point of contact of the axle with the nave will be at half a right angle from the vertical line, or the arch ef will be 45°, and ƒ H will represent the force and direction of gravity, and SH that of the power; but ƒ H is equal to CB-C i, and SH is equal to SB-fi, and because fi C is a right angle fi= Ci. If the radius of the axle CF be 1.5 inches, the sine fi, or Ci, of 45°, will be 1.06 inches: hence SH SB-1.06=13.831.06=1279, and ƒ H=C B-f g=22-106=20·94; and the power P= 500 x 12.79 20.94 =3054 lbs. ; which is 10lbs. less than that before found, without allowing for the diameter of the axle. If the power be applied to the axle in a direction not parallel to the horizon, but inclining upwards, as represented in the following figure, the resistance will be diminished, or a less power will be required; for the leverage by which the power acts is increased, while the leverage by which gravity acts is decreased, until the line of draught forms a right angle with the line drawn from the centre of the axle to the point of contact, in which case the power is a minimum. On the contrary, if the direction of the line of draught be inclined downwards from the horizontal, the leverage by which it acts will be diminished, and consequently the power must be increased as the direction of the line of draught falls below the horizontal. Formerly it was the practice to fasten on the streaks of iron, or shoeing, on the wheels of carriages with nails, the heads of which projected at least half an inch beyond the surface of the wheel. These heads formed a succession of obstacles over which the wheel had to mount; and, besides being extremely injurious to the roads, were a serious obstacle to the effective work of horses: the iron shoeing is now generally put on in hoops, and fastened by rivets, the heads of which are countersunk, and therefore form no impediment to the rolling of the wheel. When the surface is indented, or furrowed into small cavities, such as a pebble pavement or badly dressed stone (into which the wheel falls, producing a shock the noise of which is well known), the resistance which is For the moproduced arises from a different cause. mentum or velocity in the horizontal direction is partly destroyed by the descent of the wheels into the hollows and the blow or collision against the opposite side of the cavity. The resistance produced by such a surface has also been investigated by M. Gerstner. His reasoning may be briefly stated as follows: Suppose BED one of the cavities formed by two contiguous stones. M B E H Let the tangents B E and DE be drawn to the circumference of the wheel at the points of contact B and D, and suppose the velocity to be represented by A E= HE in magnitude and direction. From the point E as a centre, and with the radius A E, describe the semicircle GAH, and let fall the perpendicular A F. The velocity A E may be resolved into two others, A F and F E: of these two, one, A F, is destroyed by the shock, and the other, FE, remains acting in the direction E D; consequently, the loss of velocity is equal to A E-EF=EG-E F-G F; and this loss must be compensated by an increase in the force of traction. To avoid a complicated calculation, suppose the force of traction, K"", to be a constant accelerating force; Q, the weight of the carriage; and 2 g t, the velocity which gravity would generate in the weight Q at the end of the time t we shall then have But FG AG:: AG: 2AE; from which AG 2 FG=2AE; and from the similar triangles A EG, DCB; AG: AE::DB: BC; from which AG= AEX DB MN ; but t; r representing the velocity with น which the space MN is traversed in the time t. By making these substitutions, the former equation becomes K Q ve 4 g MN (BC). From this expression the following conclusions may be drawn: 1. That the resistance arising from a surface of this description is proportional to the load. 2. That the draught or force of traction is proportional to the square of the velocity; and consequently pebble or rough pavements are more adapted for heavy loads, with a slow velocity, than for light carriages with quick velocities. 3. That the draught increases in the inverse ratio of M N; that is, as the distance between the paving-stones diminishes, or as the stones are narrow, the cavities remaining the same. 4. That the draught increases in the ratio of the width of the cavity to the radius of the wheel. When the stones made use of for paving are of a good shape, well dressed, and sufficiently large, and laid on a firm and substantial foundation, they form the most perfect road surface for general purposes. The cavity between the stones should not exceed half an inch in width, by which means carriage wheels would pass over them without the least shock or resistance, and consequently without producing the noise often complained of in towns, at the same time that the surface would be sufficiently rough to prevent the horses from slipping. NOTE C. Page 70. THE next resistance, friction, which we shall consider, is that which arises from the wheels being forced over obstacles which break down under their weight, or when they are drawn through mud or other soft substances, or when the material of which the road is composed (such as gravel or small stones) is put on a soft or yielding substratum in layers so thin that the weight of the wheel can make an impression on it, and force it down so as to form a rut. Let A B C represent a carriage wheel resting on the horizontal road BD, the surface of which is hard and solid, but covered with mud, sand, or gravel, to the height of the line A E: if it be very soft, the wheel, as it rolls along, will press through it as if it was water, and rest on the hard and firm surface B D. If it be of a more tenacious nature, as some clays, or composed of sand or gravel which the wheels will only compress, without displacing it, the wheel will not go to the hard |