A point performs simple harmonic oscillation of period T and its equation of motion is x = a sin(ωt + π/6). After the elapse of what fraction of the time period will the velocity equal half of its maximum velocity?
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Question

A point performs simple harmonic oscillation of period T and its equation of motion is x = a sin(ωt + π/6). After the elapse of what fraction of the time period will the velocity equal half of its maximum velocity?

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G1 Physics · Accepted Answer

Step 1: Differentiate: v = aω cos(ωt + π/6). Hence maximum speed vₘₐₓ = aω. Step 2: For the first positive time when v = vₘₐₓ/2, cos(ωt + π/6) = 1/2. Step 3: The first phase to satisfy this after t = 0 is ωt + π/6 = π/3. Step 4: Thus ωt = π/6 and t = (π/6)/(2π/T) = T/12. Final answer: (D) T/12.