360 GENERALIZED COORDINATES 



8. In the last question, find the motion if the tangent at P is perpen- 

 dicular to the axis of the cylinder, but is not quite vertical. 



9. In spherical polar coordinates prove that the kinetic energy of a 

 moving particle of unit mass is given by 



T = 



Hence, prove that the acceleration of the particle has components in the 

 direction of r, 6, <f> increasing, of amounts 



1 [".I I?I\ _ #T~| 1 d_ /8T 



r \_dt\w) dO J' rsin6dt\^ 



dt\dr dr r 

 Show that the actual values of these accelerations are 



- rff 2 r sin 2 2 , -- (V 2 0) r sin cos 6 d> 2 , -- (r 2 sin 2 0). 

 dt 2 r dt r sin dt 



10. The velocity of a particle in its orbit is found to vary in the inverse 

 square of its distance from a fixed point. Apply the principle of least 

 action to find the orbit, and thence the law of attraction. 



Deduce the same results from the law of conservation of energy. 



11. Suppose that all forces are annihilated in the universe, and that 

 there is a concealed mechanism capable of possessing kinetic energy. Sup- 

 pose that the amount of this kinetic energy depends only on the positions 

 of the material bodies in the universe, being equal in magnitude except for 

 a constant, and opposite in sign, to the potential energy which the system 

 would have if the forces had not been annihilated. 



Show that the dynamical phenomena of a universe of this kind will be 

 identical with those of a universe in which both forces and kinetic energy 

 exist, the changes in the latter being determined by Newton's laws of 

 motion. 



12. A number of spheres without mass, of radii a, b, c, . . ., move in a 

 straight line through an infinite ocean of density p , the distances apart of 

 their centers being r^, r 6c , etc., and their velocities v a , v b , v c , . * .. When 

 a, b, c, . . . are small compared to r^, etc., the kinetic energy of the motion 

 of the ocean is given by 



n^W 



2T= f irpa + - + 2irp^-v a v b + - . .. 



r ab 



Show that to an observer who is unconscious of the presence of the ocean, 

 the spheres will appear to move as though having masses f 7r/>a 3 , f 7r/o& 3 , 

 etc., and as though forces of attraction acted between every pair of spheres, 

 proportional to the product of the masses, to the product of their veloci- 

 ties, and to the inverse fourth power of the distance between them. 



