Question using potential energy of a spring

Question

I found this question in my textbook:

A spring of negligible mass has force constant k = 800 N>m. (a) How far must the spring be compressed for 1.20 J of potential energy to be stored in it? (b) You place the spring vertically with one end on the floor. You then lay a 1.60-kg book on top of the spring and release the book from rest. Find the maximum distance the spring will be compressed.

The answers to the questions are below:

a) 5.48 cm b) 3.92 cm

I have found the answer to a. I do not know how to find the answer to b without knowing how far the book falls, which is not given. How do I calculate part b?

Answer 1

Before release the book only has a potential gravitational energy with the height (h) being the length of the spring (non compressed). After release the spring compresses by x maximum so the lowest height is now (h-x). At that point the total energy is the spring energy plus the gravitational energy. Conservation of energy gives:

mgh=1/2kx^2 + mg(h-x)

Solve for x:

mgh=1/2kx^2 + mgh - mgx

mgh cancels out. Bring mgx over:

mgx=1/2kx^2

x cancels out:

mg=1/2kx

Plug in given values and you get

x=0.0392m or 3.92cm

Anthon