Oscillations

JEE Main level, single correct, medium

58 of 194

Question 58 of 194

A particle moves with simple harmonic motion in a straight line. In first τs\tau s, after starting from rest it travels a distance a, and in next τs\tau s it travels 2a, in same direction, then :

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Answer: (D) Time period of oscillations is 6τ6\tau

x=−Acosωt\text{x}=-A cos\omega \text{t}

−A+a=−A cosωτ-\text{A}+\text{a}=-\text{A cos}\omega \tau ...(i)

−A+3a=−A cos(ω2τ)-\text{A}+3\text{a}=-\text{A cos}\left(\omega 2\tau \right) ...(ii)

∴−1+aA=−cosωτ∴-1+\dfrac{\text{a}}{\text{A}}=-\text{cos}\omega \tau ...(iii)

and −1+3aA=−cos(2ωτ)-1+\dfrac{3\text{a}}{\text{A}}=-\text{cos}\left(2\omega \tau \right) ...(iv)

Adding (iii) and (iv)

2=3cos(ωτ)−2cos2(ωτ)+12=3cos\left(\omega \tau \right)-2{\text{cos}}^{2}\left(\omega \tau \right)+1

∴2cos2(ωτ)−3cos(ωτ)+1=0∴2{\text{cos}}^{2}\left(\omega \tau \right)-3\text{cos}\left(\omega \tau \right)+1=0

∴cosωτ=12∴\text{cos}\omega \tau =\dfrac{1}{2}

∴ωτ=π3∴\omega \tau =\dfrac{\pi }{3}

∴T=6τ∴\text{T}=6\tau

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