150 ELEMENTARY MECHANICS OF SOLIDS. 21. Find in magnitude and direction the least force which will keep a body weighing 100 lbs. at rest on a smooth inclined plane, inclined at an angle of 45° to the horizontal. (N.B.-A graphical solution will be accepted.) (Science and Art, 1898.) Summary. A machine is an arrangement for transmitting force. The magnitude of the force is generally altered in the process. The force applied to a machine is called the Effort (sometimes called Power, but this is not a good term for it); the force overcome is called the Resistance. In any practical case the work done by the effort exceeds that done against the resistance, because some work is lost in overcoming friction, etc., in the parts of the machine. In a machine Work obtained Efficiency = Work expended If a and b are infinitesimally small distances through which the effort acts, and the resistance is overcome at the same time in any machine, then H α b Mechanical advantage Velocity ratio × Efficiency. Levers.-The general necessary and sufficient condition of equilibrium is that the moments about the fulcrum of the effort and the resistance should be equal and opposite. Levers are divided into three classes according as Fulcrum, Resistance, or Effort is in the middle. In any simple machine, if there is no friction, the work done by the effort may be shown to be equal to that done against the resistance. Or, assuming this principle, we can find the relation between the effort and the resistance. With a smooth inclined plane suppose P is the effort necessary to sustain a body of weight W on the face of the plane, and R is the normal resistance of the plane. Let the height, length, and base be h, l, b. (a) If P is parallel to the face of the plane, Suppose an effort P applied to the handle to sustain a weight W W. Then the tension in the rope is And by moments 2 EXPERIMENT 35.-A wheel-and-axle is easily made by attaching a disc of wood to a cylinder of much smaller radius, or else by turning the whole in one solid piece. The first plan is simpler. A wheel of about 8 inches diameter and an axle of 1 inch diameter and 3 inches long will do very well. The wheel should be screwed firmly on the end of the axle with three or four screws, none of which passes unrough the centre. Two wire nails or screws may then be fixed at the centre of the wheel and at the other end of the axle to form an axis of rotation, and the arrangement set so that it can turn on two upright supports. Fasten strings to the wheel and the axle, and wind them round in opposite ways. First find the velocity ratio by finding how far the wheel string must be pulled down to raise a point of the axle string by a certain amount, say 2 or 3 inches. Now Ready. In Three Stages. Globe 8vo. Price, 1s. 6d. each ELEMENTARY PHYSICS AND CHEMISTRY ADAPTED TO THE SYLLABUSES OF (1) Alternative Class Subject. Course H. (3) Practical Science in Evening Continuation Schools BY R. A. GREGORY, F.R.A.S. PROFESSOR OF ASTRONOMY, QUEEN'S COLLEGE, LONDON AND A. T. SIMMONS, B.Sc. (Lond.) ASSOCIATE OF THE ROYAL COLLEGE OF SCIENCE, LONDON London MACMILLAN AND CO., LIMITED NEW YORK: THE MACMILLAN COMPANY PRESS OPINIONS STAGE I. Guardian. -"The educational value of this first course, if the experiments are really performed, thought out, and written about, as herein indicated, is distinctly great-far greater, in fact, than Paul Bert's book. The book merits very extensive use in schools: it is a good stepping-stone to higher things." Girls and Infants' Mistress." Messrs. Gregory and Simmons have produced an excellent little work for young beginners in science." Educational News." It is certain to become a favourite." STAGES II AND III. Saturday Review.-"The same good points are to be found as in the first ample experiments, careful explanations and suggestive questions." Schoolmaster.-"These books are well worth the attention of those who are interested in the teaching of the elements of science in our elementary schools. They are useful because they contain a large amount of instruction respecting many of the chemical and physical changes which almost daily come under our notice. They are educational because the course of instruction is so arranged as to cultivate the powers of observation and reasoning.' School Guardian.-"This method is well calculated to teach pupils to observe and reason for themselves, and the habit of mind thus acquired is of infinitely more value than the mere acquisition of facts." Secondary Education.-"This is an exceedingly useful course of elementary science, and it is based on sound educational principles.” CONTENTS OF STAGE I. LESSON 1, The Senses; 2, Matter and Hardness; 3, Solids, Liquids and Gases; 4, Properties of some Common Things; 5, Properties of some Common Things, continued; 6, Measurement of Length; 7, Measurement of Area; 8, Measurement of Volume; 9, Mass and Weight; 10, Measurement of Mass; 11, The Principle of the Balance; 12, Density; 13, Density, continued; 14, Determination of Density; 15, Things which sink in Water; 16, Things which float in Water; 17, Principle of Archimedes; 18, Determination of the Density of a Solid; 19, The Air around us; 20, The Pressure of the Air; 21, Barometers; 22, Why the Height of the Barometer alters; 23, Effects of Heat; 24, Thermometers; 25, Graduation of Thermometers, Fixed Points; 26, Soluble and Insoluble Solids; 27, Soluble Liquids and Gases; 28, There is no Loss during Solution, Evaporation; 29, Saturated Solutions; 30, Solubility of Things in Acids; 31, Changes of Mass when Chemical Action accompanies Solution; 32, Crystals and Crystallisation; 33, Crystals and Crystallisation, continued; 34, Graphic Representation; 35, Graphic Representation, continued. CONTENTS OF STAGE II. SUMMARY OF FIRST STAGE. LESSON 1, Evaporation; 2, Distillation; 3, Moisture in the Air; 4, How the Amount of Water Vapour in the Air is Measured; 5, Changes of Volume and Density of Water; 6, Maximum Density of Water; 7, Heat and Temperature; 8, The Measurement of Heat; 9, Quantity of Heat; 10, Heat Capacity; 11, Relative Capacities for Heat; 12, Specific Heat; 13, Heat Absorbed in the Fusion of Ice, 14, Convection: Heat Absorbed in the Conversion of Water into Steam; 15, Increase of Mass Accompanies Burning; 16, The Rusting of Iron; 17, The Action of Phosphorus on Air; 18, Changes Produced by Burning Phosphorus in Air; 19, The Burning of a Candle; 20, The Burning of a Candle, continued; 21, The Burning of Oil and Gas; 22, Search for the Active Part of Air; 23, Preparation of Oxygen ; 24, Chemical Conduct of Oxygen. CONTENTS OF STAGE III. SUMMARY OF FIRST AND SECOND STAGES. LESSON 1, Characteristic Properties of Water; 2, Action Between Water and Iron; 3, The Composition of Water; 4, Action of Zinc upon Acids; 5, Properties of Hydrogen; 6, The Burning of Hydrogen; 7, The Action of Hydrogen on Metal Rusts; 8, Natural Waters; 9, Chalk and the Action of Heat upon it; 10, Chalk and Lime Compared; 11, Further Study of the Action of Heat on Chalk; 12, Action of Acids on Chalk; 13, Study of the Gas Obtained from Chalk: 14, Further Study of the Action of Acids on Metals and Carbonates; 15, Burning and Breathing; 16, Carbon Dioxide in the Air; 17, Hard and Soft Waters; 18, Limestones; 19, Natural Changes in which Chalk and Carbon Dioxide take part; 20,, Mortar. THINGS WHICH SINK IN WATER. 57 iron, oak, pine, and cork, one after another, into the water. Observe that (1) some sink and others float, (2) of those which float some sink further into the water than others. Take the objects which sink in water and place them in mercury. Notice that they float. Partly fill a glass cylinder, divided into cubic centimetres, and record the level of the water therein. Drop in one of the cubic centimetre solids which sinks, and again read the level of the water; put the others in in order, recording the level of the water after each such addition. It will be found that the level increases by I cubic centimetre division in each case. Take any solid, such as a glass stopper or a marble, and drop it gently into water contained in the graduated glass cylinder. Read the level of the water before and after dropping the solid in; the difference between these readings will give you the volume of the solid in cubic centimetres. READING LESSON. Some Things sink, others float in Water.-When you throw a stone into water what happens to it? It sinks to the bottom. But if you throw a piece of wood into water does it also sink? No, it floats. By noticing what happens when different sub stances are put in water you can easily divide them into two classes. Those in one division all sink, while those in the other all float. And, of those that float, some sink further into the water than others. FIG. 37. Some things sink, and others float, in water. But you must not suppose that substances which sink in water will sink in every liquid. As a matter of fact, solid iron, or even lead, will float upon mercury. Volume of Water displaced by Bodies placed into it.—If we put water into a narrow glass cylinder, and then add lumps of material of such a size and shape that they will go into |