The ends of the stretched wire of length L are fixed at x = 0 and x = L . In one experiment, the displacement of the wire is y₁ = A ( x L ) and energy is E₁ . In another experiment, its displacement is y₂ = A ( 2 x L ) and energy is E₂ . Then.

Step-by-step solution

The total energy of a standing wave in a stretched string is equal to its maximum potential energy, given by: E = 1 2 T ₀^L ( dy dx )^2 dx For the first displacement y₁ = A ( x L ) : dy₁ dx = A L ( x L ) E₁ = 1 2 T ₀^L A^2 ^2 L^2 ^2 ( x L ) dx Since ₀^L ^2 ( x L ) dx = L 2 : E₁ = 1 2 T ( A^2 ^2 L^2 ) L 2 = T A^2 ^2 4L For the second displacement y₂ = A ( 2 x L ) : dy₂ dx = 2A L ( 2 x L ) E₂ = 1 2 T ₀^L 4A^2 ^2 L^2 ^2 ( 2 x L ) dx Since ₀^L ^2 ( 2 x L ) dx = L 2 : E₂ = 1 2 T ( 4A^2 ^2 L^2 ) L 2 = 4 ( T A^2 ^2 4L ) T