D. 40). 
 
 9. What is meant by the limit of stability in a body 
 supported at three or more points ? D. 54. 
 
 10. Show that a body, displaced within its limits of stability, 
 tends to return to its original position. Prob. 
 
 11. When is a body said to be " practically stable ?" D. 54. 
 
 12. What, in general, is meant by stability? D. 42. 
 
 13. Distinguish stable, unstable, and neutral equilibrium, 
 according to the tendency of a body when displaced to return to 
 its original position of equilibrium. D. 42. 
 
36 STABILITY AND CENTRE OF GRAVITY. [XXV. 
 
 14. When is a heavy body, suspended by a point or by an 
 axis, in stable equilibrium ? in unstable equilibrium ? in neutral 
 equilibrium? State the position of the centre of gravity 
 relatively to the point or axis of suspension in each case. 
 D. 41, 42. 
 
 15. Explain the toy called " the balancer." D. 53. 
 
 1 6. Show that, in a condition of stability, the height of the 
 centre of gravity is at a minimum, in a condition of instability 
 at a maximum. D. 46. 
 
 17. State the conditions which determine whether equilibrium 
 exists, and whether it is stable or unstable, in terms of the path 
 of the centre of gravity consequent upon a given displacement 
 of the body. D. 46. 
 
 1 8. Find the state of equilibrium in a sphere or cylinder 
 rolling on a flat, on a concave, and on a convex surface. Prob. 
 (D. 46). 
 
 19. Describe and explain the action of a toy known as " the 
 tumbler," or any other similar device. D. 53. 
 
37 XXVI. 
 
 SENSITIVENESS OF A BALANCE AS RELATED TO 
 CENTRE OF GRAVITY. 
 
 i. Name the essential parts of a balance. D. 70. 
 
 2.J Describe the beam of an ordinary balance, the knife- 
 edges and their bearings. D. 75. 
 
 3.J Describe the pointer or index of a balance, the scale-pans, 
 and the means for arresting the loads, or lifting them from the 
 knife-edges. 
 
 4. Explain the importance of preserving the sharpness of 
 the knife-edges, and the ordinary precautions taken to this end. 
 (D. 75). 
 
 5. Where is the centre of gravity of a balance-beam, and how 
 is the position of the centre of gravity adjusted ? D. 73. 
 
 6. Show that the weights act as if concentrated at the knife- 
 edges. Prob. Q. XXII. i. 
 
 7. Show that if the three knife-edges be in the same straight 
 line, the centre of gravity is not disturbed by loading the scale- 
 pans so as to produce equilibrium. Prob. Q. XXII. 5. 
 
 8. Show that in a straight-arm balance, we have for the 
 angle, a, of deflection due to excess of weight, p, on aim of 
 length, /, of the beam of weight, w, with its centre of gravity at 
 a distance, d, from the axis, 
 
 tan a pi -j- wd. D. 73. 
 
 9-f What additional terms must be considered if the three 
 knife-edges are not in the same straight line ? D. 74. 
 
 10. Show that if the outer knife-edge be above the middle 
 knife-edge, the equilibrium of the balance may become unstable 
 from loading. D. 74. 
 
 11. Show that if the outer knife-edges be below the central 
 knife-edge, the equilibrium of the balance will become more 
 stable on loading. Prob. D. 74. 
 
 12. Distinguish three types of balance, according to align- 
 ment of the knife-edges. Q. 
 
 13. Show that a balance beam may belong, successively, to 
 the three several types named in the last question, through 
 bending under an increasing load. Prob. 
 
 i4.f What is meant by the sensitiveness of a balance ? 
 
38 SENSITIVENESS OF A BALANCE. [XXVI. 
 
 15. State certain conveniences in the use of a balance of 
 known sensitiveness. D. 74. 
 
 i6.f Describe the method of weighing under a constant load, 
 and state some of the advantages and disadvantages of this 
 method. D. 74. 
 
 ly.J How is the sensitiveness of a balance under a given load 
 ordinarily determined? 
 
 i8.f Explain the use and construction of a table showing the 
 sensitiveness of a balance under different loads. 
 
 ig.J Describe the use of a " rider," and show that in weighing 
 with a rider, the sensitiveness of a balance need not be kngwn. 
 
 20. J Explain the method of weighing by "oscillations" (or 
 by "vibrations "), and state some of its advantages. 
 
 2i.f Show that friction, within certain limits, does not 
 necessarily affect the sensitiveness of a balance, if the method of 
 oscillations is employed. 
 
 22. f State what advantages are to be gained by using a balance 
 with rapid oscillations, to what extent sensitiveness should be 
 sacrificed to this end, and to what extent sensitiveness is 
 consistent with rapidity. D. 71, 74. 
 
 23. f Show that it is convenient, but not necessary, that the 
 pointer should indicate how much the loads differ. 
 
 24. Correct DESCHANEL'S statement (73) that the centre of 
 gravity and axis of a balance " must not coincide." Q. 
 
 25. What qualities are sought for in a balance? D. 71, 73. 
 Q. XXVI. 1-26. 
 
 26.* Describe the ordinary "steelyard," andshowthat equality 
 of the arms of balance is not necessary in order that the true 
 weight of a body may be found. D. 76. 
 
 27. Describe (BORDA'S method of) weighing by substitution, 
 (or double weighing, according to DESCHANEL and other French 
 writers). D. 72. 
 
 28. J Describe (what is meant by most English writers by) the 
 method of "double weighing," (or GAUSS'S method, involving 
 an interchange of the loads). 
 
 29. f Show that the method of double weighing (by inter- 
 change) is twice as accurate as weighing by substitution. 
 
 30. J Describe, in detail, the processes actually employed in 
 very accurate weighings. 
 
 "-.' 
 
 Qg THE 
 
39 
 
 XXVII. 
 
 WORK, AND ITS RELATIONS TO CENTRE OF 
 GRAVITY. 
 
 i.* What name is given to the product of a force (when 
 constant in magnitude and in direction) and the displacement of 
 its point of application (in the same direction) ? 
 
 2.* Define work, and name some of the units in which it is 
 measured. 
 
 3.* Define the foot-pound and kilogram-metre. 
 
 4.f Define the foot-poundal, and erg (or dyne-centimetre). 
 
 5.* Find the work, W, necessary to raise a weight, w, 
 through the vertical distance, d. D. 45, 47. 
 
 6.* Distinguish between the work done by a force upon a 
 weight or against gravity, and that done by gravity or by the 
 weight against the force. D. 47. 
 
 7. How is work measured when the directions of the force 
 and displacement are nut the <-ame ? D. 45, 48. 
 
 8. The point of application, A, of a force, AB, moves to C. 
 What kind of work is done by the force if BAC is acute? if 
 BAG is obtuse ? if BAC is a right-angle ? D. 48. 
 
 9. Find the work done by a force, AB, in producing the 
 displacement, AC, in terms of AB, AC, and a function of the 
 angle, BAC. D. 48. 
 
 10. Show that the work done by a force in producing a given 
 displacement may be measured (i) by the product of the displace- 
 ment and the component of the force in the direction of the 
 displacement, or (2) by the product of the force and the 
 component of the displacement in the direction of the force. 
 Prob. Q. 
 
40 WORK AND CENTRE OF GRAVITY. [XXVII. 
 
 11. Show that the work necessary to move a weight in any 
 direction is the product of the weight and the vertical component 
 of the displacement. D. 48. 
 
 12. A body of weight, W, remains fixed, while another body, 
 of weight, w, moves through the distance, D. Find (by 
 geometry) the displacement, d y of the centre of gravity of the 
 two weights. Prob. 
 
 13. Compare, in the last question, the components of the 
 two displacements in the vertical (or any other) direction. Prob. 
 
 i4.f Show that the product of a weight, forming part of a 
 system, by its vertical displacement is equal to the product of the 
 weight of the whole system by the vertical displacement of its 
 centre of gravity. Q. 
 
 i5-t What is meant by the work done upon the centre of 
 gravity of a system ? 
 
 i6.f Show that the algebraic sum of the quantities of work 
 done upon the separate weights of a system (by considering these 
 one by one) is equal to the work done upon the centre of 
 gravity of the system. Prob. 
 
 17. State the "principle of work " as applied to the centre of 
 gravity of a system of weights. D. 45. 
 
 1 8. Bxplain how the calculation of work done upon a body 
 can be simplified by the principle of the last question. D. 45. 
 
 19. Find the work, W, necessary to pile a given weight, w, 
 of fragments into a rectangular, into a triangular, or into a 
 pyramidal heap of height, h, above the (mean) level of the 
 fragments. D. 45. 
 
 20. Modify the last question so as to apply to the case of 
 filling cisterns ot different shapes with liquids, or exhausting 
 wells of different depths and shapes. Prob. 
 
XXVIII. 
 WORK, AS A CRITERION OF STABILITY. 
 
 1. Find the work, W, necessary to turn a block of length, /, 
 breadth, b, thickness, /, and weight, w, from its position of 
 maximum stability into its position of medium or minimum 
 stability. Prob. D. 45. 
 
 2. Modify the last question so as to apply to an ellipsoid. 
 Prob. 
 
 3. A barrel of weight, w, rests upon a floor in stable 
 equilibrium, because its centre of gravity is at a distance, d, 
 below the axis : find the work, W, necessary to start the barrel 
 rolling. Prob. Q. 
 
 4.* State some of the most important properties of the centre 
 of gravity. Q. XXII.-XXVII. 
 
 5. Show that work must be done upon a body to displace it 
 from a position of stable equilibrium. D. 50. 
 
 6. Show that work is done by a body when it is displaced 
 from a position of unstable equilibrium. D. 50. 
 
 7. Show that no work is done upon or by a body when it is 
 displaced from a position of neutral equilibrium. D. 50. 
 
 8. State the criterion of the stability of equilibrium in terms 
 of the total work done upon or by a body when displaced from 
 its position of equilibrium. D. 50. 
 
 9f What condition of equilibrium is assumed to exist in an 
 ideal lever ? 
 
 10. t State, in general, what kind of equilibrium exists in 
 various ideal machines or " mechanical powers." 
 
XXIX. 
 
 PRINCIPLE OF WORK APPLIED TO MECHANICAL 
 
 POWERS. 
 
 , by geometry, that the work spent by a force acting 
 upon an ideal lever is equal to the work utilized in the weight 
 raised. 
 
 . 2.* State the " principle of work " (or " virtual velocities ") 
 as applied to the lever. D. 49. 
 
 3.* What assumptions (in regard to loss of work by friction, 
 etc.) are made in the investigation of ideal machines? (D. 49). 
 
 4. What distinction is made between the force known as 
 the "power" and the "weight" in various machines? D. 59. 
 
 5. What is meant by mechanical advantage ? D. 59. 
 
 6. Name some of the most important " mechanical powers." 
 D. 55- 
 
 7. Show that the wheel and axle may be regarded as an 
 endless lever. D. 60. 
 
 8. Find (by the principle of work) the relation between the 
 power, weight, and radii of a wheel and axle. Prob. 
 
 9. How would the results obtainable with a wheel and axle 
 be modified by the thickness of the ropes, if these were perfectly 
 flexible ? D. 60. 
 
 ro.f In what way would the stiffness of the ropes of a wheel 
 and axle modify the result if the ropes were perfectly elastic, so 
 that no work need be spent in bending them ? 
 
 ii. Show that a pulley may be regarded as an endless lever. 
 D. 61. 
 
 i2.f What assumption as to the tension of the cords passing 
 round one or more pulley-wheels is made in calculating the 
 mechanical advantage of ideal pulleys ? 
 
43 MECHANICAL POWERS. [XXIX. 
 
 13.* What mechanical advantage is gained by a single fixed 
 pulley ? D. 61. 
 
 i4.f What allowance must be made for the weight of a 
 movable pulley ? for the weight of the cords ? 
 
 154 How is the influence of the weight of pulleys and cords 
 eliminated in experimental demonstrations ? 
 
 1 6.* State the mechanical advantage of a single movable 
 pulley, to which the weight is attached, assuming the cords 
 parallel. D. 61. 
 
 17.* State in general the mechanical advantage of a pulley 
 niOv'ed by x cords with uniform tension in the same direction. 
 Prob. D. 62. 
 
 1 8.* Show that the principle of work applies to the case of a 
 pulley with x cords, as in the last question. Prob. D. 61. 
 
 19. Find the mechanical advantage of a series of x single 
 pulleys, each attached to the power cord of the next. Prob. 
 D. 63. 
 
 20. f What is meant by an ideal smooth surface or plane? and 
 what limitations exist in the direction of the force which such a 
 surface is capable of exerting upon a body ? 
 
 21% Describe one or more experiments with inclined planes 
 approximating to the ideal conditions. 
 
 22. Explain the resolution of forces in a body resting on a 
 perfectly smooth inclined plane. D. 64. 
 
 23. A body of weight, w, rests on an inclined plane of length, 
 AC, base, AB, and height, BC: find the force by which it is held 
 in place, (i) if this force is tangential, (2) if the force is 
 horizontal. Prob. Q. 
 
 24. Show that the principle of work applies to the forces 
 known as the " power " and the " weight " in an inclined plane. 
 D. 65, 66. 
 
 25. Show that a wedge may be regarded as a special case of 
 inclined plane. D. 67. 
 
44 MECHANICAL POWERS. [XXIX. 
 
 26. Find the mechanical advantage of an ideally smooth 
 wedge of length, /, and thickness, /. Prob. Q. 
 
 27.* Show that a screw can be treated as a special case of 
 inclined plane. D. 68. 
 
 28.* Find the mechanical advantage of a screw of circum- 
 ference, c, and distance, d, between threads measured parallel to 
 the axis of the screw, if the force is applied at the circumference, 
 at right-angles to the axis. Prob. Q. 
 
 29.* By what means is the mechanical advantage of a screw 
 greatly increased in ordinary screw presses ? D. 69. 
 
 30.* Find the mechanical advantage of a screw press with 
 arms of length, /, and threads, at the distance, d. D. 69. 
 
 3i.*fKstimate, approximately, the force which can be brought 
 to bear in an ordinary screw press, neglecting friction. 
 
 32.* Show that the principle of work applies to the screw 
 press. 
 
 33. Show that all the mechanical powers are reducible to two 
 types (the lever and the inclined plane). 
 
 34.* State, in general, the application of the principle of 
 work to mechanical powers. D. 49. 
 
 35.* What is meant by the statement that " what is gained in 
 power is lost in speed ?" Prob. Q. 
 
 36.* An ideal machine is actuated by a force, /% acting through 
 a distance, D, and produces a force,/", acting through a distance, 
 d ; express each of the four quantities, /% /" D, and d, in terms 
 of the other three. 
 
 37.* In the last question, state what must be, and what need 
 not be known about the construction of the machine, in order 
 that the principle of work may be applied. Q. 
 
45 
 
 XXX. 
 
 CONSERVATION OF WORK. 
 
 i.f A weight, w, is moved through the distances, AB, BC, 
 . . . YZ, all lying in the same vertical line. Show that the work 
 done is the same (w X AZ}, whatever may be the position of 
 the intermediate points, B, C, etc. 
 
 2. Prove that if a weight, w, is moved along any broken 
 path, A" B" C" . . . Z" , the work done is the same as along the 
 projection of this path, A' B' C' . . . Z 1 ', on a vertical line, and 
 hence, equal to w X A'Z'. Prob. 
 
 3. Show that in returning from Z" to A", the weight gives 
 out the same amount of work that it received between A" and 
 Z". Prob. D. 123. 
 
 4. Show that work done against constant forces, like those 
 exerted by gravity, depends only upon the initial and final 
 positions of a body, and is independent of the path of the body. 
 Q. (D. 123). 
 
 5. Show that when a body returns by any " closed path " to 
 its original position, the total work done upon it, or by it, is 
 zero. Q. (D. 123). 
 
 6.f Discuss, from the point of view of the last question, the 
 possibility of various proposed forms of "perpetual motion." 
 (D. 49)- 
 
 7. State the relation between the work received by a body 
 in one part of a closed path, and that given out by it in the 
 remainder of its path. Q. 
 
 8.*fWhat is meant by the "conservation of work" in 
 -mechanics ? 
 
XXXI. 
 
 LOSS OF WORK BY FRICTION. 
 
 i.* What element, entering into the working of all practical 
 machines, causes a departure from ideal conditions ? D. 49, 
 125, 131. 
 
 2.* Define (kinetical) friction (D. 131), and state the 
 peculiarity of sign in the work to which it gives rise. 
 
 3-t Describe and explain one or more experiments showing a 
 great discrepancy between the force required to start a body 
 sliding (no matter how slowly) over a surface, and that required 
 to maintain the motion. (D. 131, 132). 
 
 4. | State the results of one or more experiments showing 
 whether the force required to drag a body over a surface is or is 
 .not affected by the velocity. (D. 131). 
 
 1 5.f Why is it necessary, in measuring forces due to friction, 
 that bodies should be moved with uniform velocity ? 
 
 6. 1 State the results of one or more experiments showing the 
 relation between the force required to drag a body over a surface, 
 and the (normal) force urging the body and the surface together. 
 D. 131. 
 
 7. J What is the result of varying the ?rea of rubbing 
 surfaces, without varying either their nature or the force urging 
 them together ? 
 
 8. What is meant by a ' ' coefficient of friction ?" D. 131, 132. 
 
 9-J Describe one or more experiments in which the coefficient 
 of friction between two horizontal surfaces is determined. 
 
 10. A body of weight, W, requires a force, w, to drag it with 
 uniform velocity, along a horizontal surface ; find the coefficient 
 of friction between the body and the surface. Prob. Q. 
 
47 LOSS OF WORK BY FRICTION. [XXXI. 
 
 ii. What is meant by the "limiting angle of friction " for 
 two surf aces? D. 132. 
 
 1 2. J Explain the experimental determination of coefficents of 
 friction by means of an inclined plane. D. 133. 
 
 13. A body slides with uniform velocity down an inclined 
 plane with base, AB, and height, BC: find the coefficient of 
 friction. D. 133. 
 
 14. f Give some idea of the magnitude of coefficients of friction 
 between wooden or metallic surfaces, and the effect of grease or 
 oil upon these coefficients. 
 
 15. t Show that the mechanical advantage of a wedge cr screw, 
 no matter how fine the pitch, cannot exceed the reciprocal of the 
 coefficient of friction. 
 
 i6.f Find a relation between the weight, W, of a truck, the 
 diameters, D, and d, of the wheel and axle, the coefficient of 
 friction,/, of the (loose) bearing upon the axle, and the force, F, 
 required to pull the truck. Illustrate by a numerical example. 
 
 ry.f Find the difference between the forces, F' and F" , trans- 
 mitted by parallel cords to and from a pulley of diameter, Z?, 
 with axle of diameter, d, and coefficient of friction, /, on its 
 loose bearing, (calling the force borne by the pulley F' -\- F"). 
 
 i8.f Trace out the effect in a double block (with forr cords) 
 of a loss of 10 per cent., 20 per cent., etc., in the tension of each 
 cord passing round each wheel, on the force required to lift a 
 given weight by the block, remembering that the weight raised 
 is given by the sum of the tensions on the cords. 
 
 ig.f Find, conversely, the weight on the block necessary to 
 overcome a given force at one end of the cord. 
 
 20. f Explain the enormous loss of work in long trains of 
 clock-work, and the surprising effects of oil upon these. 
 
XXXII. 
 EFFICIENCY. 
 
 i. J Describe one or more experiments showing that the work 
 spent upon the cord of a tackle in raising a weight is greater 
 than that utilized through the pull upon this cord exerted by the 
 descending weight. 
 
 2.t What name is given to the ratio of the work spent to the 
 work utilized in a given case ? 
 
 3. Define "efficiency," as used in mechanics. Q. 2. 
 
 4.f Distinguish between the efficiency of a tackle as a 
 machine for raising weights, its efficiency as a machine for 
 utilizing work done by the descent of a weight, and its efficiency 
 as a machine for storing useful work. 
 
 5.f State a certain necessary relation between the three 
 efficiencies in the last question. 
 
 6. A man slides a box up into a wagon along a board inclined 
 at the limiting angle of friction : (Q. XXXI.) find the efficiency 
 of his combination. Prob. 
 
 7. Show that the efficiency of a screw, irreversible through 
 friction, cannot exceed 50 per cent. Prob. 
 
 8.f Show, from a point of view of efficiency, the disadvantage 
 of .a screw with too fine a thread (or of a differential screw). 
 
49 
 
 XXXIII. 
 POWER. 
 
 i. J Describe one or more experiments with an ergometer (or 
 friction-brake), showing how it is possible to measure the work 
 spent by a given agent in a given time. 
 
 2.J Describe a "transmission dynamometer," and its uses. 
 
 3.f What name is given to the quotient of work by time? to 
 the product of force and velocity ? 
 
 4.f Distinguish power, in its technical sense, from power as a 
 name for one of the forces in a machine. 
 
 5f Define pow r er in two different ways, and show that these 
 definitions are identical. Q. 3. 
 
 6.f Define "horse-power" (English or French), the erg per 
 second, and the "watt." 
 
 7-t Show that the horse-power varies slightly in different 
 latitudes, but that the erg and the watt are constant. 
 
 8.f Find the horse-power of a locomotive with two cylinders 
 100 square inches in section, and double two-foot stroke, making 
 125 revolutions per minute under a mean pressure of 33 Ibs. of 
 steam per square inch, in both the forward and backward stroke, 
 making no allowance for friction. 
 
 g.f Find the power, P, spent upon a water-motor with piston 
 of area, a (sq. cm.), making n double strokes of length, /, (cm.) 
 in every second, under a pressure, p (dynes per sq. cm.), and 
 state in what units the result is expressed. 
 
 io.f Show that the power, P, spent upon a motor (by an 
 incompressible liquid) is equal to the product of the pressure, p, 
 per unit of area, and current, c, in units of volume per unit of 
 time. 
 
 ii.t Show that (as a consequence of the principle of the last 
 question), if pressure is transmitted by an incompressible fluid 
 without loss, power must also be transmitted without loss. 
 
XXXIV. 
 
 SOLIDS AND FLUIDS, DISTINGUISHED. 
 
 1. 1 Describe an experiment (with OERSTED'S piezometer), 
 showing that liquids are compressible. D. 130. 
 
 2.* Distinguish fluids from solids (in respect to their relative 
 resistance to, and limits of recovery from, changes of form or 
 shape unaccompanied by changes of bulk or volume). (D. 129). 
 
 3.f Distinguish fluids from solids in respect to their rates of 
 yielding to tangential or transverse forces, and state whether this 
 distinction is one of kind or of degree. 
 
 4.f What name is given to that property in solids which is 
 connected with their slow yielding under transverse stresses ? 
 
 5.f When is a solid, and when is a fluid said to be especially 
 viscous? (D. 135). 
 
 6.f Show that viscosity is used in two opposite senses in its 
 applications to solids and fluids. 
 
 y.f Distinguish frictional forces within the body of a fluid 
 (whether due to viscosity or not) from forces due to the rubbing 
 of two solid surfaces, in respect to their dependence upon (i) the 
 compression, (2) the area, and (3) the relative velocity of the 
 moving parts. 
 
 8.* State some of the characteristic distinctions between 
 solids and fluids. Q. 
 
XXXV. 
 
 LIQUIDS AND GASES DISTINGUISHED. 
 
 1. 1 Describe an experiment (with a gas-bag and air-pump) 
 illustrating the indefinite expansibility of a gas. D. 194. 
 
 2.* Distinguish fluids into two classes, according to their 
 tendency toward uniform distribution throughout any space 
 within which they are confined. D. 194. 
 
 3.* How do liquids compare, as a class, with gases, in respect 
 to density ? 
 
 4.* How do liquids compare, as a class, with gases, in respect 
 to compressibility ? 
 
 5.f What is meant by the " free surface " of a fluid, and what 
 kind of fluids alone present such surfaces under ordinary 
 conditions ? 
 
 6.J Describe one or more experiments (e.g., with floating 
 needles), showing that the surfaces of liquids resist deformation 
 to a slight extent. D. 159. 
 
 7.J Describe one or more experiments (with camphor and 
 water, oil films on water, and the effects of local heat or alcohol) 
 showing that slight tangential forces are exerted by the surfaces 
 of liquids. D. 192. 
 
 8. J Describe one or more experiments (with PLATEAU'S films) 
 showing that the surface of a film is a minimum consistent with 
 its boundary. (D. 186). 
 
 9. Explain the spherical shape of rain-drops or bubbles. 
 D. 189. 
 
 10. J Describe one or more experiments showing that films of 
 liquid, when free to do so, cause their boundaries to contract. 
 
 ii. State the influence of the length, the breadth, and the 
 thickness of a film on the force with which it contracts 
 longitudinally. D. 185. 
 
52 LIQUIDS AND GASES DISTINGUISHED. [XXXV. 
 
 12. What is meant by ".surface tension ?" D. 185. 
 
 13. How many surfaces is a (soap-bubble) film considered to 
 possess ? and how is the surface tension of such a film measured ? 
 D. 188. 
 
 14.* Describe one or more experiments illustrating adhesion 
 between a liquid and a solid which it wets. 
 
 15.* Describe the surfaces of liquids of two different kinds, 
 near the edges of the vessels containing them. D. 182. 
 
 1 6.* What is meant by a capillary tube ? by capillary forces? 
 and by capillarity in general ? 
 
 17.* Describe the rise and fall of liquids caused by capillary 
 tubes. D. 182. 
 
 1 8.* When will a liquid rise in a capillary tube, and when will 
 it be depressed by it ? D. 182. 
 
 19. Describe the " meniscus " (or curved surface) of a liquid 
 accompanying (i) capillary ascensions and (2) capillary depres- 
 sions. D. 182, 190. 
 
 20. Kxplain the rise of liquids in capillary tubes which they 
 wet, and their depression by capillary tubes which they do not 
 wet. D. 1 86. 
 
 21. State some of the conditions which influence the amount 
 of the rise or fall of a liquid in a capillary tube. D. 183, 184. 
 
 22.f What is meant by the height of the meniscus (or 
 sagitta) ? 
 
 23. What is meant by the angle of contact between a liquid 
 and a solid? D. 184, 185. 
 
 24. Find the height, /i, of a column of liquid of density, d, 
 sustained by a surface tension (in gravitation units) t, in a tube 
 of radius, r, assuming that the tube is wet by a film of liquid 
 tangent to its surface. D. 186. 
 
 25. State the law of diameters governing capillary ascensions 
 and depressions. D. 184. 
 
 26. J Describe one or more experiments (e.g., with inclined 
 plates) illustrating the law of diameters. 
 
53 LIQUIDS AND GASES DISTINGUISHED. [XXXV. 
 
 27. f How high are liquids known to rise by capillary action in 
 certain vegetable structures ? 
 
 28. f Give reasons for supposing that a practical limit exists in 
 the height attainable by capillary action. 
 
 29. J Describe an experiment with an air-pump, showing that 
 capillary ascensions and depressions are not due to atmospheric 
 pressure. (D. 185). 
 
 30.J Describe one or more experiments illustrating the tensile 
 strength under certain specified conditions, of columns of liquid 
 of considerable cross section. 
 
 31.* In what class of fluids, alone, are phenomena due to 
 capillarity, cohesion, or surface tension perceptible ? Q. 
 
 32.* State some of the characteristic distinctions between 
 liquids and gases. Q. 
 
 33. 1 Describe one or more experiments illustrating effects of 
 endosmose (or diffusion through a diaphragm) and endosmotic 
 pressure. D. 193. 
 
 34. Distinguish solutions into two classes with respect to the 
 facility with which they pass through a diaphragm. D. 193. 
 
 35. Describe the process of separating colloids and crystalloids 
 by "dialysis." D. 193. 
 
 36. Describe one or more experiments illustrating the diffusion 
 of gases, and the pressures to which such diffusion may give 
 rise. D. 193, 227. 
 
54 
 
 XXXVI. 
 FUNDAMENTAL ASSUMPTIONS IN HYDROSTATICS. 
 
 i.f What assumptions are usually made in the solution of 
 problems in hydrostatics, with respect to viscosity, capillarity, 
 osmosis, etc. ? 
 
 2. Why does not viscosity enter into problems is hydrostatics? 
 D. 135- 
 
 3-t State reasons for assuming that capillary forces may be 
 neglected in the treatment of bodies of liquid of considerable 
 size, or in the case of solids of considerable size immersed in 
 these liquids. 
 
 4.J Give experimental grounds for the assumption that forces 
 or pressures due to osmosis or to diffusion are not perceptible 
 in the wide channels usually employed in hydrostatics. 
 
 5-t Show that, in the absence of viscosity and capillarity, 
 normal forces and pressures are the only ones which need be 
 taken into account. (D. 135). 
 
 6. Distinguish hydrostatics from hydrokinetics, treating 
 both as branches of hydrodynamics. D. 134. 
 
 7. Distinguish pneumatics from hydrostatics. D. 3. 
 
 8.f To what extent are the properties of gases studied under 
 the head of hydrostatics ? 
 
 Q.f State certain obvious experimental evidence of the fact 
 that fluids, when undisturbed for a long time, fall (practically) 
 into a state of equilibrium. 
 
 io.f What general principle would lead to the anticipation of 
 the result stated in the last question ? 
 
 u.f What hypothesis (as to the equilibrium of fluids) lies at 
 the foundation of hydrostatics ? 
 
55 
 
 XXXVII. 
 CONDITIONS OF EQUILIBRIUM IN FLUIDS. 
 
 i.f Show (by considerations relating to the centre of gravity) 
 that a bubble, contained in a liquid within any enclosure, seeks 
 the highest possible point. 
 
 2. Describe the construction and use of an ordinary spirit- 
 level, its attachment to telescopes with cross-hairs, the conditions 
 of, and one or more processes of testing its sensitiveness, and the 
 method of eliminating errors of adjustment. D. 180, 181. 
 
 3. State the condition of equilibrium in the free surface of a 
 liquid. D. 140. 
 
 4. Show, by the resolution of forces, that if the free surface 
 of a fluid is not horizontal, it is not in equilibrium. D. 140. 
 
 5.f Show, by considerations relating to the centre of gravity, 
 that the bounding surface between any two fluids of unequal 
 density is in equilibrium only when horizontal. (D. 145). 
 
 6. 1 Describe one or more experiments illustrating the equili- 
 brium of bounding surfaces between different fluids. D. 145. 
 
 7. Show that the equilibrium of a fluid would not be 
 disturbed if any portion of it should become rigid, or should be 
 replaced by a fixed rigid body. D. 153. 
 
 8.f Show that it is possible, by imagining rigid supports 
 substituted for certain portions of a body of fluid, to treat the 
 remainder as if contained in a /-tube, or in communicating 
 vessels of any shape, without any change of equilibrium. 
 
 9.* State the condition of equilibrium of a liquid contained in 
 communicating vessels. D. 178. 
 
 i o.*J Describe one or more experiments illustrating the fact 
 that a liquid stands at the same level in communicating vessels. 
 D. 178. 
 
56 CONDITIONS OF EQUILIBRIUM IN FLUIDS. [xXXVII. 
 
 ii. Explain the construction and use of a water-level. 
 D. 179. 
 
 i2.f Show (by the principle of action and reaction) that the 
 resultant of all the forces exerted by a fluid upon the walls of its 
 enclosure must be equal and opposite to the resultant of the forces 
 exerted by the walls of the enclosure upon the fluid, and hence 
 (by the fundamental principles of equilibrium) equal to the 
 weight of the fluid. (D. 148). 
 
 i3-t Show that the weight of any portion of a fluid, of 
 whatever shape, is equal and opposite to the resultant of the 
 forces exerted upon this portion by the body of fluid which 
 surrounds it. 
 
 14. Show that the difference between the forces exerted upon 
 the top and bottom of a column of fluid with vertical sides (from 
 which the column is supposed to derive no support) must be equal 
 to the weight of the column. D. 139. 
 
 15. Show that the resultant forces upon the two ends of a 
 fluid prism with horizontal axis must be equal and opposite. 
 D. 138. 
 
 1 6. Find by the triangle of forces, the relation between the 
 resultant forces upon the three rectangular faces of a triangular 
 fluid prism of negligible weight, and show (by similar triangles) 
 that the three forces are proportional to the areas of the three 
 faces in question. D. 137. 
 
57 
 
 XXXVIII. 
 PRESSURE. 
 
 i.f What is meant by intensity of pressure? and in what 
 units is it expressed ? D. 136. 
 
 2.f In what sense is the word " pressure," when unqualified, 
 used by most modern writers ? and how 7 does this differ from 
 another sense in w 7 hich it was sometimes employed by earlier 
 writers? D. 136. 
 
 3-f Find the (intensity of) pressure, p, (expressed in gravi- 
 tation units) exerted by a weight, w, upon a horizontal surface 
 of area, a. 
 
 4. A weight, w, of fluid is contained in a tube with vertical 
 sides, and with an area of cross-section, a. Show that if the 
 upper surface is free from force, the weight of the fluid is borne 
 by the bottom of the tube, alone ; and find the pressure due to the 
 weight of the fluid upon the bottom of the tube, supposing it to 
 be horizontal. Prob. Q. XXXVII.-XXXVIII. 
 
 5. Find the weight, w, of fluid of density, d, standing at a 
 height, h, in a tube with vertical sides and horizontal base of 
 area, a ; find also the pressure upon the bottom of the tube. 
 Prob. Q. XXXVII.-XXXVIII. 
 
 6. Find, as in the last question, the pressure, p, at the 
 bottom of a vertical column of fluid of density, d, and depth, h, 
 when in equilibrium with the surrounding fluid. Prob. Q. 
 
 7. Show (by the principle of action and reaction) that the 
 upward pressure beneath any horizontal area, a, in the body of a 
 fluid, must be equal to the downward pressure on this area, due 
 to the fluid above it. Prob. 
 
 8. \ Describe one more experiments showing that the upward 
 pressure at a given depth in a liquid is equal to the downward 
 pressure due to a column of liquid of the same depth and density. 
 D. 144. 
 
5 8 PRESSURE. [XXXVIII. 
 
 9. Prove (by the conditions of equilibrium of the pressures 
 on the faces of a small triangular prism, with axis horizontal) 
 that the pressure upon any surface of given area at a given depth 
 in a fluid is the same as upon a horizontal surface of the same 
 area, and at the same depth. D. 137. Q. XXXVII. 15. 
 
 10.* Explain the statement that the pressure of a fluid is the 
 same in all directions. D. 137. 
 
 n. Prove (by considering the conditions of equilibrium of 
 the forces on the ends of a prism with horizontal axis) that the 
 pressure at two points at the same level in a fluid must be the 
 same. D. 138. 
 
 12. Prove (by considering the conditions of equilibrium of a 
 column of fluid with vertical sides) that in descending through 
 the vertical distance, h, in a liquid of density, d, the pressure 
 (measured in gravitation units) increases by the amount, hd, 
 whether the starting point is in the free surface of the liquid or 
 not. D. 139. 
 
 I3.| Describe one or more experiments (with communicating 
 tubes) illustrating the equality of pressure between columns of 
 liquid of given vertical height in tubes having different 
 inclinations. 
 
 14. Show (by a zigzag of horizontal and vertical prisms) that 
 the pressure at two points on the same le\el in communicating 
 vessels must be the same. Prob. D. 139, 147. 
 
 15. Show (as in the last question) that the pressure of a fluid 
 of density, d, at a depth, 7z, below a given surface is greater than 
 at this surface by the amount, hd, regardless of the shape of the 
 vessel. D. 139, 147. 
 
 i6.*J Describe one or more experiments illustrating the fact 
 that the force exerted on a given area by a liquid of given depth 
 and density is independent of the shape of the vessel containing 
 the liquid. D. 147. 
 
 I7.J Describe one or more experiments {e.g. , with E. H. 
 HAUL'S pressure-gauge) illustrating the fact that the pressure 
 
59 PRESSURE. [XXXVIII. 
 
 in a liquid increases with the depth, and is the same in all 
 directions. 
 
 1 8. Discuss the applicability of the principles of hydrostatic 
 pressure to the case of liquids in capillary tubes. D. 190. 
 
 19. Show that the pressure of a liquid just within its curved 
 surface in a capillary tube must differ considerably from that just 
 outside of it, and show that the difference is explainable by 
 surface tension. D. 190, 191. 
 
 20. Find the sign of the pressure of a liquid raised in vacuo by 
 a capillary tube, on the assumption that its pressure outside of 
 this tube, is zero. D. 190. 
 
 21. Explain the attraction between two parallel plates, or 
 between two floating bodies of which (i) both are wet, or (2) 
 neither is wet by the liquid. D. 190. 
 
6o 
 
 XXXIX. 
 BALANCING COLUMNS. 
 
 i. *J Describe an experiment in which water is introduced into 
 one branch of a /-tube containing some mercury. (D. 146). 
 
 2.* If the water in the last question reaches from the level, a, 
 to the level, b, and the mercury reaches from the level, b, down 
 to the bottom of the /-tube and up again to the level, <r, what 
 is the function of the mercury below b ? and what is the relation 
 between the pressures exerted upon this portion of the mercury 
 by the column of water (ab) above it on one side, and by the 
 column of mercury (be) above it on the other side? D. 146. 
 
 3.*fShow that the depth of a /-tube, below the level where 
 two liquids abut, has nothing to do with their equilibrium. 
 
 4.*t State what portions of two liquid columns, abutting in 
 a /-tube, are considered as producing equal pressures. 
 
 5.f Show that the condition of equilibrium between two 
 balancing columns is not affected by atmospheric pressure, 
 provided that it acts upon both alike. 
 
 6.J Explain the use and function of compressed air in 
 balancing liquid columns which cannot be allowed to abut. 
 
 7. Point out the liquid columns exerting equal pressures on 
 compressed air contained in a tube shaped like a W. Prob. Q. 
 
 S.I Explain the method of balancing columns of liquid by 
 means of rarefied air. 
 
 9. Point out the liquid columns exerting equal pressures in 
 an inverted F-tube. Prob. Q. 
 
 10.* Find a relation between the densities, D and d, and the 
 vertical heights, h and H, of two balancing columns, and 
 express each of the four quantities in terms of the other three. 
 Prob. Q. 
 
6 1 BALANCING COLUMNS. [XXXIX. 
 
 ii. Describe three methods of balancing columns, and explain 
 how the relative density of two liquids can be found by each. Q. 
 
 12. f Show that the pressures of two balancing columns of the 
 heights, //and h, are not, under ordinary circumstances, exactly 
 equal, but differ by a small amount, equal to the pressure of a 
 column of air of the height, H h. 
 
 i3.f State the magnitude and sign of the correction for the 
 pressure of a column of air in some special case (e.g., when a 
 column of water differs in height by 800 mm. from its balancing 
 column, in air 800 times less dense than water). 
 
 14. t Describe a mercurial manometer, and show how it can be 
 used for measuring differences of pressure. D. 228. 
 
 15. Find the difference of pressure (in gravitation units) 
 necessary to maintain a difference of level of i cm., 100 cm., 
 etc., in a manometer containing mercury of the density 13.6. 
 Prob. Q. 
 
 1 6. Describe a " multiple branch manometer," and state its 
 advantages and disadvantages. D. 224. 
 
 OF THE 
 
 TJBIVEHSIT7 
 
62 
 
 XL. 
 ATMOSPHERIC PRESSURE. 
 
 1. 1 Describe a (BOURDON) metallic pressure-gauge, and the 
 process of calibrating such a gauge by a mercurial manometer. 
 D. 226 (225). 
 
 2.f In stating that the pressure of a liquid is proportional to 
 the depth, what assumption is made as to the pressure at the 
 surface of the liquid ? 
 
 3. If the pressure (due to atmospheric or other influences) at 
 the surface of a liquid, instead of being zero, is P, what 
 correction must be made in the last statement? D. 141. 
 
 4-t Explain how, in the method of graduating a pressure- 
 gauge, the correction in the last question might be everlooked. 
 
 5.J Describe one or more experiments with pressure-gauges, 
 showing that pressures exist lower than that of the atmosphere. 
 
 6.f State reasons why a vacuum, rather than the atmosphere, 
 should be taken as representing the zero of pressure. (D. 191 ?). 
 
 y.f What modification is necessary to convert a (BOURDON) 
 pressure-gauge into' an Aneroid barometer ? 
 
 8.J Describe the Aneroid barometer as usually constructed. 
 D. 206. 
 
 9.J Describe the construction of a simple form of mercurial 
 barometer (the ''Torricellian experiment ") and state what facts 
 it illustrates. D. 196, 197, 200. 
 
 10.* Give some idea of the average height of the barometric 
 column at the sea level in inches and in centimetres. D. 197. 
 
 ii.* How is the atmospheric pressure at a given time and 
 place measured ? Prob. D. 197. 
 
63 ATMOSPHERIC PRESSURE. [XL. 
 
 12. Find the atmospheric pressure (in gravitation units) 
 which will sustain 76 cm. of mercury of the density, 13.596. 
 Prob. Q. 
 
 i3-t Define a pressure of "one atmosphere" according to 
 French, and more or less general scientific usage. (D. 198). 
 
 14. Calculate the force on a given surface (e.g., the human 
 body) due to atmospheric pressure. Prob. Q. 
 
 15.^ Describe one or more experiments illustrating effects of 
 atmospheric pressure (e.g., Magdeburg hemispheres, burst 
 membranes, mercury forced through wood, fountain in vacuo, 
 etc.). D. 236. 
 
 16. Find the height of a column of air of density, .0012, 
 which will exert a pressure of "one atmosphere" (76 cm., 
 mercury of density, 13.6). Prob. Q. 
 
 17.. What is meant by the "height of the homogeneous 
 atmosphere?" D. 210. 
 
 1 8. Describe and explain (PASCAL'S) experiments in ascending 
 hills with a barometer. D. 199. 
 
 19. Find the difference in pressure (in gravitation units) 
 between two points on a mountain differing 10, 100, 1000, etc., 
 metres in altitude, when the mean density of the atmosphere is 
 .0012. Prob. Q. 
 
 20. Apply the last question to points higher up on the 
 mountain, where the density of the air becomes .0011, .0010, 
 .0009, .0008, .0007, or even .0006. Prob. Q. 
 
 2 1 . Explain the estimation of heights by the barometer, the 
 necessity of taking into account the density of the air, its 
 variations, if considerable, and the use of tables in this 
 connection. D. 209. 
 
 22. What relation usually exists between the fluctuations of 
 a barometer and meteorological changes ? D. 215. 
 
 23. What relation exists between the height of the mercurial 
 barometer and that of one containing water or other liquid ? 
 D. 197. 
 
6 4 
 
 XLI. 
 FLOW OF LIQUIDS AFFECTED BY AIR PRESSURE. 
 
 i.J Describe a " pipette," (or tube closed by the finger at one 
 end), and explain the agency of the atmosphere in sustaining 
 the weight of the liquid, and in transmitting it to the place where 
 it is felt. (D. 269). 
 
 2. J Describe one or more experiments illustrating the principle 
 of the pipette. D. 269. 
 
 3.| Describe and explain the action of an "intermittent 
 fountain." D. 270. 
 
 4.J Describe a " MARIOTTE'S bottle," and state the conditions 
 under which it will give rise to a flow of water under constant 
 pressure. D. 275. 
 
 5. State the uses of a MARIOTTE'S bottle for obtaining 
 constant pressure or constant suction. D. 275. 
 
 6. Find the pressure (or suction) due to a liquid of density, d, 
 issuing from a MARIOTTE'S bottle, if h is the height of the inlet 
 above the outlet. Prob. Q. 
 
 7. 1 Describe and explain the action of a "siphon." D. 271. 
 
 8. What relation exists between the height of a barometer 
 filled with a given liquid and the maximum height through 
 which that liquid can be raised by an ordinary siphon ? 
 Prob. Q. 
 
 9. Why must air be excluded from a siphon ? and through 
 what device is this accomplished in small siphons ? in large 
 siphons? D. 271. 
 
 10. Explain the "vase of Tantalus," and state the conditions 
 of intermittent flow in terms of the rates of flow into and out of 
 the vase. D. 274. 
 
 11. Give a possible explanation of certain intermittent 
 springs. D. 275. 
 
 12% Describe and explain the action of a SPRENGEL pump. 
 D. 240. 
 
XLII. 
 
 PUMPS. 
 
 1. What connection exists between the height of a barometer 
 filled with a given liquid and the maximum height to which the 
 liquid can be raised by a suction pump? D. 255. 
 
 2. Find the maximum height to which water (of density, i) 
 ^ean be sucked under atmospheric pressure (1033 grams to the 
 sq. cm.). Prob. Q. 
 
 3.J Draw a diagram of an ordinary suction-pump (showing 
 the barrel, piston, and two valves). D. 254. 
 
 4.* Name the essential parts of a suction-pump, and explain 
 its action. D. 254. 
 
 5-f Show that an ordinary suction-pump is identical in 
 principle with one of the simpler forms of air-pump, and that it 
 is use$ as an air-pump in the first stages of raising water. Prob. 
 D. 229, 254. 
 
 6.*f State the use of wetting the valves of a pump in the 
 first stages of raising water. 
 
 y.f What modifications are necessary to convert a suction - 
 pump into a force-pump ? D. 258, 259. 
 
 8. What is meant by a " plunger," and in what respect does 
 it differ from an ordinary piston ? D. 259. 
 
 9.^ Draw a diagram of an ordinary force-pump, and state its 
 advantages over a suction-pump. 
 
 10. Show that a force-pump should not be inferior to a 
 suction-pump, even when suction alone is required. Prob. Q. 
 
 n.f Show that a force-pump can be used either for exhausting 
 or for compressing air, or other gases. (D. 243). 
 
 1 2. | Draw a diagram of an air-pump with three valves, and 
 state its practical advantages. (D. 235). 
 
66 PUMPS. [xui. 
 
 i3-t Show that the two sides of a piston of any pump, raising 
 a liquid of density, d, through the vertical height, h, between 
 two levels (the atmospheric pressure being practically the same 
 at each level) are subject to a difference of pressure, hd. 
 
 i4.f Find the force due to difference of pressure on the two 
 sides of the piston in the last question, if its area is a, also the 
 work, w, necessary to move it against this force (neglecting 
 friction) through the distance, s. (D. 256). 
 
 15. Find the volume, v, and weight, w, of liquid delivered in 
 the last question by the stroke, s, and calculate the work utilized 
 in raising it to the height, h. Prob. 
 
 16. Show that the principle of work applies either to suction 
 or to force-pumps. Prob. Q. 
 
 .17. 1 Explain the construction and use of a hydraulic press. 
 D. 142, 264. 
 
 18. Find the force, F, upon the large piston of area, A, in a 
 hydraulic press, due to the force, /, on the small piston of area, 
 a, (at the same level as the large piston). Prob. Q. 
 
 19. Find the distance, d, through which the large piston of 
 area, A, moves when the small piston, of area, a, advances 
 through the distance, D, (assuming the liquid incompressible). 
 Prob. 
 
 20. Show that the principle of work applies to the hydraulic 
 press. D. 143. 
 
 2i.f What assumption (as to the levels of the two pistons) is 
 made in elementary discussions of the hydraulic press ? 
 
 22.f Show how, by the use of a counterpoise, corrections due 
 to differences of level in a hydraulic press may be eliminated. 
 
 23. What, in general, is the effect of increasing the pressure 
 at a given point in a fluid, by the amount, P, upon the pressure, 
 p, at any other point? D. 141. 
 
 24. What is meant by transmissibility of pressure in fluid 
 systems? D. 141. 
 
XLIII. 
 CENTRES OF PRESSURE AND BUOYANCY. 
 
 1. How do you find the magnitude of the force exerted by a 
 liquid upon a plane surface of variable depth? D. 150, 151. 
 
 2. Define centre of pressure. D. 150, 151. 
 
 3. How do you find the centre of pressure on any plane 
 surface of variable depth ? D. 151. 
 
 4. Where is the centre of pressure upon a dam ? and where 
 should a single prop be applied to resist this pressure? Prob. Q. 
 
 5. Where is the point of application of the forces exerted 
 by a liquid on a body immersed in it ? on a body floating upon 
 the liquid ? D. 153. 
 
 6. Define centre of buoyancy. D. 153. 
 
 7. State the conditions of equilibrium between the forces 
 due to the weight of a body and the buoyant action of the fluid 
 in which it is submerged. D. 155, 156, 157. 
 
 8. Show that the conditions in the last question do not hold 
 for a body floating on the surface of a liquid. D. 157. 
 
 9. State the conditions of stable and unstable equilibrium, 
 for bodies floating on a liquid, in terms of the path of their 
 centres of gravity, due to rotation of the bodies, with constant 
 volume immersed. Q. XXV. 17. 
 
 10. Explain the equilibrium of a boat with its centre of 
 gravity above the centre of buoyancy. D. 157. 
 
 n.f Prove that the stability of a boat is always increased by 
 lowering the centre of gravity. (D. 158). 
 
 12.* Explain the use of ballast. Q. 
 
 I3.J State the advantages of the form and distribution of 
 weight in an ordinary hydrometer. 
 
68 
 
 XLIV. 
 
 FLOTATION. 
 
 i.* A body of uniform density is immersed in a fluid ; state 
 the conditions which determine whether it will rise, sink, or 
 remain where it is, when free to move. D. 155. 
 
 2.J Explain the action of the " Cartesian diver." D. 156. 
 
 3. Show that the Cartesian diver, in the middle of a fluid, 
 is never in stable equilibrium, as far as height is concerned. 
 D. 156. 
 
 4. Under what conditions will a body, denser than a liquid, 
 float upon its surface ? 
 
 5. Show that the principle of the last question can be 
 extended to the case of bodies supported on the surface of a 
 liquid by surface tension. D. 159. 
 
 6.f A body of density, d, floats on a liquid, of density, D ; 
 find the proportion of its volume immersed. 
 
 7. A body of density, d, and volume, V, floats with a 
 volume, v, immersed in a liquid; find the density, D, of the 
 liquid. Prob. Q. 
 
 8. A body floats either with a volume, v, immersed in a 
 liquid of density, D, or with a volume, V, immersed in a liquid 
 of density, d; find the relative densities of the liquids in terms 
 of these volumes. Prob. Q. 
 
 9.J Describe the construction and use of a "specific volume- 
 nometer," (or hydrometer with uniform scale showing reciprocals 
 of density). 
 
 10. What, in general, is meant by a hydrometer of variable 
 immersion? D. 168, 171. 
 
 u.f State the advantages of using a set of hydrometers 
 instead of a single hydrometer, covering a given range of 
 density. 
 
69 FLOTATION. [XLIV. 
 
 1 2. | Describe a ''densimeter" (or hydrometer with specific 
 gravity scale), and the peculiar spacing of the scale of such an 
 instrument. D. 172. 
 
 13. Name certain arbitrary hydrometer scales, and explain the 
 use of tables in connection with these scales. D. 173, 174. 
 
 14. State the advantages, for special purposes, of instruments 
 like the " centesimal alcoholimeter." D. 175. 
 
 15. What is meant by a hydrometer of constant immersion? 
 D. 168. 
 
 1 6. J Describe NICHOLSON'S hydrometer, and the general 
 method of finding by it the weight of a body (in air or in water). 
 D. 169. 
 
 17. Show that any uniform pressure (whether due to the 
 atmosphere or not) will not affect the equilibrium of a hydro- 
 meter. Prob. Q. 
 
 1 8. Show that any cause which increases the pressure on the 
 lower part of a hydrometer more than on the upper part will 
 have a buoyant effect. Prob. Q. 
 
 I9.J Describe and explain one or more experiments illustrating 
 the buoyant effect of covering the upper portion of a hydrometer, 
 floating in water, with a liquid of lower density. 
 
 20. f State the effect of atmospheric density on a hydrometer, 
 and the advantage of reducing the dimensions of those parts of 
 a hydrometer which float above the surface of a liquid. 
 
XLV. 
 PRINCIPLE OF ARCHIMEDES. 
 
 1. Prove that the difference between the forces exerted upon 
 the top and bottom of a rigid column with vertical sides of 
 height, /z, including an area, a, by a fluid of density, d, must be 
 the same as that (hda) upon a column of the same fluid having 
 identical dimensions, and hence equal to the weight of the 
 column of fluid. D. 153. 
 
 2. Show that the buoyant action of a fluid upon a rigid 
 column, with vertical sides, is independent of the depth. D. 153. 
 
 3. Prove (by cutting up a body into prismatic columns with 
 vertical sides) that any body, suspended in a fluid, must be 
 buoyed up with a force equal to the weight of an equal bulk of 
 the fluid. D. 153. 
 
 4. Prove the statement in the last question by substituting 
 for the body, a body of the fluid having the same shape, and 
 applying the general conditions of equilibrium. D. 153. 
 
 5. What is meant by the loss of weight of a body when 
 immersed in a fluid ? D. 153. 
 
 6.* State the principle of Archimedes. D. 153. 
 
 7. Show that the principle of Archimedes holds for any fluid, 
 whether liquid or gaseous. Prob. 
 
 8. \ Explain how the weight of a body in water can be found 
 by a NICHOLSON'S hydrometer, (i) if its density is greater, and 
 (2) if its density is less than that of water. (D. 169). 
 
 9-t When is the weight of a body in a fluid considered 
 positive, and when negative ? 
 
 io.f Show that the principle of Archimedes applies to all 
 bodies, whether their density be greater or less than that of the 
 fluid in which they are immersed. 
 
 1 1. 1 Describe one or more forms of hydrostatic balance. 
 D. 164. 
 
71 PRINCIPLE OF ARCHIMEDES. [XLV. 
 
 I2.J Explain the use of a sinker in weighing bodies in a liquid 
 in which they would otherwise float, and how the weight of the 
 sinker can be allowed for. D. 165. 
 
 13. Describe the "overflow-beaker," and its use for finding 
 the weight of liquid displaced by a solid. 
 
 I4.J Show that the weight of an overflow beaker is not 
 increased when a suspended solid is immersed in it. Prob. 
 
 I5.J Show that a vessel (without overflow), into which a 
 suspended solid is immersed, gains in \veight by an amount equal 
 to the weight of liquid displaced by the solid. Prob. 
 
 1 6.* Prove (by considering the weight of a solid, as borne 
 partly by its suspension, and partly by the vessel in which it is 
 immersed) that the loss of weight of a body on immersion must 
 necessarily be equal to the gain of weight of the vessel in which 
 it is immersed. Prob. 
 
 i y.*t Describe one or more experiments illustrating the 
 principle of Archimedes. D. 154. 
 
 18.* A piece of chalk weighs (i) 5 grams in air when dry, and 
 (2) 2 grams in water ; but (3) it weighs 6 grams in air when the 
 air in its pores has been completely replaced (through long 
 soaking) by water, and (4) it weighs 3 grams under the same 
 circumstances in water ; show that if any three of the four data 
 above are given, the fourth (assuming the chalk to be inexpan- 
 sible) can be supplied. Prob. 
 
 iQ.f Discuss the truth of the statement that "water weighs 
 nothing in water." 
 
 20.* When does absorption of water increase the weight of a 
 body in water, and when does it have no effect ? Q. 
 
 2 1. *J Describe one or more experiments with a rigid vessel and 
 with an expansible vessel, illustrating the point in the last 
 question. 
 
 22. Why was ARISTOTLE'S method of detecting the w r eight 
 of air unsuccessful ? and how should it have been modified ? 
 Prob. (D. 195). 
 
72 
 
 XLVI. 
 
 ATMOSPHERIC BUOYANCY. 
 
 1. About what is the weight of a cubic centimetre of air? 
 of a litre? of a cubic metre? etc. (D. 195). 
 
 2. Calculate the lifting power of a balloon of weight, w, 
 volume, v, filled with gas of density, d, in air of density, D. 
 D. 249. 
 
 3.^ Describe an experiment with the " baroscope." D. 247. 
 
 4.J Describe a " barodeik " or graduated baroscope, and 
 explain its action. 
 
 5-f What is meant by the " effective weight" of a body in a 
 fluid? 
 
 6. Find the effective weight (in gravitation units) of a body 
 of mass, M, and density, D, in air of density, d. Prob. Q. 
 
 7. Find the effective weight (in gravitation units) of M 
 grams of brass of density 8.4 in air of density 0.0012 (grams to 
 the cubic centimetre). Prob. Q. 
 
 8. Under what circumstances are standards of weight 
 supposed to be adjusted ? 0.251. 
 
 9-f Show that all ordinary weighings consist in balancing 
 effective weights in air. 
 
 io.f What is meant by the "apparent weight" of a body (in 
 air) ? D. 251. 
 
 n.f Show that the effective weight of a body is always less 
 than its apparent weight by a certain proportional amount (about 
 i part in 7000) depending only upon the density of the air (say 
 0.0012) and of the brass weights (say 8.4) by which it is 
 counterpoised. D. i. 
 
 12. Given the apparent weight, W, and density, D, of a body 
 counterpoised by brass standards of density, D' , in air of density, 
 d, find the true weight of the body in vacuo. D. 251. 
 
73 ATMOSPHERIC BUOYANCY. [XLVI. 
 
 13. When is the correction for buoyancy positive and when 
 negative? Prob. Q. n. 
 
 i4-t Illustrate the corrections for air-buoyancy by numerical 
 examples. 
 
 15. Show that the apparent weight of a body in air varies 
 according to conditions of atmospheric density, and that, therefore, 
 all exact weighings should be reduced to vacuo. Prob. (D. 
 160, 251). Q. ii. 
 
 1 6. Prove that the effective weight of a body in water is less 
 than in air by an amount equal to the difference between the 
 weights of water and of air displaced, and hence equal to the 
 effective weight of the water displaced. Prob. Q. 
 
 17. Prove (by considering the relation between effective and 
 apparent weights) that the statement in the last question applies 
 to apparent as well as to effective weights. Prob. Q. n, 14. 
 
 1 8. Show that the principle of Archimedes applies to effective 
 or to apparent weights, as well as to true weights. Prob. Q. 
 
74 
 
 XLVIL 
 APPARENT SPECIFIC GRAVITY. 
 
 i.f What name is given to the ratio between the apparent 
 weight of a given substance and the apparent weight of an 
 equal volume of water, or other standard of reference ? 
 
 2.* Define " specific gravity," and distinguish it from density. 
 D. 1 60. 
 
 3-f Define "apparent specific gravity," and distinguish it 
 from density as well as from true specific gravity. 
 
 4.f What substance, or substances, are chosen as standards 
 of reference for specific gravity (i) in the case of liquids or solids 
 and (2) in the case of gases ? (D. 160). 
 
 5.f Criticise the common use of the term "vapor density" 
 (as synonymous with specific gravity). 
 
 6.f At what temperature did the founders of the metric system 
 intend that the density of water should be exactly i, and at 
 about what two temperatures is this condition new thought to be 
 fulfilled? (D. 1 60). 
 
 y.f Give some idea of the density of water at ordinary 
 temperatures, and of the delicacy of the apparatus necessary to 
 detect variations in its density. 
 
 8. Given that a cubic decimetre of wood weights 499 grams 
 in air, and 500 grams in vacuo ; also that a cubic decimetre of 
 water at a certain temperature weighs 997 grams in air, and 998 
 grams in vacuo ; find the density of the wood, and its true and 
 apparent specific gravities referred to water at the given tempera- 
 ture. Prob. Q. 
 
 9. * Explain the determination of (apparent) specific gravity 
 by means of a (Joiyi<Y) spring balance, or by a hydrostatic 
 balance. D. 164. 
 
75 APPARENT SPECIFIC GRAVITY. [XLVII. 
 
 10. Given the (apparent) weights, W and w, of a body in air 
 and in water, how do you find the (apparent) specific gravity of 
 the body? D. 164. 
 
 11. If W is the (apparent) weight of a body in air, how do 
 you find its (apparent) specific gravity if w is (i) the (apparent) 
 loss of weight of the body when weighed in water ? (2) the 
 (apparent) gain in weight of the vessel in which it is immersed? 
 or (3) the (apparent) weight of the water displaced ? (D. 164). 
 
 i2*fA piece of dry chalk weighs 5 grams in air, and 2 grams 
 in water, but it takes i gram of water to replace all the air in its 
 pores ; find the (apparent) specific gravity (i) of the dry chalk, 
 (2) of the wet chalk, and (3) of the material within which the 
 pores of the chalk are included. 
 
 13. How do you find the (apparent) specific gravity of a liquid, 
 if W^and w are respectively (i) the (apparent) losses of weight 
 of a given body when immersed in water and in the liquid, (2) 
 the corresponding (apparent) gains in weight on the part of the 
 vessel, or (3) the (apparent) weights of water and of liquid 
 displaced by a given body ? Prob. (D. 166). 
 
 14.*! Explain the determination of (apparent) specific gravity 
 by means of a specific gravity bottle. (D. 163). 
 
 15.* A vessel is filled by an (apparent) weight, W, of water or 
 by an (apparent) weight, w, of a given liquid ; find the 
 (apparent) specific gravity of the liquid. (D. 163). 
 
7 6 
 
 XLVIII. 
 LAW OF AVOGADRO. 
 
 i.f What is meant by the molecular constitution of matter? 
 
 2.f State the law of chemical combining proportions of gases 
 in terrns of their volumes. 
 
 3.f What conclusions have Chemists adopted in regard to the 
 number of molecules in a given volume of different gases at the 
 same pressure and temperature ? 
 
 4.f State the law of Avogadro. 
 
 5.f What is meant by the molecular weight of a gas ? 
 
 6.f Find the relation between the molecular weight of a gas 
 and its specific gravity referred to hydrogen, assuming that the 
 molecular weight of hydrogen is 2. 
 
 7. Given the density of a gas, how do you find its molecular 
 weight ? Illustrate by the following examples : 
 
 Density of hydrogen (o and 76 cm.) 0.00008952 
 " " oxygen 0.0014291 
 
 " " nitrogen 0.0012544 
 
 Prob. Q. 
 
 8.J Explain one or more methods (e.g., DUMAS') for deter- 
 mining the molecular weight of a gas (or vapor).. 
 
77 
 
 XLIX. 
 
 MOLECULAR THEORY OF THE PRESSURE OF GASES. 
 
 i.f Through what agency is the pressure of a gas believed to 
 be maintained ? 
 
 2.f Show that, if pressure is maintained by molecular impact, 
 it should vary as the number of molecules in a given space, 
 provided that the molecules are not so near together as to 
 interfere with one another. 
 
 3. State DALTON'S law for gaseous mixtures, and its applica- 
 bility in certain cases to mixtures of vapors and gases. D. 227. 
 
 4.J Describe one or more experiments illustrating DALTON'S 
 law. 
 
 5.f What necessary relation exists between the number of 
 molecules in a given space and the density of a gas ? 
 
 6.f Show the pressure of a gas, if sufficiently rare, must vary 
 as its density. 
 
 y.f State results of experimental investigation as to the truth 
 of the statement in the last question. 
 
 8. What conclusion did DULONG and ARAGO arrive at with 
 respect to the compressibility of air? D. 219. 
 
 9.f Within what limits have the results of DULONG and 
 ARAGO been confirmed by later experiments? (D. 219). 
 
 10. f Name certain gases in which the density is found to be at 
 least approximately proportional to the pressure for variations of 
 one or two atmospheres. 
 
L. 
 LAW OF BOYLE AND MARIOTTE. 
 
 1. State the necessary relation between the density and the 
 volume of a given mass of gas. Prob. 
 
 2. Prove that, according to the molecular theory of the 
 pressure of a gas, the pressure must vary inversely as the volume. 
 Prob. Q. 
 
 3.* State the law of BOYLE and MARIOTTE. D. 217. 
 
 4. Show that the product, VP, of the volume, V, and the 
 pressure, P, of a given mass of gas, obeying the law of BOYLE 
 and MARIOTTE, is constant. Prob. Q. 
 
 5.| Describe an experiment with BOYLE'S or MARIOTTE'S 
 tube, illustrating the law of BOYLE and MARIOTTE. D. 218. 
 
 6.J Describe an experiment with a mercury well, showing 
 that the law of BOYLE and MARIOTTE holds for pressures less 
 than that of the atmosphere. D. 218. 
 
 7,J Describe one or more experiments with a piece of 
 apparatus capable of illustrating the law of BOYLE and MARIOTTE 
 for pressures both greater and less than that of the atmosphere. 
 
 8. Describe and explain the use of one or more forms of 
 compressed air manometer. D. 225. 
 
79 
 
 LI. 
 UNEQUAL COMPRESSIBILITY OF GASES. 
 
 i.f What limit would the volume of an ideal gas approach if 
 the pressure should become indefinitely great ? and how would 
 this limit be modified in practice if the molecules of the gas were 
 incompressible, and occupied a certain amount of space ? 
 
 2.f Show that the size of molecules would diminish the 
 compressibity of a gas. 
 
 3.f What would be the effect of attractive forces between the 
 molecules of a gas on the volume occupied at a given pressure ? 
 
 4.f Show that forces of the nature of adhesion or cohesion 
 would tend to increase the compressibility of a gas. 
 
 5. Explain the departure of certain gases from the law of 
 BOYLE and MARIOTTE. Prob. Q. 
 
 6. Describe one or more experiments (due to DKSPRETZ, 
 POUILLET, or REGNAULT) showing the unequal compressibility 
 of different gases. D. 219, 220, 221. 
 
 y.f Give some idea of the percentage error in calculating the 
 density or pressure of the more permanent gases, by the law of 
 BOYLE and MARIOTTE, within limits of one or two atmospheres. 
 
8o 
 
 LII. 
 VAPORS AND GASES DISTINGUISHED. 
 
 i.J Describe the effects of increasing pressure upon the 
 volume of a vapor. 
 
 2.f What is meant by the "maximum pressure" (or tension) 
 of a vapor ? 
 
 3-t What is meant by the condensation of a vapor ? 
 4-t Distinguish vapors from gases in respect to phenomena of 
 condensation. 
 
 5.f Distinguish between the condensation of a vapor and the 
 absorption of a vapor by a liquid. 
 
 6. What is meant by the solubility of a gas or vapor in a 
 liquid at a given pressure and temperature ? D. 228. 
 
 y.f State the law connecting the solubility of a gas or vapor 
 in a liquid with the pressure. 
 
 8. Describe one or more experiments illustrating the absorp- 
 tion of gases or vapors by charcoal. D. 228. 
 
 9.f What is meant by the occlusion of a gas? 
 
 10. J Describe one or more experiments illustrating the 
 occlusion of hydrogen by platinum oriridium. (D. 228). 
 
 n.J Describe one or more experiments showing that a film of 
 air adheres to glass or other solids under certain conditions. 
 (D. 228). 
 
 i2.f What connection probably exists between the causes 
 giving rise to phenomena of condensation, absorption, occlusion, 
 solubility, etc., in certain gases or vapors, and their departure 
 from the law of BOYLE and MARIOTTE ? 
 
 END OF PART I. 
 
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