280
6 Oscillations
where T =
2π
ω is the time period for damped vibration and b =
ω 2
0 − ω 2 ,
where ω 0 and ω are the angular frequencies for natural and damped vibrations, respectively.
= 2π
ω 2
0
ω 2 − 1 = 2π
f 2
f 2 − 1 = 2π
20
16
2
− 1 =
3π
2
6.52 The equation for damped oscillations is 4
d 2 x
dt 2 +
r dx
dt
+ 32x = 0
Dividing the equation by 4
d 2 x
dt 2 +
r
4
dx
dt
+ 8x = 0
Comparing the equation with the standard equation
d 2 x
dt 2 +
r
m
dx
dt
+
k
m
x = 0
m = 4,
k
m
= 8 → k = 32
ω 0 =
k
m
=
√
8 = 2
√
2
The quantity b =
r
2m
represents the decay rate of oscillation where r is the
resistance constant.
(a) The motion will be underdamped if
b < ω 0 or
r
2m
<
k
m
or r < 2
√
km
i.e. r < 2
√
32 × 4 or r < 16
√
2
(b) The motion is overdamped if r > 16
√
2.
(c) The motion is critically damped if r = 16
√
2.
6.53 (a) ω 0 =
k
m
=
20
4
= 2.23 rad/s
T =
2π
ω 0
=
2π
2.23
= 2.8 s