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Math Help - Rate of change of frequency of vibrations of a violin string?

  1. #1
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    Rate of change of frequency of vibrations of a violin string?

    We're learning related rates/rates of change currently in my AP calc class. Just have a few questions, wondering if I'm understanding these problems correctly. The " " is my questions, etc.

    f = 1/2L * sq. root ( T / P )

    The frequency of vibrations of a violin string is given by the above eq. where L is the length of the string, T is its tension, and P is linear density.

    #1. Find rate of change of f with respect to: a) the length (when T & P are constant), b) the tension (when T & P are constant), c) the linear density (when T & P are density)

    " Do I take the first three derivatives of f to find these? First deriv being length, second being tension, third being linear density? "

    #2 The pitch of a note (how high or low the note sounds) is determined by the frequency F. Use the signs of the derivatives in #1 to determine what happens to the pitch of a note:

    " I don't understand this at all "

    a) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates

    b) when the tension is increased by turning a tuning peg

    c) when the linear density is increased by switching to another string
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by arkhampatient View Post
    We're learning related rates/rates of change currently in my AP calc class. Just have a few questions, wondering if I'm understanding these problems correctly. The " " is my questions, etc.

    f = 1/2L * sq. root ( T / P )

    The frequency of vibrations of a violin string is given by the above eq. where L is the length of the string, T is its tension, and P is linear density.

    #1. Find rate of change of f with respect to: a) the length (when T & P are constant), b) the tension (when T & P are constant), c) the linear density (when T & P are density)

    " Do I take the first three derivatives of f to find these? First deriv being length, second being tension, third being linear density? "
    No you take the derivative of f with respect to a) L, b) T, and c) P.

    In each case you treat the other variables as if they were constants.

    CB
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by arkhampatient View Post

    #2 The pitch of a note (how high or low the note sounds) is determined by the frequency F. Use the signs of the derivatives in #1 to determine what happens to the pitch of a note:

    " I don't understand this at all "

    a) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates

    b) when the tension is increased by turning a tuning peg

    c) when the linear density is increased by switching to another string
    When you shorten the string the frequency becomes:

    f(L-\varepsilon,T,P) \approx  f(L,T,P) -\varepsilon f_L(L,T,P)

    where f_L(L,T,P) denotes the derivative of f(L,T,P) with respect to L evaluated at (L,T,P).

    Part a) asks: Is this an increase or decrease in frequency?

    The others are similar except that the variables in question are increased rather than decreased.

    CB
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