The frequency of vibration, f, of a string of length, l, and mass per unit length, μ, depends on its tension,
T (a force), in addition to 1 and μ. By dimensional analysis determine the form of the relation between f, T, l
and μ. Make f the subject of the equation.
f , T , L , μ
f 1/s
T newtons (since F=ma that is kg meters/s^2)
L meters
μ kg/meters
so
f 1/s = k ( T kg meters/s^2 )^a * ( Lmeters )^b (μ kg/meters)^c
for 1/s on the right, a = 1/2
so we have kg^1/2 and meters ^1/2 from T
so c must be -1/2 to kg^-1/2
we have meters ^1/2 from T, meters^1/2 from c so b has to be -1
so
f = k T^1/2 * L^-1 * μ ^-1/2
f = [k/L] sqrt ( T / μ)