Thus, for equilibrium, and the denominator is 50000. example, suppose the pendulum oscillates through an angle of 10 degrees altogether, then there are 5 degrees in the angle with which we are concerned; the square of 5 is 25, and Therefore the time found by 25 = the rule of Art. 313 must be increased by so that if that rule gives one second for the time when the arc is very small the correct time when the arc corresponds to 10 degrees will be 1000 seconds. 316. The time of oscillation does not depend on the nature of the substance of which the heavy particle is composed; this corresponds with the fact that, setting aside the resistance of the air, all bodies fall to the ground from the same height in the same time. The word oscillation is used by some writers to denote the time taken by the heavy particle in passing from one end of an arc to the same point again; this amounts to twice the time which we assign to an oscillation. Also the word vibration is sometimes used instead of oscillation. 317. Instead of compelling a particle to describe an arc of a circle by means of a string we might have a fino tube made in the form of an arc of a circle, and fixed in a vertical plane; and then the particle might be placed within the tube so as to slide up and down. Theory shews that the motion is of the same kind as the other, provided the tube is smooth internally. The resistance of the tube in this case takes the place of the tension of the string in the other. 318. We may also have other cases of motion by supposing a fine smooth tube, as in the preceding Article, not in the form of an arc of a circle but in that of an arc of any other curve. One interesting result obtained by theory then is that whatever be the form of this curve the velocity of the heavy particle at any point is just the same as if it had fallen freely through a vertical space equal to the depth of this point vertically below the starting point. We have already remarked in Art. 289 that this is the case when the tube is in the form of a straight line. T. P. 9 319. Two very curious results in connexion with this subject may be noticed. Suppose the fine smooth tube made in the form of half a particular curve which mathematicians call a cycloid, and let it be placed as they would say with its base horizontal and its vertex downwards; denote the highest point of the curve by A, and the lowest point by B. Then the heavy particle would slide from A to B down this curve in less time than down any other curve from A to B. And if C denote any point of the curve between A and B the particle would slide down the portion of the curve from C to B, starting at C, in the same time as down the whole curve from A to B. The two statements can be well demonstrated experimentally by constructing tubes or troughs on a large scale. In particular the truth of the second statement can be very effectively shewn; a man takes a ball in each hand, and by stretching out his arms he can put one ball at a point of the trough far above the point at which he puts the other, and let both start at the same instant; then the upper ball just overtakes the lower ball at the bottom of the curve. 320. It is easy to give a notion of the curve which we call a cycloid. It is the curve which a point in the circumference of a carriage wheel would trace out as the wheel turns once round in rolling along the ground; the point being supposed the lowest point of the wheel at the beginning and at the end of the turning. The curve thus formed will bear some resemblance to the outline of a very flat arch of a bridge. The curve must be supposed turned upside down and half of it taken when used in the manner of Art. 319. XXIII. FRICTION. 321. WE have hitherto supposed that all bodies are smooth, but practically this is not the case, and we must now examine the results which follow from the roughness of bodies. 322. The ordinary meaning of the words smooth and rough is well known, and a little explanation will settle the sense in which these words are used in Natural Philosophy. Let there be a fixed plane horizontal surface formed of polished marble; place on this a piece of marble having a plane polished surface for its base. If we attempt to move this piece of marble by a horizontal force we find that there is some resistance to be overcome; the resistance may be very small, but it always exists. The same thing will appear if we change the material with which we make the experiment, as for instance if we use wood instead of marble, or if we have the fixed plane of one material and the moveable body of another. We say then that the surfaces are not perfectly smooth, or we say that they are to some extent rough. Thus surfaces are called smooth when no resistance is caused by them to the motion of one over the other, and they are called rough when such a resistance is caused by them; this resistance is called friction, and it always acts in the contrary direction to that in which motion takes place or is about to take place. Although we may imagine smooth bodies to exist, yet strictly speaking there is always some degree of roughness in practice. 323. The following is another method of explaining the meaning of the words smooth and rough in our subject. When bodies are such that if they are pressed together the force which each exerts on the other must be at right angles to the two surfaces the bodies are called smooth; when this is not the case they are called rough. If the two surfaces which are pressed together are both plane surfaces this definition is immediately applicable, but if one or each of the surfaces is a curved surface some explanation is required. Suppose that one surface is curved and the other plane, as for example when a sphere is pressed against a plane; then a straight line at right angles to the plane at the point of contact is to be considered as also at right angles to the curved surface. Next suppose that each surface is curved, as for example when one sphere is pressed against another sphere; then a plane must be supposed to touch each surface at the point of contact, and a straight line at right angles to this plane is to be considered as also at right angles to the curved surfaces. 324. Suppose we want to support a Weight by the aid of a machine; then friction may be said to help the l'ower, for the Weight may be increased beyond the value which according to theory the Power would support, and yet motion may be prevented by the friction. But suppose we want to give motion; then friction may be said to oppose the Power, for the Power must be increased beyond the value which according to theory would move the Weight in order to overcome the friction. Suppose we increase the Power sufficiently then we actually overcome the friction and produce the motion which we desire. Thus there is in every case a limit to the friction, and experiments have been made in order to obtain information with respect to the extreme amount of friction which can be brought into action between two surfaces when they are pressed together. The following Laws have been thus obtained. (1) The friction varies in the same proportion as the force with which the bodies are pressed at right angles to the surfaces in contact, so long as the materials of the bodies in contact remain the same. (2) The friction remains the same whatever may be the extent of the surfaces in contact so long as the force pressing the bodies at right angles to the surfaces is the same. These two Laws are true not only when motion is just about to take place, but when there is sliding motion. But in sliding motion the friction is not always the same as in the state bordering on motion; when there is a difference the friction is greater in the state bordering on motion than in actual motion. (3) The friction is the same whatever may be the velocity when there is sliding motion. 325. Coefficient of Friction. Let two bodies be pressed together by any force at right angles to the surfaces in contact, and let us try to make one body slide on the other by a force parallel to the surfaces, increasing the force we apply until it is just sufficient for the purpose; then the proportion of this transverse force to the force at right angles to the surfaces is called the Coefficient of Friction. For example suppose that two bodies are pressed together by a force of 10 pounds, and that we can just make one body slide on the other by a force of 3 pounds; then the 326. The following results have been obtained by experiment; they apply to the case of actual motion. The 3 coefficient of friction for iron on stone is between and 10 7 10 3 ; for timber on timber between and ; for metals 3 on metals between and. Thus, for example, if two 20 4 metallic bodies are pressed together with a force of 100 pounds, then in order to keep one in motion over the other we must exert a force between of 100 pounds 1 3 20 and of 100 pounds, that is between 15 pounds and 25 pounds. The precise amount of force will depend on the nature of the metals and the degree of smoothness of their surfaces. 327. Angle of Friction. Let a body be placed on an Inclined Plane; if the plane were perfectly smooth the body would not remain in equilibrium. Let W denote the weight of the body, which acts vertically downwards; let R denote the Resistance of the Plane, which acts at right angles to the Plane; let F denote the W Friction, which acts along the Plane. Now, by Art. 246, we know that so long as the body remains in equilibrium, the Weight, the Resistance, and the Friction are in the proportion of the length, the base, and the height of the Plane respectively. Thus the Friction is to the Resistance in the same proportion as the height of the Plane is to its base. Let the Plane be gradually tilted until the body just begins |