94 



HI. GENERAL PRINCIPLES. WORK AND ENERGY. 



The loss of potential energy is M g times the vertical distance 

 f alien , ssina, where a is the angle of inclination of the plane to 

 the horizontal. Our equation thus becomes 



l iTtf/ds\ 2 l K {ds\z\ 

 4 ) T \ M (di) + a* U) ) - 



- const. 





If = V when s = 0, determining the constant we have 



Thus the motion is the same (cf. 18) as that of a particle falling 

 freely with the acceleration diminished in the ratio 



Fig. 23. 



Thus by increasing K y 

 which may be done 

 by symmetrically attach- 

 ing heavy masses to a bar 

 fastened to the cylinder 

 in such a way as not to 

 interfere with the rolling 

 of the cylinder (Fig. 23), 

 we may make the motion 

 as slow as we please and 

 thus study the laws of 

 constant acceleration. 



33. Moment of Momentum. Under the supposition that the 

 equations of constraint were compatible with the displacement of 

 the system parallel to itself and that the force -function was thereby 

 unchanged we obtained the principle of the conservation of motion 

 of the center of mass. We will now suppose that the equations of 

 constraint are compatible with a rotation of the system about the 

 axis of X and that the force -function is thereby unaffected. This 

 will be the case in a rigid system or in a free system left to its 

 own internal forces (if conservative). 



If we put 



y r = r r COS C0 r , 



ob) 



2 r = r r sin 



such a displacement is obtained by changing all the ro r 's by the 

 same amount do, leaving the r's unchanged. We have then 



dx r = 0, dy r = 



^7 ) 



dz r = r r cos G) r do = 



