

THEORY OF VIBRATIONS 



plane. The horizontal excursions x, y of M, m respectively 

 being supposed small, the tensions of the upper and lower 

 strings will be (M + m)g and mg, approximately. The equa- 

 tions of motion are therefore 



d*x 



m ^ = ~ m 9 



x 



(15) 



To find the possible modes of simple-harmonic vibration we 

 assume 



x = A cos (nt + e), y B cos (nt + e) ....... (16) 



The equations are satisfied provided 



...(17) 



.(18) 



where /ji = 



Eliminating the ratio A : B, we find 



.(19) 



which is a quadratic in n z . The condition 

 for real roots, viz. 



r 

 ab 



.(20) 



is obviously always fulfilled. It is further 

 easily seen that both roots are positive, so 

 that n also is real. 



The problem includes a number of inter- 

 esting special cases, but we will only notice 

 one or two. If the ratio /*, = m/(M 4- m), 

 be small, the two roots of (19) are nf^g/a, 

 nf=g/b, approximately. In the former 

 case M oscillates like the bob of a simple 

 pendulum of length a, whilst m executes 

 what may be regarded as a forced oscillation 



