Question

The frequency of vibration (v) of a string may depend upon length (l) of the string, tension (T) in the string and mass per unit length (m) of the string. Use method of dimensions for establishing the formula for frequency v .

Answer 1

I hope this will help

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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