Asked by admin @ in Physics viewed by 310 People
the frequency of vibration of a 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.
Answered by admin @
In dimensional analysis,
Length is represented as [L]
Mass is represented as [M]
=> Mass per unit length, m = [ML⁻¹]
Time is represented as [T]
=> Frequency = 1/time = [T⁻¹]
Force = mass x acceleration = [M] x [LT⁻²] = [MLT⁻²]
Let
f = kmᵃLᵇFⁿ
where k is a dimensionless constant
=> [T⁻¹] = k[ML⁻¹]ᵃ[L]ᵇ[MLT⁻²]ⁿ
=> [T⁻¹] = k[Mᵃ⁺ⁿ] [ Lᵇ⁺ⁿ⁻ᵃ] [T⁻²ⁿ]
from the equation we get,
-2n = -1
=> n = 1/2
a+n = 0
=> a = -1/2
b + n -a = 0
=> b + 1/2 + 1/2 = 0
=> b = -1
Hence the formula for frequency
f = k/L x √F/m
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