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.

By K Balaji|Updated : November 12th, 2022

The expression for its frequency from dimensional analysis is f = K/L √F/M

Frequency

The number of full oscillations that any wave element performs in one unit of time is how we determine the frequency of a sinusoidal wave. According to the definition of frequency, if a body is moving periodically, it has completed one cycle after going through a number of situations or postures and then returning to its initial position. The rate of oscillation and vibration is thus described by the parameter frequency.

The following equation shows how frequency and period are related:

The following equation demonstrates the relationship between frequency and period:

f=1/T

Using the equation to represent a sinusoidal wave:

y (0,t) = -a sin (ωt)

Expression of frequency is given as follows:

Frequency,

f ∝ [La] …. (1)

f ∝ [Fb] ….. (2)

f ∝ [Mc] …. (3)

Combining above equation (1) (2) and (3) we can say:

f = [KLaFbMc]

where M = Mass / unit length

L = Length

F = Tension (Force)

Dimension of f = [T-1]

Dimension of right side:

Dimension of force, F = [MLT-2]b = [Mb Lb T-2b]

Dimension of mass per unit length, = [ML-1]c = [McL-c]

So [T-1] = [La] [Mb Lb T-2b] [McL-c]

[M0L0T-1] = Mb+c La+b-c T-2b

 

Equating the dimensions of both sides, we get

b + c = 0 …. (1)

- c + a + b = 0 …. (2)

-2b = -1 …. (3)

Solving the equations, we get,

a = -1, b = ½, c = -½

f = KL-1F1/2M

Hence expression of frequency will be as follows:

f = K/L √F/M

Summary:-

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.

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, the expression for its frequency from dimensional analysis is f = K/L √F/M

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