Problem 3-8. Show that for small

k

, the two values of

\\\\omega

from

Eq

(3-18) are given approximately by\ Eqs.

(3-20)

and (3-21) You will need the relations,

sinx~~x

and

\\\\sqrt(1+x)~~1+(1)/(2)x

, for small

x

.\ Solution:\

\\\\omega ^(2)=\\\\alpha ((1)/(m_(1))+(1)/(m_(2)))+-\\\\alpha \\\\sqrt(((1)/(m_(1))+(1)/(m_(2)))^(2)-(4sin^(2)(k(a)/(2)))/(m_(1)m_(2)))\ =\\\\alpha ((1)/(m_(1))+(1)/(m_(2)))+-\\\\alpha ((1)/(m_(1))+(1)/(m_(2)))\\\\sqrt(1-(k^(2)a^(2))/(m_(1)m_(2))((1)/(m_(1))+(1)/(m_(2)))^(-2))3-18\ =\\\\alpha ((1)/(m_(1))+(1)/(m_(2)))[1+-1(k^(2)a^(2))/(2m_(1)m_(2))((1)/(m_(1))+(1)/(m_(2)))^(-2)]

\ 2 steps!\ top sign:

\\\\omega ^(2)=2\\\\alpha ((1)/(m_(1))+(1)/(m_(2)))=>\\\\omega =\\\\sqrt(2\\\\alpha ((1)/(m_(1))+(1)/(m_(2))))

\ bottom sign:

\\\\omega ^(2)=\\\\alpha ((1)/(m_(1))+(1)/(m_(2)))(k^(2)a^(2))/(2m_(1)m_(2))((1)/(m_(1))+(1)/(m_(2)))^(-2)=(\\\\alpha k^(2)a^(2))/(2(m_(1)+m_(2)))

\

=>,\\\\omega =ka\\\\sqrt((\\\\alpha )/(2(m_(1)+m_(2))))

\ Lower curve near

k=0

\ Upper curve near

k=0

\

\\\\omega ~~\\\\sqrt(2\\\\alpha ((1)/(m_(1))+(1)/(m_(2))))

.

Problem 3-8. Show that for small k, the two values of from Eq (3-18) are given approximately by Eqs. (320) and (3-21) You will need the relations, sinxx and 1+x1+21x, for small x. Solution: 2=(m11+m21)(m11+m21)2m1m24sin2(ka/2)318=(m11+m21)(m11+m21)1m1m2k2a2(m11+m21)2=(m11+m21)[112m1m2k2a2(m11+m21)2] details in this 2 steps! top sign: 2=2(m11+m21)=2(m11+m21) bottom sign: 2=(m11+m21)2m1m2k2a2(m11+m21)2=2(m1+m2)k2a2 =ka2(m1+m2) Lower curve near k=0ka2(m1+m2) Upper curve near k=02(m11+m21)