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% .