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I don't understand this problem. I see that to solve this Conservation of Energy and N2L are merged using x, and that h from Ugrav, and a from F=ma are thus related, but why ? I understand the algebra, but how does this merger model the problem ?

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An experimental apparatus with mass m is placed on a vertical spring of negligible mass and pushed down until the spring is compressed a distance x. The apparatus is then released and reaches its maximum height at a distance h above the point where it is released. The apparatus is not attached to the spring, and at its maximum height it is no longer in contact with the spring. The maximum magnitude of acceleration the apparatus can have without being damaged is a, where a > g.

(a) What should the force constant of the spring be?

(b) What distance x must the spring be compressed initially?

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