930

THEORY AND PRACTICE OF ENGINEERING.

Book II.

weighed. In constructing a steelyard for practical purposes, the arm should be madesufficiently strong to prevent its bending, and it is necessary that the scale be graduatedby experiment: supposing it is required to graduate the bar to indicate pounds, let onepound be suspended at P, and let the counterpoise be moved until the bar rests in thehorizontal position; then try two pounds in the same manner : these two divisions beingestablished, let the whole length of the bar be divided into equal divisions, and each divisionwill give the number of pounds it will sustain ; care being taken that the centre of gravityof the bar is at or under the fulcrum on which it is suspended.

The Danish and Su-edish Steelyard is a bar or lever, with a constant fixed weight at oneextremity, and a scale for holding the body to be weighed at the other. The point ofsuspension, therefore, varies, and is usually a ring moving along the bar, and placed in theposition where the equilibrium takes place: for instance, suppose the weight, W, is thatof the whole apparatus, with the exception of the ring, to be united in its centre of gravity ;D and P, to be that of the body to be weighed. An equilibrium will take place, whenW x C D = P x C B, which, since CD = BD — CB, gives

CB =

Wx_BD

wTp“'

from which the lever may be readily graduated. There are many applications of theprinciples of the lever to the balance for commercial purposes, and the sensibility of a pairof scales is made much more delicate, the more the length of the arms is increased, and theweight of the bar is diminished. The bent lever balance is graduated upon a quadrantattached to it, and an index which is acted upon by the body to be weighed. Supposingthe scale be loaded successively, the divisions on the quadrant may be made to exhibit therespective weights by noticing the successive positions of the index.

The Pulley will have an equilibrium when each of the two powers is to the weight, asthe radius of the pulley is to the arch of the circumference, which the rope surrounds.Let the powers, P, P, attached to the ropes which pass over the circles, i\I,N, and roundthe pulley, CE, whose centre is D, be in cquilibrio with the weight W;

then P : P' : : \V=CD : D F, CE.

P and P' being equal, and in equilibrio, may be represented by the equal lines W Aand 1VB; then drawing AF parallel to MW, and BF parallel to NW, the diagonal,WF, will represent the resultant of thetwo powers, P, P': but as this resultantis in equilibrio with the weight, W,acting in the same direction F W, weshall have W F for the measure of theweight, W. By joining C D, D E, anddrawing C E, two isosceles triangles areformed,WB F and C D E, which are simi-lar, because the angles A and D comprisedbetween equal sides are both the supple-ment of the same angle, C W E. Thefigure C D E W, which is quadrilateral,having C and E right angles, we haveWB:BF::WF-DC: D E : C E ;therefore P : W or P' : W as radius,is to the chord of the arc embraced bythe rope. Each of the powers is to theweight, as radius is to twice the cosineof the angle which each rope formswith the direction of the weight. Asthe angle D C H is equal to CWD,

C E is equal to twice its cosine, ortwice C H.

The pulley is a cylindrical piece ofwood, or any other hard substance, with a groove cut out of its rim for the rope to passover. The cylinder moves round an axis, which works in a frame called the block. Asystem of pulleys, or a number combined together, called a muffle, is either fixed or move-able, according as the block which contains the pulleys is so. When a rope used to trans-mit force has its direction altered, if the angle be too sudden, the probability is that it willbreak ; instead, therefore, of bending it over a single angle, the change of direction isgenerally produced by deflecting it over several angles, each of which is less sharp than asingle one could be, and this is most perfectly done w T hen the rope is bent over the surfaceof a curve or pulley. If the rope were employed to sustain a fixed body, and not to movea weight, no inconvenience could arise from its rigidity ; but when motion is required, it is

Fig. 1255.