Answer to Question #206294 in Mechanics | Relativity for KABIR

Question #206294
  • A sinsoidal wave with amplitude A= 0.1m, period T=1/2 s, and wave length λ=10m. When t=0, the element at the wave source happened to be at the positive maximum displacement position. If the wave source is the origin point of the x axis and the wave is travelling along the positive OX axis, find: (1) The wave function of this wave? (2) The displacement of the element at x₁=λ/4 when T₁=T/4? (3) The velocity of the element at x₁=λ/4 when T₂=T/2?
1
Expert's answer
2021-06-16T14:26:59-0400

(1) The wave function of this wave?

"y=Asin(kx\u00b1 \\omega t + \\phi)"

Given x=0 and t= 0, y = A

"y=Asin(k*0- \\omega *0+ \\phi)"

"1=sin(\\phi)"

"\\phi = \\frac{\\pi}{2}"

Therefore the equation becomes "y=Asin(kx-\\omega t + \\frac{\\pi}{2})"

"y=Asin(kx- \\omega t + \\frac{\\pi}{2})"

"y=Asin( \\frac{2 \\pi}{\\lambda}x- \\frac{2 \\pi}{T} t + \\frac{\\pi}{2})"

"y=Asin( \\frac{ \\pi}{5}x- 4 \\pi t + \\frac{\\pi}{2})"

(2) The displacement of the element at x₁=λ/4 when T₁=T/4?

"y=Asin( \\frac{2 \\pi}{\\lambda}*\\frac{ \\lambda}{4}- \\frac{2 \\pi}{T}* \\frac{T}{4} + \\frac{\\pi}{2})"

"y=Asin( \\frac{\\pi}{2}- \\frac{\\pi}{2} + \\frac{\\pi}{2})"

"y=Asin( \\frac{\\pi}{2})"

"y=A=0.1 m"

(3) The velocity of the element at x₁=λ/4 when T₂=T/2?

"\\frac{d}{dx}\\left(A\\sin \\left(\\frac{\\pi }{2}\\right)\\right)"

"\\frac{d}{dx}\\left(A\\right) =0"

"v=0m\/s"


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