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% .