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6 September, 04:52

A damped harmonic oscillator loses 6.0% of its mechanical energy per cycle.

a. by what percentage does its frequency (equation 14-20) differ from its natural frequency?

b. after how many periods will the amplitude have decreased to 1/2 of its original value?

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Answers (1)
  1. 6 September, 06:02
    0
    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?
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