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

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

(ii)
`(df)/(dT)=(1)/(4L√(T\rho))`

(iii)
`(df)/(d\rho)=-\frac{√(T)}{4L\rho^{-(3)/(2)}}}`

Explanation:

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

`f=(1)/(2L)\sqrt{(T)/(\rho)}`

(i)

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

`(df)/(dL)=(d)/(dL)(1)/(2L)\sqrt{(T)/(\rho)}`

Taking out constant terms.

`(df)/(dL)=(1)/(2)\sqrt{(T)/(\rho)}(d)/(dL)(1)/(L)`

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

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

(ii)

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

`(df)/(dT)=(d)/(dT)(1)/(2L)\sqrt{(T)/(\rho)}`

Taking out constant terms.

`(df)/(dT)=(1)/(2L)\sqrt{(1)/(\rho)}(d)/(dT)√(T)}`

`(df)/(dT)=(1)/(2L)\sqrt{(1)/(\rho)}((1)/(2√(T)))`

`(df)/(dT)=(1)/(4L√(T\rho))`

(iii)

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

`(df)/(d\rho)=(d)/(d\rho)(1)/(2L)\sqrt{(T)/(\rho)}`

Taking out constant terms.

`(df)/(d\rho)=(√(T))/(2L)(d)/(d\rho)(\rho)^{-(1)/(2)}}`

`(df)/(d\rho)=(√(T))/(2L)(-(1)/(2)(\rho)^{-(3)/(2)}})`

`(df)/(d\rho)=-\frac{√(T)}{4L\rho^{-(3)/(2)}}}`