A point performs simple harmonic oscillation of period T and the equation of motion is given by x $$=$$ $$asin($$ωt $$+$$ π$$6)$$. After the elapse of what fraction of the time period the velocity of the point will be equal to half of its maximum velocity?

Question

A point performs simple harmonic oscillation of period T and the equation of motion is given by x $$=$$ $$asin($$ωt $$+$$ π$$6)$$. After the elapse of what fraction of the time period the velocity of the point will be equal to half of its maximum velocity?

Infinity Learn · Accepted Answer

Velocity is the time derivative of displacement.Writing the given equation of a point performing SHMx $$=$$ a $$sin$$$$($$ω$$t+$$π$$6)-------(i)Differentiating$$ Eq.(i) w.r.t time, we obtainv $$=$$ dxdt $$=$$ a ω $$cos$$$$($$ω$$t+$$π$$6)It$$ is given that v $$=$$ aω$$2$$, so thataω$$2$$ $$=$$ a ω $$cos$$$$($$ωt $$+$$ π$$6)or$$ $$12$$ $$=$$ $$cos$$$$($$ω$$t+$$π$$6)or$$ $$cos$$$$π$$$$3=$$ $$cos$$$$($$ω$$t+$$π$$6)or$$ ωt $$+$$ π$$6$$ $$=$$ π$$3$$  ⇒ ωt $$=$$ π$$6or$$ t $$=$$ π$$6$$ω$$=$$ π×π$$6$$×$$2$$π $$=$$ $$T12Thus$$, at $$T12$$ velocity of the point will be equal to half of its maximum velocity.

A point performs simple harmonic oscillation of period T and the equation of motion is given by $x = asin (ωt + \frac{π}{6})$ . After the elapse of what fraction of the time period the velocity of the point will be equal to half of its maximum velocity?

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A point performs simple harmonic oscillation of period T and the equation of motion is given by x = asin(ωt + π6). After the elapse of what fraction of the time period the velocity of the point will be equal to half of its maximum velocity?

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Ready to Test Your Skills?

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A point performs simple harmonic oscillation of period T and the equation of motion is given by $x = asin (ωt + \frac{π}{6})$ . After the elapse of what fraction of the time period the velocity of the point will be equal to half of its maximum velocity?