Kinetic Theory of Solids. 



217 



of the kinetic energy, a value of the frequency n is obtained, 

 which is approximately identical with that given by the com- 

 plete theory of the lateral vibrations of a bar. Accordingly 

 it is only necessary to add to the potential energy of bending 

 that due to gravity to get an approximate solution. Suppose 

 clamped at either the top or bottom a uniform vertical bar of 

 length I, density p, section <j, radius of gyration of section k, 

 mass m, loaded with mass M f which has its centre of mass at 

 distance I' from the clamped end, and suppose the free end 

 deflected with force F. Now the tendency of the weight of 

 M is to increase the bending effect of F when M is up, and to 

 diminish it when M is down ; but for small displacements this 

 difference in the effect of the weight of M in the two cases 

 can be neglected. If y is the deflexion of the point at dis- 

 tance x from the clamped end, then the form of the rod is 

 given by 



where B = ^Fcr, in which q is Young's modulus. 



Then, according to the approximate method, the form at 

 time t is 



y = £-5 (x 2, — Six 2 ) cos 27rnt. 

 The potential energy of bending is 



WjB(g)^. 



To find the potential energy due to gravity we have to 

 determine how much each point of the rod is vertically dis- 

 placed on bending. Let ds be an element of length in the 

 bent rod, then 



ds = dx \/l + (dy/dxy = dx\J\ + (~Y (x 2 - 2lx) 2 

 = dx< 1 + J ( oT> ) (x 2 — 2lx) 2 j-approximately; 



••' = * + MibXt-^ + -3-> 



so that s — x, the required vertical displacement of the element 

 at length s along the rod, 



-i(m-«+m 



