165-167] MOTION EXPRESSED IN TERMS OF FORCE. 149 



mf being the central attractive force, and thus the equation of energy is 



fr 



^mv 2 + I mfdr= const. 

 This equation can be written 



i>- //*+<* 



and can thus be identified with the equation (1) at the top of p. 63. 



167. Production of simple harmonic motion. We have 

 explained in Article 127 that the tension in an extensible string is 

 proportional to the extension, so that a body attached to such a 

 string will be subject to a force in the line of the string, of the 

 amount indicated, so long as the string remains stretched. Thus 

 if the string is attached to two fixed points whose distance apart 

 is greater than its natural length, and the body is attached to any 

 point of it, then, when the body is displaced in the line of the 

 string from its position of equilibrium, there will be a force urging 

 it towards its position of equilibrium, and proportional to its dis- 

 placement. It will therefore have a simple harmonic motion. 



Again, as we have explained in Article 129, the stress in a 

 spring is tension proportional to the extension when the spring is 

 extended, and pressure proportional to the contraction when the 

 spring is compressed, so that a body attached to the spring will 

 be acted upon by a force proportional to its displacement from 

 the position in which it would rest. It will therefore have a 

 simple harmonic motion. 



We may also state here that in very small vibrations of a 

 system about a position of equilibrium each particle executes a 

 motion compounded of simple harmonic motions in various direc- 

 tions. 



Taking the case of the body attached to the spring, moveable 

 in the line of the spring, and under no force except that arising 

 from the action of the spring, let x be the displacement of the end 

 of the spring to which the body is attached, the other end being 

 fixed, the stress in the spring is tension fjuv when the spring is 

 extended through x, and pressure JJLX when the spring is com- 

 pressed through x, where //, is a constant depending only on the 

 spring. This constant is known as the " strength " of the spring. 



