MECHANICS. 



11 



wo will neglect it now, and roppo** that the plane is perfectly 

 , nn.1 thut the weight is just kept in its position by the 

 We found in our third losson that, if we draw a 

 line, o B, downwards from o, the centre of gravity of w, and 

 make it of uoh a length as to represent the weight of w, and 

 then through K draw r parallel to o D, and just long enough 

 to meet the line o F, which is perpendicular to the surface of 

 the plane, that then c F represents in magnitude the power p. 

 We hare, in fact, a triangle of forces, the three aides of which 

 represent the throe forces which act on the weight and keep it 

 at rest But the angles of the triangle K F a are equal to those 

 of the triangle c B A. This is easily seen, for the angle a F G 

 is equal to c B A, each being a right angle, a K F is also equal 

 to A c B ; for, if we continue K F till it meets B c, we shall have 

 a parallelogram, and these will be opposite angles, and so 

 must be equal ; the third angles are equal too, since o F and 

 B o are perpendicular to A c and A B. The angles of one tri- 

 angle are equal, then, to those of the other, and therefore the 

 aides f the triangle K F a bear the same proportion to one 

 another that those of c B A do. Of this you can satisfy your- 

 self by actual measurement, and you will find the rule always 

 hold good. The proper mode of proving it you will learn from 

 Euclid. 



The three sides of A B c represent, then, the three forces which 

 act on w ; AC representing the weight, B c the power, and A B 

 the resistance of the plane, or the part of the weight which is 

 supported by it. Hence we see that if the incline be 1 foot in 

 20, a man in rolling a weight up will only have to support & 

 of it. 



We can easily arrive at this result in another way. Suppose 

 a person wants to lift a weight of 200 pounds to a height of one 

 foot, he will have to exert a force of that amount if ho lift it 

 straight up, and will then move it through just one foot. But 

 if, instead of this, he moves it up this incline, when he has 

 passed over one foot in length of its surface, he will only have 

 raised it ^ of a foot, and will have to move it over the whole 

 twenty feet of the plane in order to raise it the one foot. That 

 is, he will have to move it twenty times the space he would if 

 he lifted it direct, and will therefore sustain only : * of the 

 weight at any moment. Still, he must sustain this portion 

 twenty times as long. This supplies ns with another illustration 

 of the law of virtual velocities which wo explained in the last 

 lesson. 



The general rule for the gain in the inclined plane when the 

 power acts in a direction parallel to it, may be stated as follows : 



The power bears the same ratio to the weight it will sustain 

 that the perpendicular elevation of the plane does to the length 

 of its surface. 



If the power, instead of acting along the plane, acts at an angle 

 to it, whether it be parallel with the base or in any other direc- 

 tion, as o K, we have merely to draw E H parallel to the line of 

 action of the force, instead of parallel to the plane, and, as before, 

 we shall obtain a triangle of forces, the three sides of which 

 represent the three forces, and thus we can calculate the power 

 required to support the weight. 



If we have two inclined planes meeting back to back, like the 

 letter V inverted, and a weight resting on each, the weights 

 being connected by a cord which passes over a fixed pulley at 

 the summit, we can see from this principle that there will be 

 equilibrium when the weights bear the same proportion to each 

 other as the lengths of the inclines on which they rest : for 

 it is clear that, the steeper the plane, the less is tho portion of 

 the resistance borne by it. If, for example, one indino is 15 

 inches long, and tho ether 21 inches, a weight of 5 pounds 

 on the former will balance one of 7 pounds on the latter. 

 For, supposing tho vertical height of tho summit to be 6 inches, 

 the portion of the force of 5 pounds which acts downwards, and 

 tends to raise the other, is of 5 pounds, which equals 2 

 pounds ; while tho portion of the other which acts downwards is 

 fc of 7 pounds, which is also equal to 2 pounds. 



This system of two inclines is often used in mining districts, 

 a train of loaded trucks running down from the pit's mouth to 

 the stoith, being mado to drag a train of empty ones up the 

 incline. Many familiar instances of the nse of the inclined 

 plane are met with every day, though they often escape notice, 

 unless we are specially looking for them. For example, our 

 knives, scissors, bradawls, chisels, needles, and nearly all 

 cutting and piercing tools, act on this principle. 



Fig. 78. 



THB WKDOB. 



We pass on now to notice the wedge, which essentially oon- 

 usts of two inclined planes of small inclination placed with 

 their bases one against tho other. 



Sometimes one side only of the wedge is sloping, and it is 

 then simply a movable inclined plane. In using this, it is so 

 placed that it can only be moved in the direction of the length, 

 ind the weight to be raised is likewise prevented from moving 

 in any direction except vertically. If 

 pressure be applied to the bead of the 

 wedge, the weight will be raised. The 

 gain is the same here as in the in- 

 clined plane. 



The wedge, however, usually con- 

 sists of a triangular prism of steel, 

 or some very hard substance, and is 

 used as shown in Fig. 78. The point 

 is inserted into a crack or opening, 

 and the wedge is then driven, not by a 

 constant pressure, but by a series of 

 blows from a hammer, or some similar 

 instrument. It is usual to consider 

 the wedge as kept at rest by three forces first, a pressure 

 acting on the head of the wedge, and forcing it vertically down- 

 wards, as at P ; secondly, the mutual resistance of it, and the 

 obstacle which acts at right angles to the surface of the wedge, 

 as at B B ; and thirdly, the force which opposes the motion, and 

 acts at right angles to the direction in which the object would 

 move, as at c. 



As, however, the resistance to be overcome varies very much 

 from moment to moment, both in direction and intensity, and as 

 the force is usually supplied by impact or blows, and not by 

 pressure, such calculations afford very little help towards deter- 

 mining the real gain. 



The other mechanical powers are usually employed in sustain- 

 ing or raising a weight, or offering a continuous resistance ; a 

 continuous force is therefore used with them. In the wedge, 

 the resistance to which it is applied is usually one which, when 

 once overcome, is not again called into play. In splitting timber, 

 for instance, when the wedge is driven in, the particles of timber 

 are forced apart, their cohesion is overcome, and they do not 

 join again. So in dividing large stones, when once a crack has 

 been made through them, no continued application of force is 

 needed to keep them from re-uniting. When continuous force is 

 required, the wedge having been driven forward is kept from 

 slipping back by friction. 



As, then, we cannot calculate the force generated by a blow, 

 we must be content with the general statement that the smaller 

 the angle of the wedge the greater is the power gained. 



THE SCBKW. 



This is the last of the mechanical powers, and, like the wedge, 

 acts on the principle of the inclined plane. If we stretch a cord 

 so as to represent the slope of an 

 inclined plane, and then, holding a 

 ruler, or some cylindrical body, ver- 

 tically, we roll up the cord upon it, 

 we shall have a screw, the spiral 

 line traced out by the cord being 

 called its thread. It is easy to see 

 that the thread has at every point 

 the samo inclination as the inclined 

 plane, and that a particle in tra- 

 velling up the screw will pass over 

 tho same distance as if it moved up 

 the plane. 



A screw, then, is a cylinder with 

 a spiral ridge raised upon it ; this 

 ridge is sometimes made with a 

 square edge (Fig. 79 o), and then 

 has more strength ; but usually it 

 is sharp, as seen in a common screw, and this way of making 

 it reduces friction. 



To use the screw, it is necessary to have a hollow cylinder 

 with a groove cut on the inside of it (Fig. 79 6), so that tho 

 thread of the screw (Fig. 79 c) exactly fits into it, and the screw 

 will rise or fall according to which way it is turned. Thia 

 hollow cylinder is called tha nut or female screw. 



