934
THEORY AND PRACTICE OF ENGINEERING.
Boon 11.
ferenccs are adapted to the length of rope which passed over them at the same time ;this will be evident if we consider the circumference of one wheel to be 1 foot, and that itmake one revolution while the lower block is raised 1 foot: in this time 3 feet of ropewould pass over the circumference of the corresponding wheel; and if the circumferenceof the latter be 3 feet, it will revolve once during the ascent of the weight: from thence itfollows, that the several wheels will revolve in the same time, if their circumferences be asabove stated.
In the single movable pulley when the power descends through aspace of 2 feet, the rope will have passed over in the same length orbe shortened as much ; thus the weight ascends through a foot whilethe power descends 2 feet, or the velocity of the power is twice that ofthe weight.
The laws of the equilibrium of pulleys are sometimes supposed to bethe same as those which belong to the lever; but this can only be the casewhere two equal weights suspended by a rope pass over a single pulley:a pulley ought rather to be considered as a cylinder moving on an axis,the sole object being to change the direction of the rope without anyunnecessary friction, for when a rope has its course altered by passingover a smooth surface, it communicates to it the entire force which isacting upon it without any friction, and the resistance of such a surfacebeing in a direction perpendicular to itself and to the rope, the force ap-plied remains undisturbed : a fixed pulley, therefore, on this accounthas no mechanical advantage. By a movable pulley the weight maybe suspended by two forces, each equivalent to half the weight appliedin a vertical direction to the extremities of the rope ; and these forcesmay be derived from two weights if the rope passes over two fixed pul-leys in a proper direction, and if one of the ends l»c attached to a fixedpoint, and the other to the weight, the equilibrium will still be main-tained.
When two ropes are wound round two cylinders which move on the same axis, there willbe an equilibrium when the weights which are suspended to them are inversely as the radiiof the cylinders ; this will be apparent, if we consider that every section of the cylinderperpendicular to the axis is a circle, and fne ropes, being tangents to the circles, will be atthe distances of the radii from the vertical plane: therefore by similar triangles the rightline joining the weights will be divided in the ratio ofthe radii, and the centre of gravity will be in the verticalplane ; and the point of the axis immediately over it isthe centre of suspension. When a rope is wound rounda cylinder in contrary directions, the first perpendicular,and the other in an oblique position, there will be anequilibrium when the forces are as the perpendiculardistance of any point of the oblique rope from the axis, toits distance from the point of contact.
The common pulley consists of a wheel with a groovemade on its edge or circumference which confines therope; this wheel is usually put into an iron frame,where it turns with a metai pin, the diameter of whichis about a twelfth that of the wheel; the thickness of thewheel is made a fifth of its diameter.
Sometimes the groove ✓->.
on the wheel is made of \ y
portions of cones, soarranged that they holdmore securely the ropewhich is passed overthem.
When there is a greatstrain upon machinery,and its motions are re-quired to be regular,belts and ropes cannotbe safely employed, andchains are more advan-tageous, which act asropes by simple frictionon the surfaces around which they pass. When ropes are used, they generally move ingrooves cut in the circumference of the sheaves or wooden wheels.
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Fig. 1270.
Fig. 1271.