WORK OF RAISING BODIES AGAINST GRAVITY 153 



119. Work of raising a system of bodies against gravity. If a 



particle of mass m is moved a distance ds along a path making an 

 angle < with the vertical (upwards), the work done is mg cos < ds. 

 Since the distance through which the particle is raised is ds cos <, 

 we may say that the work done is equal to the weight of the 

 body (mg) multiplied by the distance through which the particle 

 is raised. 



By taking the particle along any path, and adding together the 

 amounts of work done on the successive elements of the path, we 

 find that the total work done against gravity is equal to the weight 

 of the particle multiplied by the total vertical distance through 

 which the body has been raised. 



120. Let us suppose that we move a number of particles of 

 masses m l} m 2 , . Let their heights above the ground before the 

 motion be h lt h 2 , - , and let their heights at the end of the motion 

 be h[,h' 2 ,--. The work done against gravity on the first particle 

 is ffi^gil^ 7^); by addition of such quantities, the total work 

 done against gravity 



( 36 ) 



Now let M be the total mass of the particles, and let H, H' 

 denote the heights of the center of gravity of all the particles 

 above the ground before and after the motion respectively. Then, 

 by the formula of 86, we have 



so that 



and, similarly, ^m^J = MH'. 



Thus the total work, as given by expression (35), becomes 

 g(MH J MH) = Mg(H r H). 



