241. Third System of Pullies. The diagram represents a series of Pullies in which each string is attached to the Weight, and all the strings are parallel; it is usually called the Third System of Pullies. In this system the string which passes round any Pully, except the lowest, has one end attached to the block of the next lower Pully; the string which passes round the lowest Pully has one end attached to the Weight, and the other end supported by the Power. The highest Pully is fixed, and the others are moveable. The condition of equi

librium for the Third System of Pullies can be expressed most easily by stating what proportion the sum of the Weight and the Power must bear to the Power. If there is one Pully the sum is twice the Power, if there are 2 Pullies the sum is 4 times the Power, if there are 3 Pullies the sum is 8 times the Power, if there are 4 Pullies the sum is 16 times the Power, and so on. Thus for every additional Pully the proportion which the sum bears to the Power is doubled. If the sum is twice the Power the Weight is equal to the Power; if the sum is 4 times the Power the Weight is 3 times the Power; if the sum is 8 times the Power the Weight is 7 times the Power; if the sum is 16 times the Power the Weight is 15 times the Power, and so on.

242. There is no difficulty in the reasoning by which the condition of equilibrium for the Third System of Pullies is established. Suppose that there are four Pullies. Let W denote the Weight to which all the strings are fastened, and P the Power which acts vertically downwards at the end of the string which passes over the lowest Pully. The tension of the string which passes over the lowest Pully is P, hence this Pully is drawn downwards by a force equal to 2P, and consequently this must be the tension of the string which is fastened to it, and draws it upwards, passing over the second Pully. Hence the second

Pully is drawn downwards by a force equal to 4P, and consequently this must be the tension of the string which is fastened to it, and draws it upwards, passing over the third Pully. In like manner 8P is the tension of the string which passes over the fourth Pully, that is the highest in our diagram. Now all the strings are fastened to the Weight, and so help to support it; thus W must be equal to the sum of P, 2P, 4P, and 8P; that is, W must be equal to 15P. Or we might shorten the process a little thus. The tension of the string which goes over the highest Pully is 8P, so that this Pully is drawn downwards by a force equal to 16P ; but the whole Weight supported at K must be equal to the sum of W and P; therefore the sum of W and P is equal to 16P, and consequently W is equal to 15P. A remark similar to that of Art. 238 may be repeated here.

243. We have hitherto supposed that the weights of the Pullies themselves are neglected, but in practice it may be necessary to take these weights into account; it will be sufficient to treat one case as an example. Consider the Third System of Pullies, and suppose, as in Art. 241, that there are four Pullies. The weight of the lowest Pully here assists the Power, and acts just like the Power, except that it has a system of three Pullies above it instead of four; thus it will support 7 times its own weight. Similarly the weight of the next Pully will support 3 times its own weight, and the weight of the next to that will support just its own weight. The weight of the highest Pully will not give any aid. Thus finally we have the following result: the Weight W' is equal to the sum of 15P together with 7 times the weight of the lowest Pully, 3 times the weight of the next, and the weight of the next to that.

244. The Pully is one of the most useful of the simple machines, on account of its portability, the cheapness of its construction, and the ease with which it may be applied in almost any situation. It is much used in building when weights are to be raised to great heights. But its chief employment is in connexion with the rigging of ships, where almost every arrangement is accomplished by its aid. In practice however it is found that the mechanical advantage is far less than that which theory assigns: this arises

from the stiffness of the string or rope and the friction between the wheels and the blocks: it appears that in most cases, owing to these causes, the Power produces only one-third of its theoretical effect.

XVI. THE INCLINED PLANE, THE WEDGE, AND THE SCREW.

245. An Inclined Plane in Mechanics is a smooth plane supposed to be made of wood or metal or some other rigid material, and fixed in a position inclined to the horizon. It is supposed to be capable of resisting in a direction perpendicular to its surface, to any required amount. When an Inclined Plane is used as a Mechanical Power the straight lines indicating the directions in which the Power and the Weight act are supposed to be both in one vertical plane, namely in the plane perpendicular to the straight line in which the Inclined Plane meets the horizon. Thus the Inclined Plane is represented by a right-angled triangle such as ABC; the horizontal side AC is called the base, the vertical side BC is called the height, and the hypotenuse AB is called the length. The angle BAC is the inclination of the Inclined Plane to the horizon.

A

B

C

246. Suppose a heavy body placed on an Inclined Plane. The Weight of the body tends vertically downwards, but owing to the resistance of the Plane the body cannot move in that direction; it will however slide down the Plane unless prevented by a suitable force, and the amount of the force which we must use will depend on the direction in which it acts. We will suppose that the force acts along the Plane, or parallel to it; the proposition which applies to this case is the following: When a Weight is put on an Inclined Plane, and kept in equilibrium by a Power acting parallel to the Plane, the Power is to the Weight in the same proportion as the height of the Plane is to its length.

B

M

L

N

247. The preceding statement may be taken as an experimental truth, or it may be established by reasoning, as we will now shew. Let W denote the Weight, and P the Power. From any point L in the Inclined Plane draw LN at right angles to the Plane, meeting the base at N; and draw NM vertical, meeting the Plane at M. The body on the Inclined Plane is kept in equilibrium by three forces, the Power which is supposed to act along the Plane, the Weight of the body which acts vertically downwards, and the Resistance of the Plane which acts at right angles to the Plane. Now the sides of the triangle LMN are parallel to the directions of these three forces, namely LM to that of the Power, MN to that of the Weight, and NL to that of the Resistance. Hence, by Art. 155, the sides of this triangle are in the proportion of the forces, so that the Power is to the Weight in the same proportion as LM is to MN, and the Resistance is to the Weight in the same proportion as LN is to MN. But by measurement, or by theory, it may be shewn that the triangles LMN and CBA are similar; so that LM is to MN in the same proportion as CB is to BA, and NL is to MN in the same proportion as AC is to BA. Hence finally the Power is to the Weight in the same proportion as CB is to BA, and the Resistance is to the Weight in the same proportion as AC is to BA._Strictly speaking we required only the proportion of the Power to the Weight; but the proportion of the Resistance to the Weight will be useful hereafter.

248. If we suppose the Power to be a little greater than is necessary for equilibrium the Weight will be moved along the Plane. Suppose the Weight to be drawn along the Plane from A to B, so that the Power has passed over the length of the Plane; then the Weight has passed over as much space as the Power, but the vertical height through which the Weight has passed is BC. Thus we have here a fresh illustration of the important principle of Art. 208, and at the same time an indication of the way in which the principle is to be understood: the motion of the Weight estimated in the direction of the Weight bears the

T. P.

7

same proportion to the motion of the Power estimated in the direction of the Power, as the Power bears to the Weight in equilibrium.

249. There is another case with regard to the Inclined Plane which it is usual to notice, namely that in which the Power acts horizontally; the proposition which applies to this case is the following: When a Weight is put on an Inclined Plane and kept in equilibrium by a Power acting horizontally, the Power is to the Weight in the same proportion as the height of the Plane is to its base.

A

M

B

N

C

250. The preceding statement may be taken as an experimental truth; or it may be established by reasoning, as we will now shew. Let W denote the Weight, and P the Power. From any point L in the Inclined Plane draw LN at right angles to the Plane, meeting the base at N; and draw NM vertical, meeting at M the horizontal straight line drawn through L. The body on the Inclined Plane is kept in equilibrium by three forces, the Power which acts horizontally, the Weight of the body which acts vertically downwards, and the Resistance of the Plane which acts at right angles to the Plane. Now the sides of the triangle LMN are parallel to the directions of these three forces, namely LM to that of the Power, MN to that of the Weight, and NL to that of the Resistance. Hence, by Art. 155, the sides of this triangle are in the proportion of the forces, so that the Power is to the Weight in the same proportion as LM is to MN. But by measurement, or by theory, it may be shewn that the triangles LMN and BCA are similar, so that LM is to MN in the same proportion as BC is to CA. Hence finally the Power is to the Weight in the same proportion as BC is to CA.

251. If we suppose the Power to be a little greater than is necessary for equilibrium the Weight will be moved along the Plane. Suppose the Weight to be drawn along the Plane from A to B, so that the Power has passed