A mass attached to a spring performs S.H.M. whose displacement is x = 3 × 1 0 − 3 cos 2 π t m. The time taken to obtain maximum speed for the first time is (in s)

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The displacement of the particle is given by $x = 3 \times 1 0^{- 3} cos (2 π t)$ .

The velocity of the particle is $v = \frac{d x}{d t} = - 6 π \times 1 0^{- 3} sin (2 π t)$ .

The speed is maximum when the magnitude of velocity is maximum, which occurs when $∣ sin (2 π t) ∣ = 1$ .

For the first time, this happens when $2 π t = \frac{π}{2}$ .

$t = \frac{1}{4}$ s.

Alternatively, the particle starts from the extreme position at $t = 0$ . The speed is maximum at the mean position, which it reaches for the first time at $t = \frac{T}{4}$ .

Comparing with $x = A cos (ω t)$ , we get $ω = 2 π$ .

Time period $T = \frac{2 π}{ω} = \frac{2 π}{2 π} = 1$ s.

Time taken to reach maximum speed $t = \frac{T}{4} = \frac{1}{4}$ s.

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