6.3 Solutions
285
6.63 The time elapsed between successive maximum displacements of a damped
harmonic oscillator is represented by T , the period.
T
=
2π
ω =
2π
ω 2
0 − b 2
=
2π
k
m
−
r 2
4m 2
=
4π m
√
4km − r 2
= constant
6.64 Force = mg = kx
∴
k
m
=
g
x
=
980
9.8
= 100
ω 0 =
k
m
=
√
100 = 10 rad/s
= bT
=
2π b
ω 2
0 − b 2
(1)
Substituting = 3.1 and ω 0 = 10 in (1), b = 4.428
T
=
2π
ω 2
0 − b 2
=
2π
10 2 − (4.428) 2
= 0.7 s
6.65 d 2 x
dt 2 +
2dx
dt
+ 8x = 16 cos 2t
(1)
This is the equation for the forced oscillations, the standard equation being
m
d 2 x
dt 2 + r
dx
dt
+ kx = F cos ωt
(2)
Comparing (1) and (2) we find
m = 1 kg, r = 2, k = 8, F = 16 N , ω = 2
(a) ω 0 = 2π f 0 =
k
m
=
8
1
= 2
√
2
∴ f 0 =
2
√
2
2π
=
√
2
π
/s
(b) ω = 2π f = 2
∴ f =
2
2π
=
1
π
/s
285
6.63 The time elapsed between successive maximum displacements of a damped
harmonic oscillator is represented by T , the period.
T
=
2π
ω =
2π
ω 2
0 − b 2
=
2π
k
m
−
r 2
4m 2
=
4π m
√
4km − r 2
= constant
6.64 Force = mg = kx
∴
k
m
=
g
x
=
980
9.8
= 100
ω 0 =
k
m
=
√
100 = 10 rad/s
= bT
=
2π b
ω 2
0 − b 2
(1)
Substituting = 3.1 and ω 0 = 10 in (1), b = 4.428
T
=
2π
ω 2
0 − b 2
=
2π
10 2 − (4.428) 2
= 0.7 s
6.65 d 2 x
dt 2 +
2dx
dt
+ 8x = 16 cos 2t
(1)
This is the equation for the forced oscillations, the standard equation being
m
d 2 x
dt 2 + r
dx
dt
+ kx = F cos ωt
(2)
Comparing (1) and (2) we find
m = 1 kg, r = 2, k = 8, F = 16 N , ω = 2
(a) ω 0 = 2π f 0 =
k
m
=
8
1
= 2
√
2
∴ f 0 =
2
√
2
2π
=
√
2
π
/s
(b) ω = 2π f = 2
∴ f =
2
2π
=
1
π
/s