Question

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.

Answer 1

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

Answer 2

Explanation:

I hope you understand this solution