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The motion of a particle is given by x = A.sin⍵t+B.cos⍵t. The motion of the particle is (a) not simple harmonic (b) simple harmonic with amplitude A+B (c) simple harmonic with amplitude (A+B)/2 (d) simple harmonic with amplitude √(A²+B²). |
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Answer» (d) simple harmonic with amplitude √(A²+B²). Explanation: The given equation can be written as, x = √(A²+B²){(A/√(A²+B²))sin⍵t+(B/√(A²+B²))cos⍵t} Since the magnitudes of A/√(A²+B²) and B/√(A²+B²) are less than 1 and the sum of their square is also 1 we can find an angle α between 0 and 2π such that sinα=B/√(A²+B²) and cosα=A/√(A²+B²) So, x = √(A²+B²){sin⍵t.cosα+cos⍵t.sinα} →x = √(A²+B²).sin(⍵t+α) This is an equation of simple harmonic motion with amplitude √(A²+B²). Hence option (d). |
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