a clamped string is oscillating in nth harmonic, then (a) total energy of oscillating will be n^2 times that of fundamental frequency. (b) total energy of oscillations will be (n-1)^2 times that of fundamental frequency. © average kinetic energy of the string over a complete oscillations is half of that of the total energy of the string (D) none of these More than one correct type.

Question

Saurabh Koranglekar · Answer

Dear student

the total energy of oscillations will be n^2 times that of the fundamental frequency

the average kinetic energy of the string over complete oscillations is half of that of the total energy of the string.

Regards

Arun · Answer

For a sine wave, y = A sin ( k x − Ω t ) Velocity equation for this wave is V y ​ = Ω A cos ( k x − Ω t ) Kinetic energy = d ( K E ) = 1 / 2 ( V y 2 ​ × d m ) = 1 / 2 ( V y 2 ​ × μ d x ) , μ is the linear mass density. => 1 / 2 ( μ × Ω 2 × A 2 × c o s 2 ( k x − Ω t ) ) d x integrating at t = 0 , with limits as 0 and λ , we have K . E = 1 / 4 ( μ × Ω 2 × A 2 × λ ) Potential energy, d U = 1 / 2 ( Ω 2 × y 2 × μ ) d x integrating at t = 0 , with limits as 0 and λ , we have U = 1 / 4 ( μ × Ω 2 × A 2 × λ ) Total energy E = K . E + U => E = 1 / 2 ( μ A 2 λ ) Therefore, for the first and fundamental frequency, energy is E 1 ​ = ( 1 / 2 ( μ A 2 λ ) ) / n 2 And clearly from the above derivation, we have, K.E is half the total energy.

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