60.] RECTILINEAR MOTION. 3I 



potential energy of the moving particle m. Denoting this by 

 V, and the kinetic energy by T, the last equation becomes 



or 7> V= r o + r o =const. ; 



i.e. the sum of the kinetic and potential energies remains con- 

 stant during the motion. This is the principle of the conservation 

 of energy for this particular problem. 



58. It is easy to see that the principle of the conservation 

 of energy holds generally whenever the resulting force F is a 

 function of the distance s alone. 



Indeed, if F= F(s), the principle of kinetic energy gives 



mv*-mv*= \ F(s)ds; (13) 



/ 



hence, putting J F(s)ds=f(s), where f(s) is called the force- 



function, or potential function, while f(s) is the potential 

 energy, we have 



\mT?-\m V *=f(s)-f( S ^ (14) 



or, with the notation of Art. 57, 



=r o +F = const. (15) 



59. When the resultant force F is an attraction directly proportional 

 to the distance s from a fixed centre O, say 



the potential energy is, by Art. 58, 



Hence, the principle of the conservation of energy gives 



z/ 2 -f- KV = const, j 

 or, if the initial velocity is zero when s = s , 



60. Tension of an Elastic String. According to Hooke's law, the 

 tension of an elastic string is, within the limits of elasticity (i.e. as long 



