A progressive wave on a string of linear mass density ρ is y = A sin(2πx/λ − ωt). Given ρ = 3 × 10⁻² kg m⁻¹, A = 1 mm, ω = 100 rad s⁻¹ and λ = 16 cm, find the total energy, in μJ, crossing the origin from t = 0 to t = π/(2ω).
Physics · Wave Motion
JEE Mains Advanced

Question

A progressive wave on a string of linear mass density ρ is y = A sin(2πx/λ − ωt). Given ρ = 3 × 10⁻² kg m⁻¹, A = 1 mm, ω = 100 rad s⁻¹ and λ = 16 cm, find the total energy, in μJ, crossing the origin from t = 0 to t = π/(2ω).

Physics · Wave Motion

G1 Physics · Accepted Answer

Step 1: Compare with y = A sin(ωt ± kx + φ): f = ω/(2π), λ = 2π/k and wave speed v = ω/k. Step 2: The sign of kx fixes the propagation direction; particle motion follows ∂y/∂t. Step 3: Apply these to the given coefficients. Final: 6.