Question

The frequency (f) of a stretched string on a)the length (l)of the string b)mass per unit length(m) of string and tension (T) in the string. Derive an expression for frequency of vibration of string using
dimensional analysis.​

Answer 1

Answer:

Suppose, that the frequency f depends on the tension raised to the power a, length raised to the power b and mass per unit length raised to the power c. 

Then, f∝[F]a[l]b[μ]c

or, f=k[F]a[l]b[μ]c ...(i)

Here, k is a dimensionless constant. 

Thus, [f]=[F]a[l]b[μ]c

or, [M0L0T−1]=[MLT−2]e[L]b[ML−1]c

or, [M0L0T−1]=[Ma+cLa+b−cT−2a]

For dimensional balance, the dimensions on both sides should be same.

Thus, a+c=0 ...(ii)

a+b−c=0 ...(iii)

−2a=−1 ...(iv)

Solving these three equations, we get

a=21,c=−21andb=−1

Substituting these values in Eq. (i), we get

f=k(F)1/2(l)−1(μ)−1/2orf=l

Explanation:

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