The vibration frequency of a string depends on the length L between the nodes, the tension F in the string, and its mass per unit length m . Determine the expression for its frequency using dimensional analysis, assuming k is a dimensionless constant.

Step-by-step solution

Let the frequency f be given by f = k L^a F^b m^c . Writing the dimensions of each physical quantity: [f] = [T⁻¹] [L] = [L] [F] = [M L T⁻²] [m] = [M L⁻¹] Substituting the dimensions into the assumed relation: [T⁻¹] = [L]^a [M L T⁻²]^b [M L⁻¹]^c [T⁻¹] = [M]^ b+c [L]^ a+b-c [T]^ -2b Equating the powers of M , L , and T on both sides: b + c = 0 a + b - c = 0 -2b = -1 Solving these equations: b = 1 2 c = -b = - 1 2 a = c - b = - 1 2 - 1 2 = -1 Substituting the values of a , b , and c back into the expression: f = k L⁻¹