Question

(violin string) The frequency of vibrations of a vibrating violin string is given by \[ f=\frac{1}{2 L} \sqrt{\frac{T}{\rho}} \] where \( L \) is the length of the string, \( T \) is its tension, and \( \rho \) is its linear density. [See Chapter 11 in D. E. Hall, MusicaI Acoustics, 3rd ed. (Pacific Grove, CA, 2002).] a. Find the rate of change of the frequency with respect to i. the length (when \( T \) and \( \rho \) are constant), ii. the tension (when \( L \) and \( \rho \) are constant), and iii. the linear density (when \( L \) and \( T \) are constant). b. The pitch of a note (how high or low the note sounds) is determined by the frequency \( f \). (The higher the frequency, the higher the pitch.] Use the signs of the derivatives in part (a) to determine what happens to the pitch of a note i. when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates, ii. when the tension is increased by turning a tuning peg, iii. when the linear density is increased by switching to another string.

          (violin string)
The frequency of vibrations of a vibrating violin string is given by
\[
f=\frac{1}{2 L} \sqrt{\frac{T}{\rho}}
\]
where \( L \) is the length of the string, \( T \) is its tension, and \( \rho \) is its linear density. [See Chapter 11 in D.
E. Hall, MusicaI Acoustics, 3rd ed. (Pacific Grove, CA, 2002).]
a. Find the rate of change of the frequency with respect to
i. the length (when \( T \) and \( \rho \) are constant),
ii. the tension (when \( L \) and \( \rho \) are constant), and
iii. the linear density (when \( L \) and \( T \) are constant).
b. The pitch of a note (how high or low the note sounds) is determined by the frequency \( f \). (The higher the frequency, the higher the pitch.] Use the signs of the derivatives in part (a) to determine what happens to the pitch of a note
i. when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates,
ii. when the tension is increased by turning a tuning peg,
iii. when the linear density is increased by switching to another string.

(violin string)
The frequency of vibrations of a vibrating violin string is given by

f=(1)/(2 L)√((T)/(ρ))

where L is the length of the string, T is its tension, and ρ is its linear density. [See Chapter 11 in D.
E. Hall, MusicaI Acoustics, 3rd ed. (Pacific Grove, CA, 2002).]
a. Find the rate of change of the frequency with respect to
i. the length (when T and ρ are constant),
ii. the tension (when L and ρ are constant), and
iii. the linear density (when L and T are constant).
b. The pitch of a note (how high or low the note sounds) is determined by the frequency f. (The higher the frequency, the higher the pitch.] Use the signs of the derivatives in part (a) to determine what happens to the pitch of a note
i. when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates,
ii. when the tension is increased by turning a tuning peg,
iii. when the linear density is increased by switching to another string.

Added by Steve A.

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