Question

3The frequency (in Hz) of a vibrating violin string is given by f = 1 2L s T rho where L is the length of the string (in meters), T is the tension of the string (in Newtons), and rho is the linear density of the string. a) Find the derivative of f with respect to: i. L (assuming T and rho are constants) ii. T (assuming L and rho are constants) iii. rho (assuming L and T are constants)

Answer 1

Answer:

(i) $\dfrac{df}{dL}=-\dfrac{1}{2L^2}\sqrt{\dfrac{T}{\rho}}$

(ii) $\dfrac{df}{dT}=\dfrac{1}{4L\sqrt{T\rho}}$

(iii) $\dfrac{df}{d\rho}=-\dfrac{\sqrt{T}}{4L\rho^{-\frac{3}{2}}}}$

Step-by-step explanation:

Let as consider the frequency (in Hz) of a vibrating violin string is given by

$f=\dfrac{1}{2L}\sqrt{\dfrac{T}{\rho}}$

(i)

Differentiate f with respect L (assuming T and rho are constants).

$\dfrac{df}{dL}=\dfrac{d}{dL}\dfrac{1}{2L}\sqrt{\dfrac{T}{\rho}}$

Taking out constant terms.

$\dfrac{df}{dL}=\dfrac{1}{2}\sqrt{\dfrac{T}{\rho}}\dfrac{d}{dL}\dfrac{1}{L}$

$\dfrac{df}{dL}=\dfrac{1}{2}\sqrt{\dfrac{T}{\rho}}(-\dfrac{1}{L^2})$

$\dfrac{df}{dL}=-\dfrac{1}{2L^2}\sqrt{\dfrac{T}{\rho}}$

(ii)

Differentiate f with respect T (assuming L and rho are constants).

$\dfrac{df}{dT}=\dfrac{d}{dT}\dfrac{1}{2L}\sqrt{\dfrac{T}{\rho}}$

Taking out constant terms.

$\dfrac{df}{dT}=\dfrac{1}{2L}\sqrt{\dfrac{1}{\rho}}\dfrac{d}{dT}\sqrt{T}}$

$\dfrac{df}{dT}=\dfrac{1}{2L}\sqrt{\dfrac{1}{\rho}}(\dfrac{1}{2\sqrt{T}})$

$\dfrac{df}{dT}=\dfrac{1}{4L\sqrt{T\rho}}$

(iii)

Differentiate f with respect rho (assuming L and T are constants).

$\dfrac{df}{d\rho}=\dfrac{d}{d\rho}\dfrac{1}{2L}\sqrt{\dfrac{T}{\rho}}$

Taking out constant terms.

$\dfrac{df}{d\rho}=\dfrac{\sqrt{T}}{2L}\dfrac{d}{d\rho}(\rho)^{-\frac{1}{2}}}$

$\dfrac{df}{d\rho}=\dfrac{\sqrt{T}}{2L}(-\dfrac{1}{2}(\rho)^{-\frac{3}{2}}})$

$\dfrac{df}{d\rho}=-\dfrac{\sqrt{T}}{4L\rho^{-\frac{3}{2}}}}$