The frequency of vibrations of a vibrating violin string is given by f = where L is the length of the string, Tis its tension, and p is its linear density.t (a) Find the rate of change of the frequency with respect to the following. (i) the length (when T and p are constant) (ii) the tension (when L and p are constant) (iii) the linear density (when L and T are constant)
The frequency of vibrations of a vibrating violin string is given by f = where L is the length of the string, Tis its tension, and p is its linear density.t (a) Find the rate of change of the frequency with respect to the following. (i) the length (when T and p are constant) (ii) the tension (when L and p are constant) (iii) the linear density (when L and T are constant)
Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![The frequency of vibrations of a vibrating violin string is given by
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\rho}} \]
where \( L \) is the length of the string, \( T \) is its tension, and \( \rho \) is its linear density.
(a) Find the rate of change of the frequency with respect to the following:
(i) the length (when \( T \) and \( \rho \) are constant)
[Answer box]
(ii) the tension (when \( L \) and \( \rho \) are constant)
[Answer box]
(iii) the linear density (when \( L \) and \( T \) are constant)
[Answer box]
(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 for the following.
(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
\[ \frac{df}{dL} \] [Dropdown menu: >, <, =] 0 and \( L \) is [Dropdown menu: decreasing, increasing] \(\Rightarrow f\) is [Dropdown menu: decreasing, increasing, constant] \(\Rightarrow\) [Dropdown menu: lower pitch, higher pitch, unchanged]
(ii) when the tension is increased by turning a tuning peg
\[ \frac{df}{dT} \] [Dropdown menu: >, <, =] 0 and \( T \) is [Dropdown menu: decreasing, increasing] \(\Rightarrow f\) is [Dropdown menu: decreasing, increasing, constant] \(\Rightarrow\) [Dropdown menu: lower pitch, higher pitch, unchanged]
(iii) when the linear density is increased by switching to another string
\[ \frac{df}{d\rho} \] [Dropdown menu: >, <, =] 0 and \( \rho \) is [Dropdown menu: decreasing, increasing] \(\Rightarrow f\) is [Dropdown menu: decreasing, increasing, constant] \(\Rightarrow\) [Dropdown menu: lower pitch, higher pitch, unchanged]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb787237-32bd-4647-baa5-49b05320a55c%2F2eaaa9a5-a99d-444a-9be5-ccc0f95880f4%2F1dtuokf_processed.png&w=3840&q=75)
Transcribed Image Text:The frequency of vibrations of a vibrating violin string is given by
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\rho}} \]
where \( L \) is the length of the string, \( T \) is its tension, and \( \rho \) is its linear density.
(a) Find the rate of change of the frequency with respect to the following:
(i) the length (when \( T \) and \( \rho \) are constant)
[Answer box]
(ii) the tension (when \( L \) and \( \rho \) are constant)
[Answer box]
(iii) the linear density (when \( L \) and \( T \) are constant)
[Answer box]
(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 for the following.
(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
\[ \frac{df}{dL} \] [Dropdown menu: >, <, =] 0 and \( L \) is [Dropdown menu: decreasing, increasing] \(\Rightarrow f\) is [Dropdown menu: decreasing, increasing, constant] \(\Rightarrow\) [Dropdown menu: lower pitch, higher pitch, unchanged]
(ii) when the tension is increased by turning a tuning peg
\[ \frac{df}{dT} \] [Dropdown menu: >, <, =] 0 and \( T \) is [Dropdown menu: decreasing, increasing] \(\Rightarrow f\) is [Dropdown menu: decreasing, increasing, constant] \(\Rightarrow\) [Dropdown menu: lower pitch, higher pitch, unchanged]
(iii) when the linear density is increased by switching to another string
\[ \frac{df}{d\rho} \] [Dropdown menu: >, <, =] 0 and \( \rho \) is [Dropdown menu: decreasing, increasing] \(\Rightarrow f\) is [Dropdown menu: decreasing, increasing, constant] \(\Rightarrow\) [Dropdown menu: lower pitch, higher pitch, unchanged]
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