Question

The frequency of vibration of a string depends on the length L between the nodes, the tension F in the string and its mass per unit length m. Guess the expression for its frequency from dimensional analysis.

Answer 1

The expression for its frequency from dimensional analysis is given by $\underline{v}=\frac{k}{L} \sqrt{\frac{f}{m}}$.

Explanation:

Frequency (f) $=\frac{1}{T I M E}=\left[T^{-1}\right]$

Force = mass (m) x acceleration (a)

          = [M] x [LT⁻²] = [MLT⁻²]

      F = ma

Here,     F = force

             m = mass

Length is represented as [L]

Mass is represented as [M]

Mass per unit length, m = [ML⁻¹]

Time is represented as [T]

Let ,

                 f = kmᵃLᵇFⁿ

where, k is a dimensionless constant

            [T⁻¹] = k [ML⁻¹]ᵃ [L]ᵇ[MLT⁻²]ⁿ

            [T⁻¹] = k [Mᵃ⁺ⁿ] [ Lᵇ⁺ⁿ⁻ᵃ] [T⁻²ⁿ]

From the equation we get,

              -2n = -1

                 n = 1/2

            a + n = 0

                  a = -1/2

        b + n - a = 0

       $b+\frac{1}{2}+\frac{1}{2}=0$

Where,      b = -1

The formula for frequency  $\underline{v}=\frac{k}{L} \sqrt{\frac{f}{m}}$

Therefore, frequency is $\underline{v}=\frac{k}{L} \sqrt{\frac{f}{m}}$.

Answer 2

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