Question

a uniform wire of length L and mass M is streched between two fixed points , having tension F . A sound of frequency v is impressed on it . Then the maximum energy is developed in wire when v=?
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Answer 1

The frequency of vibration may depend upon length of the string (L), tension(F) and mass per unit length(m=M/L).

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Answer 2

The frequency of the wire depends on the length of the string (L), tension (F) and the mass per unit length of the wire (m = M/L)

Let Abegin mathsize 12px style proportional to end styleLaFbmc ........ (Equation 1)

Writing the dimensions of each in the above equation, we get

[M0L0T-1] = La [M1L1T-2]b [M0L-1]c

                = Mb+cLa+b-cT-2b

Here, b + c = 0 ;  

 a + b − c = 0  and

  − 2b = − 1

On solving, we get

a = −1 , b = 1/2 and c = −1/2

Substituting these values in equation (1), we get

begin mathsize 12px style straight A space equals space straight k space straight L to the power of negative 1 end exponent straight F to the power of 1 divided by 2 end exponent straight m to the power of negative 1 divided by 2 end exponent

space space space space equals space straight k 1 over straight L square root of straight F over straight m end root

or space straight A space equals space straight k 1 over straight L square root of fraction numerator straight F over denominator left parenthesis straight M divided by straight L right parenthesis end fraction end root

or space straight A space equals space space straight k square root of straight F over ML end root

Experimentally space straight k space equals space 1 divided by 2

therefore straight A space equals space space 1 half square root of straight F over ML end root end style