A longitudinal wave is represented by x = 10 sin 2π ( nt - x/λ ) cm. The maximum particle velocity will be four times the wave velocity if the determined value of wavelength is equal to :

Solution

Physics sir · 3 steps

Read amplitude, ω and k off the equation, then set Aω equal to four times ω/k.

1. Read the equation. Here the x on the left is the particle's displacement along the wave, and the x inside the bracket is position. Compare with A sin(ω t - kx): A = 10 cm, ω = 2π n and k = 2π/λ.
2. Two speeds. The maximum particle velocity is Aω = 10 × 2π n = 20π n. The wave velocity is ω/k = nλ.

   20π n = 4 nλ ⇒ λ = 20π/4 = 5π

3. So. The frequency n cancels, and λ = 5π cm. That is option (b).

The idea to keep. Maximum particle speed is Aω and wave speed is ω/k. Their ratio is kA = 2π A/λ, so any condition comparing the two speeds fixes λ in terms of A.

Answer from the NTA final answer key. Solution by PrepWiser.