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Show that for a wave traveling on a string yₘₐₓ/vₘₐₓ = vₘₐₓ/aₘₐₓ where the symbols have usual meanings. Can we use Componendo and Dividendo taught in algebra to write (yₘₐₓ+vₘₐₓ)/(yₘₐₓ-vₘₐₓ) = (vₘₐₓ+aₘₐₓ)/(vₘₐₓ-aₘₐₓ)? |
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Answer» y = A.sin2π(t/T-x/λ) ....... (i) yₘₐₓ = A. {Because maximum value of sine = 1} v = ∂y/∂t = A.cos 2π(t/T-x/λ)*2π/T →vₘₐₓ = 2πA/T. {} Because also maximum value of cosine =1} a = ∂²y/∂t² = A.sin2π(t/T-x/λ)*4π²/T² →aₘₐₓ = 4π²A/T² Now yₘₐₓ/vₘₐₓ =A/(2πA/T) = T/2π and vₘₐₓ/aₘₐₓ = (2πA/T)/(4π²A/T²) =T/2π Hence, yₘₐₓ/vₘₐₓ = vₘₐₓ/aₘₐₓ. We cannot use Componendo and Dividendo because addition and subtraction are allowed only between quantities having same unit or dimension. Here yₘₐₓ, vₘₐₓ and aₘₐₓ all have different units and dimensions. |
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