THEORY OF VIBRATIONS 



If we put n* = gll, (2) 



d?x 



this becomes rf^ +w2a? = ^' ^ 



and the solution is 



x A cos nt + B sin nt, (4) 



where the constants A, B may have any values. That this 

 formula really satisfies (3) is verified at once by differentiation ; 

 and since it contains two arbitrary constants A, B, we are able 

 to adapt it to any prescribed initial conditions of displacement 

 and velocity. Thus if, when < = 0, we are to have #=o? , 

 dxjdt = u Q , we find 



Un . 



cos nt + sin nt. 

 n 



.(5) 



It is of course necessary, in the application to the pendulum, 

 that the initial conditions should be such as are consistent with 

 the assumed " smallness " of the oscillations. Thus in (5) we 

 must suppose that the ratios x /l and u /nl are both small. In 

 virtue of (2) the latter ratio is equal to */(u */gl), so that u 

 must be small compared with the velocity " due to " half the 

 length of the pendulum. 



5. Simple-Harmonic Motion. 

 If in 4 (4) we put 



A ~D /"I \ 



as is always possible by a suitable choice of a and e, we get 



The particular type of vibration represented by this formula 



is of fundamental importance. 



It is called a "simple-harmonic," 



or (sometimes) a "simple" 



vibration. Its character is best 



exhibited if we imagine a 



geometrical point Q to describe 



a circle of radius a with the 



constant angular velocity n. 



The orthogonal projection P of 



Q on a fixed diameter AOA' 



will move exactly according to 



