80.

(a) From Eq. 33-25,

dq

dt

=

d

dt

Qe

−Rt/2L

cos(ω



t + φ)



=

−

RQ

2L

e

−Rt/2L

cos(ω



t + φ)

− ω



Qe

−Rt/2L

sin(ω



t + φ)

and

d

2

q

dt

2

=



R

2L



e

−Rt/2L

 

RQ

2L



cos(ω



t + φ)

− ω



Q sin(ω



t + φ)



+

e

−Rt/2L



RQω



2L

sin(ω



t + φ)

− ω

2

Q cos(ω



t + φ)



.

Substituting these expressions, and Eq. 33-25 itself, into Eq. 33-24, we obtain

Qe

−Rt/2L



− ω

2

L

−



R

2L



2

+

1

c



cos(ω



t + φ) = 0 .

Since this equation is valid at any time t, we must have

−ω

2

L

−



R

2L



2

+

1

C

= 0

=

⇒ ω



=



1

LC

−



R

2L



2

=



ω

2

−



R

2L



2

.

(b) The fractional shift in frequency is

∆f

f

=

∆ω

ω

=

ω

− ω



ω

= 1

−



(1/LC)

− (R/2L)

2



1/LC

= 1

−

1

−

R

2

C

4L

=

1

−



1

−

(100 Ω)

2

(7.30

× 10

−6

F)

4(4.40 H)

= 0.00210 = 0.210% .