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प्रश्न
Choose the correct option:
A particle performs linear S.H.M. starting from the mean position. Its amplitude is A and time period is T. At the instance when its speed is half the maximum speed, its displacement x is ______.
पर्याय
`sqrt3/2 A`
`2/sqrt3 A`
`A/2`
`1/sqrt2A`
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उत्तर
A particle performs linear S.H.M. starting from the mean position. Its amplitude is A and time period is T. At the instance when its speed is half the maximum speed, its displacement x is `underlinebb(sqrt3/2 "A")`.
Explanation:
Express the relation for the velocity of a particle executing S.H.M.
`v = omegasqrt(A^2 - x^2)`
The displacement of the particle in motion is given as x = A sin (ωt).
When differentiating the displacement with respect to time we obtain velocity.
So, `"d"/"dt" (x) = "d"/"dt" ("A" sin (omega "t"))`
v = Aω cos (ωt)
The speed is maximum when cos(ωt) = 1.
v = Aω
The displacement for the time when speed is half of the maximum is:
v = `("A"omega)/2`
Now,
`(Aomega)/2 = omegasqrt(A^2 - x^2)`
`=> A^2/4 = A^2 - x^2`
`=> x^2 = A^2 - A^2/4`
`=> x^2 = (3A^2)/4`
Square on both sides
`x = sqrt(3)/2 A`
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