Question

The frequency v of vibration of a stretched string depend upon:-1) its length 'l' 2) its mass per unit length 'm' 3)the tension 't' in the string. Obtain dimensionally an expression for frequency

Answer 1

Hope this will help you

Answer 2

The formula is

$\boxed{v=\frac{k}{l}\sqrt{\frac{T}{m}}}$

Explanation:

Let the frequency of vibration v depends upon, length l, tension T and mass per unit length m in the following way

$v=kl^aT^bm^c$  , where k is a constant

Dimensions of v = [T⁻¹]

Dimensions of l = [L]

Dimensions of T = [MLT⁻²]

Dimensions of m = [ML⁻¹]

Thus,

LHS Dimensions =RHS Dimensions

$[T^{-1}]=[L]^a[MLT^{-2}]^b[ML^{-1}]^c$

$\implies [T^{-1}]=[M]^{b+c}[L]^{a+b-c}[T]^{-2b}$

Comparing the dimensions on both sides

We get

$b+c=0$   ............. (1)

$a+b-c=0$  ..............(2)

$-2b=-1$  ................... (3)

From eq (3)

$b=\frac{1}{2}$

Thus, from eq (1)

$c=-\frac{1}{2}$

And from eq (2)

$a+\frac{1}{2}-(-\frac{1}{2})=0$

$\implies a+1=0$

$\implies a=-1$

Thus, the equation becomes

$v=l^{-1}T^{1/2}m^{-1/2}$

$\implies v=\frac{k}{l}\sqrt{\frac{T}{m}}$

Thus us the required equation.

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