A block of mass \( m \) is suspended from one end of a light spring as shown. The origin length of the spring from the ceiling and vertical downward direction as positive \( y \)-axis. When the system is in equilibrium, a bullet of mass \( m / 3 \) moving in vertical upward direction with velocity \( v_{0} \) strikes the block and embeds into it. As a result, the block (with bullet embedded into it) moves up and \( m / 3 \) starts oscillating.
Based on the given information, answer the following questions:
The amplitude of oscillation would be
(1) \( \sqrt{\left(\frac{4 m g}{3 k}\right)^{2}+\frac{m v_{0}^{2}}{12 k}} \)
(2) \( \sqrt{\frac{m v_{0}^{2}}{12 k}+\left(\frac{m g}{3 k}\right)^{2}} \)
(3) \( \sqrt{\frac{m v_{0}^{2}}{6 k}+\left(\frac{m g}{k}\right)^{2}} \)
(4) \( \sqrt{\frac{m v_{0}^{2}}{6 k}+\left(\frac{4 m g}{3 k}\right)^{2}} \)
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