EXAMPLES. 225 



160. The resistance of a medium is KV* ; prove that the orbit described in 

 it by a particle under a central attraction /u/r 2 will be an equiangular spiral if 

 the velocity of projection is that in a circle at the same distance, and the 

 angle of projection is cos~ 1 (2/i^). 



161. Two particles move in a medium the resistance of which is pro- 

 portional to the velocity, under the action of their mutual attraction, which 

 is any function of their distance. Prove that their centre of inertia either 

 remains at rest or moves in a straight line with a velocity which diminishes 

 in geometric progression as the time increases in arithmetic progression. 



162. A particle acted on by a central force and moving in a resisting 

 medium in which the resistance is K (velocity) 2 describes an equiangular 

 spiral whose pole is the centre of force ; prove that the force is proportional 

 to r~ 3 e~** raeca , w h ere a j s the angle of the spiral. 



163. A particle of unit mass moves in a resisting medium, of which the 

 resistance at any point is R, under the action of a radial force F and a 

 transversal force G. Prove with the usual notation of central orbits that 



164. A particle of mass m moves in a field of force having a potential V 

 in a medium in which the resistance is k times the velocity. Prove that, if 

 D is the quantity of energy dissipated in time t, 



-j- H (D V) = const. 



cat m 



If the resistance is k (velocity) 2 , and if ds is the element of arc of the path 

 of the particle, then 



-j- -\ (D-V} = const. 



as m 



165. A smooth straight tube rotates in one plane with uniform angular 

 velocity o> about a fixed end, and a particle moves within it under a resistance 

 equal to K times the square of the relative velocity. Prove that if the particle 

 is projected so as to come to rest at the fixed end the relative velocity at 

 a distance r from that end is 



166. A particle is suspended so as to oscillate in a cycloid whose vertex 

 is at the lowest point, and starts at a distance a from that point measured 

 along the curve. Prove that, if the medium in which it moves gives a small 

 resistance K (velocity) 2 per unit of mass, then before it next comes to rest 

 energy equal to ka of the original energy will have been dissipated. 



L. 15 



