MECHANICS. 



Fig. 17. 



is found by drawing a perpendicular from the 

 fulcrum or centre to the direction of the force. 

 In fig. 16, a and b, CA' and CB' are the virtual 

 arms, and P . CA' = Q . CB'. 



When a force thus exerts an effort to turn a 

 lever or any system about a centre, the value of 

 the effort is found by multiplying the force by the 

 perpendicular drawn from the centre upon the 

 direction of the force. This product is called the 

 moment of the force. 



THE WHEEL AND AXLE. 



Let the larger circle in the fig. represent the 

 rim of a wheel, and the 

 smaller the section of the 

 cylindrical axle on which it 

 is fixed. A small weight, P, 

 attached to a cord wound 

 round the wheel, is sufficient, 

 by its descent, to wind up a 

 large weight, W, attached to 

 another cord wound round 

 the axle. It is evident that 

 the two forces here act on 

 the unequal arms, FA, FB, 

 of a lever whose fulcrum 

 is F ; therefore P X AF 

 = W X FB. As the system 

 turns round, the points A and B, at which the 

 forces act, are constantly changing, and there is 

 a succession of new levers, as it were, the arms, 

 however, continuing always of the same length ; 

 the machine is therefore sometimes called the 

 perpetual lever, 



We can conceive the wheel removed, and noth- 

 ing left but the axle and one radius or spoke, FA. 

 If now a projecting handle is inserted into the end 

 of FA, the weight P may be dispensed with, and 

 the force of a hand of equal amount applied 

 at A will turn the axle round and wind up W. 

 Such a spoke and handle attached to an axle is 

 called a winch, and is another form of perpetual 

 lever. 



The principle of the wheel and axle, or perpetual 

 lever, is introduced into various mechanical con- 

 trivances, which are of great use in many of the 

 ordinary occupations of life. One of the simplest 

 machines constructed on this principle is the com- 

 mon windlass for drawing water from wells by a 

 rope and bucket. Coal is lifted from the pits in 

 which it is dug by a similar contrivance, wrought 

 by horse or steam power. 



The capstan in general use on board ships 

 for hauling or drawing up anchors, and for other 

 operations, is an example of the wheel and axle, 

 constructed in an upright or vertical, instead of 

 a horizontal, position. 



There are combinations of the wheel and axle, 

 as there are combinations of the simple lever. 

 Cranes, watch and clock work, and wheel-ma- 

 chinery in general, mostly consist of such com- 

 binations. The parts are usually made to act 

 upon one another by means of projections called 

 teeth or cogs; or, when at a distance, by straps. 

 In any case, the advantage is calculated in the 

 same way as in combinations of the lever. For 

 illustration, let us take the combination of three 

 wheels and axles represented in fig. 18 ; and sup- 

 pose that the radii of the wheels A, B, and C are 



Fig. 18. 



12, 15, and 20 inches respectively ; those of the 

 pinions F and 

 G, 3 inches 

 each, and that 

 of the axle E 

 2 inches the 

 radii of the 

 toothed -wheels 

 and pinions be- 

 ing measured 

 from the centre 

 to the point of 

 contact of the 

 teeth. By the 

 principle of the 

 lever, the down- 

 ward force of P becomes an upward pressure on 

 the teeth of wheel B, and is increased as 3 to 12, 

 or made fourfold. Similarly, the upward pressure 

 on the teeth of wheel B becomes a downward 

 pressure on those of C, and is increased as 3 to 

 15, or five times ; that is, it is equal to 20 times P. 

 Again, this downward pressure of 20 times P upon 

 the teeth of C becomes an upward force at E, 

 increased as 2 to 20, or 10 times ; so that W is 

 sustained by a force equal to 200 times P. 



The object of wheel-work is often, not to gain 

 power, but to gain velocity at the expense of 

 power. In this case the moving force is applied 

 as at W, and the gain in velocity in the revolution 

 of the last wheel in the system, might be calcu- 

 lated from the radii, in much the same way as the 

 gain in power is calculated. It may also be calcu- 

 lated from the number of teeth in the several 

 wheels and pinions. 



THE I'ULLEY. 



The pulley consists of a small wheel, called a 

 sheaf, turning on an axis in a block, with a flexible 

 cord resting in a groove in the circumference of 

 the wheel. There are two kinds of pulleys the 

 fixed and the movable. 



The annexed cut represents a pulley, A, fixed 

 by its block or frame to a beam or roof, B, and 

 having two weights attached to 

 the ends of the cord. In order _ 



that P and W may be in equili- 



brium in this case, it is evident 

 that they must be equal. This 

 appears from the wheel acting 

 as a lever with equal arms, so 

 that neither P nor W has any p A 

 advantage. It is also plain that 

 the cord, being free to move 

 either way, must be equally Fig. 19. 

 stretched, or have the same ten- 

 sion throughout its whole length. A fixed pulley, 

 therefore, does not increase the power ; it only 

 serves to change the direction .in which the power 

 acts. This is often as great a gain as increase of 

 power would be. A force, for instance, at P, pull- 

 ing downward, can raise W upward. 



The wheel is not an essential part of the pulley, 

 in theory at least. The same object might be gained 

 by bending the cord over the axis of the wheel, or 

 over any bar, and making it slide on it ; but in 

 this case the friction of the cord on the surface of 

 the bar would cause great resistance, and would 

 chafe the cord : and it is to obviate this that the 



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