Question

The frequency of vibration of a wire under tension depends upon the tension (T), mass per unit length (m) and vibrating length (l) of the wire. Using dimensional analysis, obtain the dependence of frequency on these quantities.

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Answer 1

$\huge\boxed {\sf{ANSWER!!!!}}$

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

HOPE IT HELPS U

Answer 2

SOLUTION

Refer to the attachment.

hope it helps ☺️