A block of mass M rests on a table. It is fastened to the lower end of a light, vertical spring. The upper end of the spring is fastened to a block of mass m. The upper block is pushed down by an additional force 3mg, so the spring compression is 4 mg/k. In this configuration, the upper block is released from rest. The spring lifts the lower block off the table. what is the greatest possible value for Mm?

If the spring is just barely able to lift the lower block from the table, the spring lifts it through no noticeable distance, but exerts on the block a force equal to its weight Mg. The extension of the spring, from F→s=kx, must be mg/k Between an initial point at release and a final point when the moving block first comes to rest, we have Ki+Ugi+Usi=Kf+Ugf+Usf0+mg−4mgk+12k4mgk2=0+mgMgk+12kMgk2−4m2g2k+8m2g2k=mMg2k+M2g22k4m2=mM+M22M22+mM−4m2=0M=−m±m2−412−4m2212=−m±9m2Only a positive mass is physical, so we takeM=m(3−1)=2m Hence, Mm=2.