262 Prof. Barton and Miss Browning on Coupled 



And these values put in (66) and (67) give the special 

 solution 



(m 2 — q 2 )u . , (»* — m 3 )?4 . 



9 (p 2 -q 2 )p r (p 2 -q 2 )q *' I f83) 



(p 2 — m 2 )(m 2 — q 2 )u . , (m 2 — q 2 ) (p 2 — m 2 )u . I 



s= — ^ — o-Ts * ; sincH 2 * g — „, ' sm a£. J 



m z a(p z — q z )p 1 m z oi(p z — q 2 )q * J 



So 



F-(p 2 ~m> and H" p * ' • (84) 

 Note the contrast of (84) with (80). 



(iii.) Upper Bob Displaced : Lower Free. — This case may 

 be represented by 



y = ub, z = b, | 7 =0, J=0, for *=0. . . (85) 

 These put in (66) to (69) give equations satisfied by 



2 T 2 p 2 —g 2 p^—q 2, ' 



These values in (66) and (67) give the special solution 



*q 2 b ufb s 



V— ^ — 2 C0S P*+ 2 — - 2 cos qt, 1 



p 2 -q 2 p 2 -q 2 I r87) 



(p 2 — m 2 )q 2 b (m 2 — q 2 )p 2 b \ A / 



F-'/^Hr^-sv ' ' ' ( } 



(iv.) Lower Bob Displaced : Upper Free. — Let the dis- 

 placement (a) of the lower bob P be produced by a horizontal 

 force. Then the corresponding value of the displacement 

 (z) of Q when at rest can be found statically We thus 

 obtain 



These conditions inserted in (66) to (69) give equations 

 satisfied by 



_7T (l + ot)m 2 -(l + a + * 2 )q 2 



€ ~2> *" (l + . + a% u 2 -2 2 ) "' | 

 7T - (l + ^ + a> 2 -(l + ^)m 2 [ 



2' (1 + * + **)' (i**-**) 



