The equation $\displaystyle f= \frac{1}{2L} \sqrt{\frac{T}{\rho}}$ represents the frequency of vibrations of a vibrating violin string. Where $L$ is the length of the string, T is its tension, $\rho$ is its linear density

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 is determined by the frequency $f$. Use the signs of the derivative 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 another string.

$ \begin{equation} \begin{aligned} \text{a.) } & (i) & \frac{df}{dL} &= \frac{1}{2} \sqrt{\frac{T}{\rho}} \frac{df}{dL} \left( \frac{1}{L} \right)\\ \\ & \phantom{x} & \frac{df}{dL} &= \frac{1}{2} \sqrt{\frac{T}{\rho}} \left( \frac{-1}{L^2} \right)\\ \\ & \phantom{x}& \frac{df}{dL} &= \frac{-1}{2L^2} \sqrt{\frac{T}{\rho}}\\ \\ \\ \\ & (ii) & \frac{df}{dT} &= \frac{1}{2L(\rho)^{\frac{1}{2}}} \cdot \frac{d}{dT} (T)^{\frac{1}{2}} \\ \\ & \phantom{x} & \frac{df}{dT} &= \frac{1}{2L(\rho)^{\frac{1}{2}}} \cdot \frac{1}{2} T^{\frac{-1}{2}}\\ \\ & \phantom{x} & \frac{df}{dT} &= \frac{1}{4L\sqrt{\rho T}} \\ \\ \\ & (iii) & \frac{df}{d\rho} & = \frac{\sqrt{T}}{2L} \cdot \frac{d}{d\rho} (\rho)^{\frac{-1}{2}}\\ \\ & \phantom{x} & \frac{df}{d\rho} &= \frac{T}{2L} \cdot -\frac{1}{2}(\rho)^{\frac{-3}{2}}\\ \\ & \phantom{x} & \frac{df}{d\rho} &= \frac{-1}{4L} \sqrt{\frac{T}{\rho^3}} \end{aligned} \end{equation} $

b.) $(i)$ A decrease in length $L$ is associated with the increase in the frequency since $\displaystyle \frac{df}{dL}$ from part(a) is negative.

$(ii)$ An increase in the tension $T$ is associated with an increase in frequency since $\displaystyle \frac{df}{dT}$ from part (a) is positive.

$(iii)$ An increase in the linear density $\rho$ is associated with a decrease in frequency since $\displaystyle \frac{df}{d\rho}$ from part (a) is negative.

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