22 



MECHANICS. 



about the cylinder, and is the means by 

 which the weight is to be raised, or the 

 resistance overcome. 



Fig. 32. 



It is evidently only a modification of 

 the lever, as will appear by reference to 

 (7,) and the condition of its equilibrium 

 immediately follows from the general 

 principle there established. Let R be 

 the radius of the wheel, and r that of 

 the axle, and the condition of equili- 

 brium is P x R =Wxr; that is, the 

 power multiplied by the radius of the 

 wheel is equal to the weight multiplied 

 by the radius of the axle. 



Thus, if the power be two pounds, 

 and the radius of the wheel 1 2 inches, 

 and the weight 8 pounds, and the radius 

 of the axle 3 inches, there will be equi- 

 librium, each product being 24. 



This condition is sometimes express- 

 ed otherwise ; thus, that the power 

 must have to the weight the same ratio 

 as the radius of the wheel has to the 

 radius of the axle, or 



P : W : : r : R. 



(5 1 .) It is easy to perceive that this 

 machine furnishes an instance of the 

 principle of virtual velocities already 

 alluded to (5.) In one revolution of 

 the wheel, the power descends to a space 

 equal to the circumference of the wheel, 

 and the weight is raised through a space 

 equal to the circumference of the axle. 

 Hence, the velocity of the power is to 

 the velocity of the weight, as the cir- 

 cumference of the wheel is to the cir- 

 cumference of the axle. But the cir- 

 cumferences of circles are as their radii, 

 and therefore the velocity (V) of the 

 power is to the velocity (v) of the 

 weight as the radius of the wheel is to 

 that of the axle, that is 



V : v : : R : r . 



By what has been already proved, the 

 weight is to the power in the same pro- 

 portion, and therefore we have 



W : P : : V : v. 



or P x V = W x v ; 

 that is, the power multiplied by the ve- 

 locity of the power is equal to the 

 weight multiplied by the velocity of the 

 weight. 



(52.) The axle in the wheel has been 

 not improperly called the continual or 

 perpetual lever, because the motion 

 which it communicates to the weight is 

 constant and not intermitting, as we 

 have explained to be the case in the 

 common lever. In order that it may be 

 possible to suspend the action of the 

 power without suffering the weight to 

 descend or recoil, and thus lose the ad- 

 vantage which has been gained by its 

 elevation, a contrivance called a ratchet- 

 wheel is sometimes annexed to the cy- 

 linder or axle. 



Fig. S3. 



In fig. 33, this apparatus is repre- 

 sented at G D. It is a wheel furnished 

 with teeth upon its edge. The teeth are 

 not presented directly from the centre, 

 but are curved and all bent in one di- 

 rection, which is that direction in which 

 the rope is coiled upon the cylinder. A 

 curved bolt or catch working on a pivot 

 at H, falls by its weight between the teeth 

 of the wheel ; and the effect is, that the 

 wheel and cylinder to which it is at- 

 tached, are suffered to revolve in that 

 direction in which the weight is raised, 

 but are not permitted to revolve in the 

 other direction. So that the weight can 

 never descend unless the bolt H be pre- 

 vented from acting on the wheel GD. 

 By such an apparatus the action of the 

 power may be suspended at pleasure, 

 and yet the effects of its past action 

 maintained, 



