A string fixed at x = 0 and x = L vibrates first as y₁ = A sin(πx/L) sinωt with energy E₁, and then as y₂ = A sin(2πx/L) sin2ωt with energy E₂. Find E₂/E₁.
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Question

A string fixed at x = 0 and x = L vibrates first as y₁ = A sin(πx/L) sinωt with energy E₁, and then as y₂ = A sin(2πx/L) sin2ωt with energy E₂. Find E₂/E₁.

G1 Physics · Accepted Answer

Step 1: For the same string and same amplitude, total energy of a normal mode is proportional to ω²A². Step 2: The second mode has angular frequency 2ω, so E₂/E₁ = (2ω)²/ω² = 4. Final: C.