There are the three particles shown in the Fig. 8.16, with \(m=1.00 \mathrm{~kg}\) and \(M=2 m\). The three particles are aligned in the direction of their centers, and no sources of friction are present. The two particles each of mass \(M\) are attached to a spring of negligible mass and elastic constant \(k=200 \mathrm{~N} / \mathrm{m}\), and are at rest. The \(m\) particle is initially thrown toward the left \(M\) particle with a velocity \(v_{0}=6.00 \mathrm{~m} / \mathrm{s}\). Assuming that the collision between \(m\) and \(M\) is completely inelastic, determine:

1. the energy \(\Delta E\) lost in the collision;

2. the velocity of the center of mass \(v_{c m}\) of the system;

3. the maximum compression \(\Delta x\) of the spring after the impact;

4. the pulsation \(\omega\) with which the system oscillates about its center of mass.

Fig. 8.16

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