Question

Assuming that the frequency γ of a vibrating string may depend upon i) applied force (F) ii) length (l) iii) mass per unit length (m), prove that ϒ ∝ `1/lsqrt(F/m)` using dimensional analysis.

Answer 1

Given:

The frequency v of a vibrating string depends

(i) applied force (F)

(ii) length (l)

(iii) mass per unit length (m)

Solution:

v ∝ F^x l^y m^z ∝ v = K F^x l^y m^z  ............................(1)

Substitute the dimensional formulae of the above quantities

[M⁰L⁰T^-1] = [MLT^-2]^x [L] [ML^-1]^z

[M⁰L⁰T^-1] = `["M"^{"x + z"} "L"^{"x + y - z"} "T"^{-2"x"}]`

Comparing the powers of M, L, T on both sides,

x + z = 0, x + y – z = 0, -2x = -1

Solving for x, y, z, we get

x = `1/2`

y = -1

z = `-1/2`

Substitute x, y, z values in equ(1)

v ∝ F^1/2 l^-1 m^-1/2

∴ v ∝ `1/l sqrt(F/m)`