164 MOTIONS OF FREE PARTICLES. [CHAP. IX. 



14. A particle is describing a circular orbit of radius a under a force to 

 the centre producing an acceleration f(r) at distance r, and a small increment 

 of velocity Aw is given to it in the direction of motion. Prove that the 

 apsidal distances of the disturbed orbit are 



a and a+4Ait n ,.,\ J. , > . 

 3/(a) + af'(a) 



Prove also that, if the increment of velocity imparted to the particle is 

 directed radially, the apsidal distances are approximately 



Jfo 



15. A particle moves under a central force //.(l + 8KCOs20)/r 2 being 

 projected from an apse on the initial line at distance c with velocity >/OV c ) ; 

 show that the next apsidal distance is c/(l+3*). 



16. A particle moves under a central force proportional to u 2 (cu + cos &}"'' 

 towards the centre. Show that the orbit is one of the conies given by the 

 equation (cu + cos 0) 2 = a + b cos 2 (6 + a). 



17. A particle moves in a plane under a radial force P and a transverse 

 force T, where 



P=- ^3(3+500820), T=pu s am20; 



prove that a first integral of the differential equation of the path can be 

 expressed in the form 



V (sin 0^-ucoa0\ + -f~(sm 30- sin 0) j+u cos 301 = C 9 

 where h Q 2 and C are constants. 



18. A particle moves under the action of a central force P and a transverse 

 disturbing force -/(*) Prove that 



where F(t) = 



19. Prove that in a plane field of force of which the potential referred to 

 polar coordinates is 



' 



a particle, if projected in the proper direction with the velocity from infinity, 

 will describe a curve of the form 



(r a sin 6)(r b sin 6) = ab, 



20. A particle of mass m describes a circle (centre C) in period T under 

 the action of a force to a fixed point 8. Prove that the force can be resolved 

 into two directed to inverse points 0, O in CS and equal respectively to 

 16mrr 2 CO^ OP 6 16m7r 2 CO'* CP 



T 2 CS* OP* ~W 



