Question

A uniform string of length l is stretched under a force T. If the frequency of vibration of the string (n) is assumed to depend upon load applied (T), length of the string (l) and mass per unit length (m), then show that: n = 1/2l √T/m

Answer 1

Suppose that n ∝ m^a T^b l^c

Or. n= K m^a T^b l^c, where K is a dimensional constant and a, b and c are unknown quantities to be determined.

Writing done the dimensions on both sides of the above equation, we have

[M⁰L⁰T^-1] = [ML^-1]^a [MLT^-2]^b [L]^c

or, [M⁰L⁰T^-1] = [M^a+b L^-a+b+c T^-2b]

Because dimensions on both sides of the equation must be equal, hence we get

a + b = 0, -a + c + c = 0 and -2b = -1

Therefore, a = -1/2 b = 1/2 and c = -1

Hence, n = K m^-1/2 T^-1/2 l^-1 = K/l √T/m

The value of K is found to be 1/2 experimentally.