MOTION ABOUT A CENTER OF FORCE 273 



4. A particle is acted on by attractions to a number of centers of force, each 

 Deing proportional to the distance. Show that it describes an ellipse. 



How could a mechanical model be constructed to illustrate this motion ? 



5. A particle is acted on by a repulsion proportional to its distance from a 

 ter of force. Show that it describes a hyperbola. 



6. Show that in the last question the radius vector joining the particle to 

 center of force sweeps out equal areas in equal times. 



GENERAL THEORY OF MOTION ABOUT A CENTER OF FORCE 



218. Suppose that we have a particle acted on only by a force 

 directed towards a fixed center of force, the magnitude of this force 

 being any function of the distance from the center. 



Let be the center of force, P the position of the particle at 

 any instant, and PP' the direction of the velocity of the particle 

 at this instant. Then 

 the plane OPP' con- 

 tains the velocity of the 



particle, which is along ( o 



PP' y and also the accel- 

 eration, which is along 

 PO. Hence, after any 



, , . , , , , ' FIG. 135 



short interval the veloc- 

 ity of the particle will still be in the plane OPP 1 . The particle is 

 still in this plane, say at P 1 , so that the acceleration which is 

 along P'O is also in this plane. 



Hence we can show that, after a further small interval, the posi- 

 tion, velocity, and acceleration of the particle are all in the plane 

 OPP', and so we can proceed indefinitely. 



It follows that the particle will never leave the plane OPP 1 . 

 We accordingly have the theorem : 



The orbit described by a particle about a fixed center of force lies 

 entirely in one plane. 



This theorem has already been ^emplified in 214 by the 

 case of the orbit described under an attraction proportional to the 

 distance from the center of force. 



