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

B

C

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.

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.

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 From Weight, and P the Power. 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

horizontally over the space AC; then the Weight has passed vertically over the space BC. Hence the space passed over by the Weight estimated in the direction of the Weight is to the space passed over by the Power estimated in the direction of the Power as the Power is to the Weight in the state of equilibrium. Thus we have here a fresh illustration of the important principle of Art. 208, and an indication, as in Art. 248, of the way in which it is to be understood.

252. The Wedge is a hard solid body bounded by five plane figures, two of which are triangles and the others are four-sided figures. The four sided-figures are often rectangles, and then the triangles are in parallel planes.

B

253. The Wedge may be employed to separate bodies. We may suppose the Wedge urged forwards by a force A P acting on one of the four-sided faces, and urged backwards by two resistances Q and R arising from the bodies which the Wedge is employed to separate, and acting on the other four-sided faces. These forces will be supposed all to act in one plane which is perpendicular to the edge of the Wedge; and we shall assume that the Wedge is smooth, so that the force on each face is at right angles to the face. Let the triangle ABC represent a section of the Wedge made by a plane perpendicular to its edge; and suppose the Wedge kept in equilibrium by the forces P, Q, R at right angles to AB, BC, CA respectively: then by reasoning which we do not give here it is shewn that P, Q, R are in the same proportion to each other as AB, BC, CA respectively. If AC and BC are equal the Wedge is called an Isosceles Wedge; in this case Q and R must be equal, and each of them be in the same proportion to P that AC is to AB.

254. There is very little value or interest in the preceding Article, because the circumstances there supposed 284865B

7-2

scarcely ever occur in practice. A nail is sometimes given as an example of the Wedge; but when the nail is at rest the resistances on its sides are balanced by friction, and not by a Power at the head. The nail is indeed driven into its place by blows on the head; but the discussion of the motion produced by such blows in conjunction with the resistances and the friction is too difficult for a work like the present.

255. The Screw. Everybody is familiar with the use of a Screw for fastening pieces of wood together, and this will

W
B

supply great help towards understanding the action of a Screw as a Mechanical Power. The Screw consists of a right circular cylinder AB, with a uniform projecting thread abcd... traced round its surface, making always the same angle with straight lines parallel to the axis of the cylinder. This cylinder fits into a block C pierced with an equal cylindrical aperture, on the inner surface of which is cut a groove, the exact counterpart of the projecting thread abcd...Thus when the block is fixed and the cylinder is introduced into it, the only manner in which the cylinder can move is backwards or forwards by turning round its axis.