Determining Potential Energy Values, Conservation of Mechanical Energy,?

Question

A block of mass m = 2.7 kg is dropped from height h = 55 cm onto a spring of spring constant k = 1190 N/m (Fig. 8-38). Find the maximum distance the spring is compressed.

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Answer 1

Using algebra, assume that the distance the spring is compressed to be x (m). Thus the block would have a potential drop of mgh = 2.7*g*(x + 0.55) = 26.46(x + 0.55). The spring would store up the spring potential energy of 0.5kx^2. Since mechanical energy is conserved in this case, we have:
26.46(x + 0.55) = 0.5kx^2 = 595 x^2
or: 595 x^2 - 26.46 x - 26.46*0.55 = 0
The solutions of this quadratic equation is x = 0.18 and x = -0.14.
Ignore the negative x, and the solution is that the distance the spring is compressed to be 0.18m, of course under the assumption that this spring can be pressed for 0.18m elastically.