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A string tied between x = 0 and x = ℓ vibrates in fundamental mode. The amplitude A, tension T and mass per unit length μ is given. Find the total energy of the string.

A string tied between x = 0 and x = ℓ vibrates in fundamental mode. The amplitude A, tension T and mass per unit length μ is given. Find the total energy of the string.

Grade:10

1 Answers

Jitender Pal
askIITians Faculty 365 Points
9 years ago
Hello Student,
Please find the answer to your question
Here ℓ = ℷ/2 or ℷ = 2 ℓ Since, k = 2π/ℷ = 2π/ ℓ = π/ ℓ
The amplitude of vibration at a distance x from x = 0 is given by A = a sin k x
Mechanical energy at x of length dx is
dE =1/2 (dm) A2 ω2 = 1/2 (μdx) (a sin k x)2 (2πv)2
= 2π2 μv2 a2 sin2 kx dx
But v = vℷ
∴ v = v/ℷ ⇒ v2 = v2/ℷ2 = T/μ / 4 ℓ2 [∴ v = √T/μ ]
∴ dE = 2π2 μ T/μ / 4ℓ2 a2 sin2 {( π/ℓ )x} dx
∴ Total energy of the string
E = ∫ dE = ∫ℓ0 2π2 μ T/μ / 4ℓ2 a2 sin2 (π x/ ℓ) dx
= π2 Ta2/4 ℓ

Thanks
Jitender Pal
askIITians Faculty

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