250 



THE LEVER AND WHEELWORK. 



difficult to show that the total expenditure of force in the motion of one pound 

 from P to P' is exactly the same as in the motion of ten pounds from W to W'. 

 If the space P P 7 be ten inches, the space W W' will be one inch. A weight 

 of one pound is therefore moved through ten successive Inches, and in each 

 inch the force expended is that which would be sufficient to move one pound 

 through one inch. The total expenditure of force from P to P 7 is ten times the 

 force necessary to move one pound through one inch, or, what is the same, it 

 is that which would be necessary to move ten pounds through one inch. But 

 this is exactly what is accomplished by the opposite end, W, of the lever ; for 

 the weight W is ten pounds, and the space W W 7 is one inch. 



If the weight W of ten pounds could be conveniently divided into ten equal 

 parts of one pound each, each part might be separately raised through one inch, 

 without the intervention of the lever or any other machine. In this case, the 

 same quantity of power would be expended, and expended in the same manner 

 as in the case just mentioned. 



It is evident, therefore, that when a machine is applied to raise a weight, or 

 to overcome resistance, as much force must be really used as if the power were 

 immediately applied to the weight or resistance. All that is accomplished by 

 the machine is to enable the power to do that by a succession of distinct efforts 

 which should be otherwise performed by a single effort. These observations 

 will be found to be applicable to all other machines. 



Weighing-machines of almost every kind, whether used for commercial or 

 philosophical purposes, are varieties of the lever. The common balance, 

 which, of all weighing-machines, is the most perfect, and best adapted for or- 

 dinary use, whether in commerce or experimental philosophy, is a lever with 

 equal arms. In the steelyard, one weight serves as a counterpoise and meas- 

 ure of others of different amount, by receiving a leverage variable according to 

 the varying amount of the weight against which it acts. 



We have hitherto considered the power and weight as acting on the lever, in 

 directions perpendicular to its length, and parallel to each other. This does 

 not always happen. Let A B, fig. 5, be a lever whose fulcrum is F, and let A 



ftp 



R be the direction of the power, and B S the direction of the weight. If the 

 lines R A and S B be continued, and perpendiculars F C and F D drawn from 

 the fulcrum to those lines, the moment of the power will be found by multiply- 

 ing the power by the line F C, and the moment of the weight by multiplying 

 the weight by F D. If these moments be equal, the power will sustain the 

 weight in equilibrium. 



It is evident that the same reasoning will be applicable when the arms of 

 the lever are not in the same direction. These arms may be of any figure or 

 shape, and may be placed relatively to each other in any position. 



In the rectangular lever the arms are perpendicular to each other, and the 

 fulcrum F, fig. 6, is at the right angle. The moment of the power, in this case, 

 is P multiplied by A F, and that of the weight W multiplied by B F. When 

 the instrument is in equilibrium these moments must be equal. 



When the hammer is used for drawing a nail, it is a lever of this kind. The 



