Respuesta :
1. The problem statement, all variables and given/known data
A damped harmonic oscillator loses 6.0% of it's mechanical energy per cycle. (a) By what percentage does it's frequency differ from the natural frequency f
0
=
(
1
2
π
)
√
k
m
? (b) After how many periods will the amplitude have decreased to
1
e
of it's original value?
2. Relevant equations
natural frequency
f
0
=
(
1
2
π
)
√
k
m
damped frequency
f
′
=
1
2
π
√
k
m
−
b
2
4
m
2
displacement for lightly damped harmonic oscillator
x
=
A
e
(
−
b
2
m
)
t
c
o
s
ω
′
t
Total mechanical energy
E
=
1
2
k
A
2
=
1
2
m
v
2
m
a
x
And I know the mean half life,
2
m
b
is the time until oscillations reach 1/e of original.
3. The attempt at a solution
I used the A^2 expression for E and the A decay term,
A
e
(
−
b
2
m
)
t
,said it loses 6% of E when A^2 = .94A^2 (original) or in other words when
A
e
(
−
b
2
m
)
t
=
√
0.94
A
so,
e
(
−
b
2
m
)
t
=
√
.94
−
b
2
m
t
=
1
2
ln(.94)
t =
− 1. The problem statement, all variables and given/known data
A damped harmonic oscillator loses 6.0% of it's mechanical energy per cycle. (a) By what percentage does it's frequency differ from the natural frequency f
0
=
(
1
2
π
)
√
k
m
? (b) After how many periods will the amplitude have decreased to
1
e
of it's original value?
2. Relevant equations
natural frequency
f
0
=
(
1
2
π
)
√
k
m
damped frequency
f
′
=
1
2
π
√
k
m
−
b
2
4
m
2
displacement for lightly damped harmonic oscillator
x
=
A
e
(
−
b
2
m
)
t
c
o
s
ω
′
t
Total mechanical energy
E
=
1
2
k
A
2
=
1
2
m
v
2
m
a
x
And I know the mean half life,
2
m
b
is the time until oscillations reach 1/e of original.
3. The attempt at a solution
I used the A^2 expression for E and the A decay term,
A
e
(
−
b
2
m
)
t
,said it loses 6% of E when A^2 = .94A^2 (original) or in other words when
A
e
(
−
b
2
m
)
t
=
√
0.94
A
so,
e
(
−
b
2
m
)
t
=
√
.94
−
b
2
m
t
=
1
2
ln(.94)
t =
−
m
b
ln(.94)
But this is time and I need it to be one cycle so do I plug the period in for t?
T = 1/f or 2∏ ω?
m
b
ln(.94)
But this is time and I need it to be one cycle so do I plug the period in for t?
T = 1/f or 2∏ ω?
