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TI1JO POPULAE EDUCATOR 



riot generally into thoae of equilibrium. The speed or velocity 

 of a cannon-ball must be considered at every varying moment of 

 its flight ; but the strains and pressures among and on the 

 beams of the roof of a railway station are the same at all 

 moments. Time does not affect the latter unless by wear and 

 tear; With statics, therefore, we commence, and, of course, 

 with the simplest class of questions, those which relate to a 

 force or forces acting on a single point. But here I must turn 

 back to the notion of force, and endeavour to fix it with greater 

 accuracy in your minds. I must show you how it is said to be 

 applied and measured to the body it moves or strains ; and tins 

 will best be done under the three following heads : 



1. The Direction of a Force. 



2. The Point of Application of a Force. 



3. The Magnitude of a Force. 



1. The Direction of a Force. In Mechanics, forces are assumed 

 to act in, right lines. The assumption is made for the best of 

 reasons namely, that of experience. All the simpler cases of 

 motion confirm it, and all the more complicated can be 

 aacounted for by it. A ball falls to the ground in a right line - 

 that which points to the centre of the earth, whence the force 

 of attraction which moves it acts. The billiard-ball moves in a 

 right line ; and the calculations of the skilful player, which are 

 based on the supposition that it so moves, are never found to 

 be wrong. A ship, with her sails square set and wind aft, moves 

 in a right line ; and to make it leave that line the steersman 

 must put the helm to port or starboard, and by turning the 

 face of the rudder against the water, cause another force to be 

 applied to the ship across the line of its course, and at her stern, 

 turning her round. It is true that the stone thrown obliquely 

 into the air moves in a curved path ; but in this case we know 

 that there are two forces not one only acting on it, namely, 



;ho original impulse, which makes it move in a right line, 

 and the earth's attraction, which pulls it from that line into a 

 curved course. Moreover, all the calculations on which are 

 based the predictions of astronomers as to the places in which 

 the sun, moon, and planets will be on a certain day, hour, and 

 minute, are based on this assumption, that forces act in right 

 lines ; and the predictions invariably prove true. Our first 

 mechanical axiom may, therefore, on the ground of experience 

 be assumed to be true -namely, that the direction in which a 

 force acta is that of a right line. Indeed, it is not easy to 

 conceive how it could act otherwise. 



2. The Point of Application of a Force. The direction of a 

 force being disposed of, we must fix our ideas as to its point of 

 application. The rule is, that any point on the line of its direc- 

 tion may be considered such ; but this you must understand 

 with a limitation, or exception, which should not be forgotten. 

 The point of application can only be on so much of the line of 

 direction as lies within the body. For instance, suppose a person 

 to push with an iron rod, which he holds in his hand, at the 

 point A (as in the diagram), against a block of iron which lies 

 on a table. Then, clearly A is the point of application of the 

 force with which he pushes. Let now a hole be drilled through 

 the block in the direction of the push from A to E, into which the 

 rod may fit closely but freely ; and also other holes, downwards, 

 b B, c c, d D, to meet the passage, A E, into which thumbscrews, 

 b, c, d, are fitted. Let the rod now be passed through the 

 block so as to emerge at the other side, and clamp it down 

 firmly by the thumb-screw, b. If it be now pushed against the 

 block with the same force as before, it is clear that the force 

 will be arrested by the thumb-screw, b, at B, and that B will 

 become its point of application to the body. So, in like manner, 

 may it be applied to c and D, by tightening in succession each 

 screw, while the others are left loose. In all these cases the 

 force is the same, and the direction the same ; but the points of 

 application are different. But will the effects in the several 

 cases bo different ? No ; for the portion of the rod within the 

 block, and extending from A to any of the points of application, 

 performs the same part in transmitting the force from A to the 

 point within, as the iron which was removed did when the force 

 was first applied directly at A. The removed iron has its place 

 filled by an equivalent of that metal in rod, and the body is 

 virtually in its original condition. The force of the hand may 

 atill be considered applied at A, thence to be transmitted to B, 

 or C, or D, as we please, by the portion of rod within. The 

 second case becomes identical with the first, and the effects, 

 therefore, must be identical in every respect; and, nothing 



being changed, intensity, direction, nor effect of the force, 

 it is clearly indifferent which point we make the point of 

 application. 



Another instance is the raising of a weight by a rope. Weight 

 and rope together make one body ; and whether the lifting 

 power be applied by engine, by horse, or by man, whether it 

 acts over a pulley or not, every point of the strained rope 

 may be considered a point of application. Or let the case be 

 tliat of three strings attached to a ring, and pulled in different 

 directions by three persons. It makes no difference, in 

 this compound bodj of ring and strings, whether the hold 

 taken of the latter be long or short all their points are points 

 of application of their respective forces. 



We thus see that, in all cases, we may assume that the point 

 of application of a force is any point on so much of its line of 

 direction as lies within the body. To suppose it applied to a 

 point outside would be absurd ; for, as we have shown, though 

 a force may act or push through a point of empty space, it can 

 make no impression on that point, either in the way of strain 

 or motion, and therefore cannot come under the consideration of 

 Mechanics. 



3. Tlie Magnitude of a Force. To find a suitable measure of 

 the intensity or magnitude of a force, we must also look to 

 experience. It would be very convenient to measure forces 

 by comparing them with weights ; but this is not always 

 practicable, and, even if it were, it would not answer all the 

 purposes of Mechanics. I may as well, therefore, explain to you 

 the perfect method, as that is as simple as any other. Experi- 

 ence teaches that a double force produces a double velocity, a 

 treble force a treble velocity, and so on, in any body to which it 

 is applied. But then a difficulty occurs : the sama force will 

 produce different velocities in bodies of different sizes. If \t 

 make a ball of one pound weight move at a certain rate, it will 

 give double that speed to a half-pound ball, and half to one of 

 two pounds. As a general rule, the greater the mass of the 

 body, the less the speed produced. Everybody is familiar with 

 this fact. We see, then, that if we desire to measure forces by 

 the velocities they produce, we must try them on bodies of some 

 fixed weight or mass. Tried on this particular mass, experience 

 teaches that that which produces the greater velocity is the 

 greater force. Now, the mass of matter which mechanicians 

 choose for this purpose is that of any substance which is equal 

 in weight to a cubic inch of distilled water. That much matter 

 is designated the Unit of Mass, and for. a reason I shall 

 hereafter more fully explain. Imagine, then, a round ball, say 

 of ivory, whose weight is that of a cubic inch of pure water, 

 and suppose that several forces are in succession applied to it ; 

 the velocities they produce will be accurate measures of their 

 intensities, or of their magnitudes. 



But, then, how are the velocities to be ascertained ? Clearly 

 by the spaces the ball would move over in any given time, say 

 the unit of time -a second on the force being applied to it. 

 Suppose, then, the unit ivory ball, put on a perfectly smooth 

 floor, and then suddenly struok by a blow equal to the force you 

 want to measure. By some means and there are many which 

 may be devised manage to ascertain the distance the ball moves 

 over in one second. That space, or length of line, will be the 

 measure of the force ; and if any number of such forces be tried 

 in the same way and on the same ball, that which causes it to 

 move over the greater space is the greater force, over a double 

 space a double force, and so on. 



The final result, then, is that, in considering a force in 

 Mechanics, we must first suppose drawn within the body a line 

 representing its direction. Then, on that line, let any point be 

 taken for its point of application. Thirdly, on the line of 

 direction so fixed, let as many inches be measured from the 

 point of application as, on any scale you agree to use, represents 

 the space the force would cause the unit ivory ball to move over 

 in one second. Then you have a line which also in tnagnitude 

 represents the force. Or in fewer words 



A FOJSCE is represented, both in magnitude and in direc- 

 tion, by a finite right line passing through its point of appli- 

 cation. 



If in the above explanations I have succeeded in giving you 

 clear notions of the aim of Mechanics, and of the nature and 

 effects of force, you are prepared for the consideration of a 

 force, or forces, applied to a single point, which will be the 

 Bubjeot of our next Lesson. 



