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Consider an elastic string stretched between two fixed ends. The string is plucked in the middle and is held in a triangular form with a maximum height $h < < l$ at its middle point. At time t = 0, the plucked string is released from rest. All effects due to gravity may be neglected. If linear mass density of the string is $μ$ and the speed of propagation for transverse waves in the string is ‘c’, find the total mechanical energy of vibrating string. (Assume that tension in the string does not change during oscillation)

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$\frac{μ c^{2} h^{2}}{l}$

$\frac{μ c^{2} h^{2}}{2 l}$

$\frac{μ c^{2} h^{2}}{4 l}$

$\frac{2 μ c^{2} h^{2}}{l}$

answer is C.

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Detailed Solution

$d U = \frac{1}{2} T \partial x (\frac{\partial y}{\partial x})$

$(\frac{\partial y}{\partial x}) = \frac{2 h}{l}$

$U = \frac{1}{2} (μ c^{2}) \times \partial x \times {(\frac{2 h}{l})}^{2} = \frac{2 μ c^{2} h^{2}}{l^{2}} \int_{0}^{l} d x = \frac{2 μ c^{2} h^{2}}{l}$

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