Answer:
[tex]f = \frac{1 }{L}\sqrt{\frac{T}{m} }[/tex]
Explanation:
Let the frequency ,
[tex]f = L^{x}T^{y}m^{z}[/tex]
Now the unit of frequency is hertz = s⁻¹ = T⁻¹ where T is time, tension T = kgm/s² = MLT⁻¹ and linear density m = kg/m = ML⁻¹.
So,
[tex]T^{-1} = L^{x}[MLT^{-2} ]^{y}[ML^{-1} ]^{z}\\[/tex]
collecting the like bases, we have
[tex]T^{-1} = [L^{x + y -z}][M^{y + z} ][T^{-2y} ] \\L^{0} M^{0} T^{-1} = [L^{x + y -z}][M^{y + z} ][T^{-2y} ][/tex]
Equating powers on both sides, we have
x + y - z = 0 (1)
y + z = 0 (2)
-2y = -1 (3)
From (3), y = 1/2
From (2), z = -y = -1/2
Substituting y and z into (1), we have
x + y - z = 0
x + 1/2 - (-1/2) = 0
x + 1/2 + 1/2 = 0
x + 1 = 0
x = -1
So,
[tex]f = L^{x}T^{y}m^{z}\\f = L^{-1}T^{\frac{1}{2} }m^{\frac{-1}{2} }[/tex]
[tex]f = \frac{1 }{L}\sqrt{\frac{T}{m} }[/tex]
