THE INCLINED PLANE, WEDGE, AND SCREW. 



285 



for ever, without any further action of the power. But if the plane be inclined, 

 the power will be expended in raising the weight through the perpendicular 

 height of the plane? Thus, in a road which rises one foot in ten, the power 

 is expended in raising the weight through one perpendicular foot for every ten 

 feet of the road over which it is moved. As the expenditure of power depends 

 upon the rate at which the weight is raised perpendicularly, it is evident that 

 the greater the inclination of the road is, the slower the motion must be with 

 the same force. If the energy of the power be such as to raise the weight at 

 the rate of one foot per minute, the weight may be moved in each minute 

 through that length of the road which corresponds to a rise of one foot. Thus 

 if two roads rise, one at the rate of a foot in fifteen feet, and the other at the 

 rate of one foot in twenty feet, the same expenditure of power will move the 

 weight through fifteen feet of the one, and twenty feet of the other at the same 

 rate. 



From such considerations as these, it will readily appear that it may often 

 be more expedient to carry a road through a circuitous route than to continue 

 it in the most direct course ; for, though the measured length of road may be con- 

 siderably greater in the former case, yet more may be gained in speed with the 

 same expenditure of power, than is lost by the increase of distance. By at- 

 tending to these circumstances, modern road-makers have greatly facilitated 

 and expedited the intercourse between distant places. 



If the power act oblique to the plane, it will have a twofold effect : a part 

 being expended in supporting or drawing the weight, and a part in diminishing 

 or increasing the pressure upon the plane. Let W P, fig. 1, be the power. 

 This will be equivalent to two forces, W F', perpendicular to the plane, and 

 W E', in the direction of the plane. In order that the power should sustain 

 the weight, it is necessary that that part W E x of the power which acts in the 

 direction of the plane, should be equal to that part W E, fig. 1, of the weight 

 which acts down the plane. The other part W F, of the power acting perpen- 

 dicular to the plane, is immediately opposed to that part W F of the weight 

 which produces pressure. The pressure upon the plane will therefore be di- 

 minished by the amount of W F'. The amount of the power, which will 

 equilibrate with the weight, may, in this case, be found as follows : Take W E' 

 equal to W E, and draw E' P perpendicular to the plane, and meeting the 

 direction of the power. The proportion of the power to the weight will be 

 that of W P to W D. And the proportion of the pressure to the weight will 

 be that of the difference between W F and W F 7 to W D. If the amount of 

 the power have a less proportion to the weight than W P has to W D, it will 

 not support the body on the plane, but will allow it to descend. And if it had 

 a greater proportion, it will draw the weight up the plane toward A. 



It sometimes happens that a weight upon one inclined plane is raised or 

 supported by another weight upon another inclined plane. Thus, if A B and 

 A B', fig. 2, be two inclined planes, forming an angle at A, and W W' be two 

 weights placed upon these planes, and connected by a cord passing over a 

 pulley at A, the one weight will either sustain the other, or one wHl descend, 

 drawing the other up. To determine the circumstances under which these 



Fig. 2. 



