Question

The frequency of vibration of string depends on the length L between the nodes, the tension F in the string and its mass per unit length m. Guess the expression for its frequency from dimensional analysis.

Answer 1

Frequency, ` f=KL^aF^bM^c
where M=Mass/unit length
L=Length
F=Tension (Force)
Dimension of f=[T^-1]`, Dimension of right side
`L^a=[L^b], F^b=[MLT^-2]^b
M^a=[ML^-1]^c
[T^-1]=K[L]^a[MLT^-2]^b[ML^-1]^c
[M^0L^0T^-1]=KM^(b+c)L^(a+b-c)T^(-2b)
Equating the dimensions of both sides
we get `b+c=0...........i
-c+a+b=0................ii
-2b=-1`
Solving the equatiions
we get
`a=-1
b =1/2
and a=-1
b=1/2
and c=(-1/2)
frequency f=KL^-1F^(1/2)M^(-1/2)
=K/L.F^(1/2)M^((-1/2))ltbrgeK/Lsqrt((F/M))`