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The displacement of an elastic wave is given the function y = 3 sin ωt + 4 cos ωt.When y is in cm and t is in second. Calculate the resultant amplitude. |
Answer» y =3sin ωt +4 cos ωt …(i) Let, 3 = a cos ϕ …(ii) 4 = a sin ϕ …(iii) ∴ = a cos ϕ sin ωt + a sinϕ cos ωt …(iv) y = a sin(ωt + φ) From eq. (s) (i) & (iii)- tan ϕ = \(\frac{3}{4}\) or ϕ = tan-1\(\frac{3}{4}\) Squaring (ii) & (iii), then adding- a2cos2ϕ+ a2sin2ϕ = 32 + 42 a2(cos2ϕ + sin2ϕ) = 9 + 16 a2 = 25 or a = 5 or amplitude, = 5 cm ∴ y1 = 5 sin (ωt + ϕ), ∵ ϕ = tan-1\(\frac{4}{3}\) Hence, new amplitude, a = 5cm. |
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