46 KINETICS OF A PARTICLE. [36.. 



direction at any point coincides with the direction of the result- 

 ant force at that point. Such a line is called a line of force. 



The lines of force evidently form the orthogonal system ta 

 the system of equipotential surfaces. The differential equations 

 of the lines of force are therefore : 



dx _ dy _ dz 



WJ~WJ~W (13) 



dx dy 82 



86. Exercises. 



(1) Show that a force-function exists when the resultant force is 

 constant in magnitude and direction. 



(2) Find the force-function in the case of a free particle moving 

 under the action of the constant force of gravity alone (projectile in 

 vacua} ; determine the equipotential surfaces and the potential energy. 



(3) Show the existence of a force-function when the direction of the 

 resultant force is constantly perpendicular to a fixed plane, say the 

 .xy-plane, and its magnitude is a given function /(z) of the distance z 

 from the plane. 



(4) Find the force-function, the equipotential surfaces, and the 

 kinetic energy when the force is a function f(f) of the perpendicular 

 distance r from a fixed line, and is directed towards this line at right 

 angles to it. 



(5) Show that a force-function always exists for a central force, i.e. 

 a force passing through a fixed point and depending only on the dis- 

 tance from this point. 



(6) Show the existence of a force-function when a particle moves 

 under the action of any number of central forces. 



(7) A homogeneous sphere of mass m( attracts a free particle P of 

 mass m with a force F= urnm 1 /r*, where K is a constant, and r OP is 

 the distance of /'from the centre of the sphere. Show that the poten- 

 tial is V Kmm 1 /r, and that the equipotential surfaces are spheres 

 whose common centre is at O. 



(8) In Ex. (7), assume Kmm'= i, and draw the intersections of the 

 equipotential surfaces with a plane passing through O, from r = i centi- 

 metre to r = 2 centimetres, with a difference of potential = T ^. 



