. 



MOTION ABOUT A CENTER OF FORCE 269 



MOTION OF A PARTICLE ABOUT A CENTER OF FORCE 

 Force Proportional to the Distance 



214. Let us suppose that a particle moves under no forces ex- 

 cept an attraction to a fixed point 0, the force of attraction being 

 directly proportional to its distance from 0. 



Taking as origin, let the coordinates of the point P y the posi- 

 tion of the particle at any instant, be denoted by x, y, z. Let the 

 force acting on the particle be /* OP directed along PO, where /z. 

 is a constant. The components of this force along the three 

 coordinate axes are _^ _^ _^ 



The components of acceleration are, as in 177, 



d*x tfy cPz 

 ~dt^ W d?' 



Thus the equations of motion of the particle are 



/72 r 



m = -^x, (100) 



TO = -^> (101) 



(102) 



These three equations are all of the same type, namely the type 

 of equation which denotes simple harmonic motion. Thus the foot 

 of the perpendicular from the moving point on to each of the 

 coordinate axes moves with simple harmonic motion. 



The solution of equation (100) has already been seen to be 



x=Acosp(t e), 

 where p* = fi/m. This can be written 



x = A cos pe cos pt + A sin pe sin pt, 

 or again x C cos pt + D sin pt, 



