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A block of mass m attached to a spring of force constant k oscillates with angular velocity ω in a horizontal plane without friction. If initially (t = 0 ), it is pulled to a distance x₀ and pushed towards the centre with a velocity v₀, then find the amplitude of oscillations in terms of ω, x₀ and v₀

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$\sqrt{\frac{v_{0}^{2}}{ω^{2}} - x_{0}^{2}}$

$\sqrt{ω^{2} v_{0}^{2} + x_{0}^{2}}$

$\sqrt{\frac{x_{0}^{2}}{ω^{2}} + v_{0}^{2}}$

$\sqrt{\frac{v_{0}^{2}}{ω^{2}} + x_{0}^{2}}$

answer is D.

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Detailed Solution

let the displacement of the block at instant of time t be, $x = Acos (ω t + ϕ)$

At t = 0, $x = x_{0}$

$∴ x_{0} = Acos ϕ \dots (I)$

Velocity, $ν = \frac{dx}{dt} = - A ω \sin (ω t + ϕ)$

At t = 0, $ν = - ν_{0}$

$∴ - ν_{0} = - A ω \sin ϕ$

or $Asin ϕ = \frac{ν_{0}}{ω} .$ …. (ii)

squaring and adding (i) and (ii), we get

$\begin{array}{l} A^{2} (\sin^{2} ϕ + \cos^{2} ϕ) = \frac{ν_{0}^{2}}{ω^{2}} + x_{0}^{2} \\ A = \sqrt{\frac{ν_{0}^{2}}{ω^{2}} + x_{0}^{2}} \end{array}$

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