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), an…

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asked Nov 11, 2020 187k views

1 Answer

RobWhistler · Answer 1 · 2020-11-15T03:45:51+0000

Answer:

(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.

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

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

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