32] RELATIVE KINETIC ENERGY. 93 



and inserting these in the equation of energy for a conservative 

 system, T+W=h, 



In this case accordingly the principle of conservation of energy holds 

 also for the relative kinetic energy, the constant h heing changed. 

 Inasmuch as we know of no absolutely fixed system of axes of 

 reference it is obvious that the kinetic energy of any system contains 

 an indeterminate part. But in virtue of the above principle if we 

 consider the center of mass of the solar system to be at rest all our 

 conclusions with regard to energy will hold good. The effect in 

 general of referring motions to systems of axes which are not at 

 rest will be dealt with in Chapter VII. 



As a simple example of the above principle let us consider the 

 case of a rigid sphere or circular cylinder, with axis horizontal, 

 rolling without sliding down an inclined plane under the action of 

 gravity. If the distance that the center of the body has moved 



parallel to the plane be s, the first part of T is -- -^M ^) If the 



angle that a plane through the horizontal axis parallel to the inclined 

 plane makes with the normal to the inclined plane be # (Fig. 23), 



j f\ 



the velocity of a particle with respect to the center is f-^t where r 



is its distance from the horizontal axis. The relative kinetic energy 

 is thus 



7 f\ 



or since -j-> the angular velocity of rolling is the same for all terms 

 of the summation, 



The factor 2 r m r rl is called the moment of inertia of the system about 

 the horizontal axis through the center of mass and will be denoted 

 by K. Thus we have 



If the rolling takes place without sliding we have the geometrical 



condition of constraint, 



- d& ds 



where E is the radius of the rolling body. 



