## Question on spring constant w/potential energy

Hi all. I'm having trouble understanding a homework problem. Here it is...

1. The problem statement, all variables and given/known data

The question, from my book, is: "When a mass m sits at rest on a spring, the spring is compressed by a distance d from its undeformed length. Suppose instead tht the mass is released from rest when it barely touches the undeformed spring. Find the distance D that the spring is compressed before it is able to stop the mass. Does D = d?"

In my own words: there is a spring attached to the ground, sticking up vertically. A person holds an object at the top of the spring, in contact with the spring but not compressing it at all (Position 1). The person lets go, and the mass drops, compresses the spring, until the spring stops the mass (Position 2). Then it goes back and forth until it settles (Position 3).

2. Relevant equations / The attempt at a solution

The chapter is "Conservation of Energy" so that's the formula I'm using. I start by finding K, Ugrav and Uel at each position:

Position 1:
K = 0 (velocity = 0)
Ugrav = mg * 0 (we choose the undeformed length of the spring as x = 0)
Uel = 1/2 * k * 02 (again, x = 0)

Position 2:
K = 0 (velocity = 0)
Ugrav = -mgD (we choose the maximum compression before the spring stops the object as x = D. D is downward so we use a negative sign. Our goal is to find D in terms of d.)
Uel = 1/2 * k * D2

Position 3:
K = 0 (velocity = 0)
Ugrav = -mgd
Uel = 1/2 * k * d2

So far, so good, right? Now, the total mechanical energy at one position is equal to that at any other, right? I might have the terminology wrong, but what it means is:

E1 = E2 = E3
∴ K1 + U1 = K2 + U2 = K3 + U3
∴ 0 = 1/2 * k * D2 - mgD = 1/2 * k * d2 - mgd

Now, in order to solve for D in terms of d, I need to eliminate the other unknown: k. I do it like this:

E1 = E3
∴ 0 = 1/2 * k * d2 - mgd
∴ k = 2mg / d

BUT

I can also do this:

Since F = ma, and F = kx
then mg = kx
∴ k = mg / d

I have two different possibilities for k.

According to the book, the k = mg / d result is correct; plugging that into the previous formula using E1 = E2 results in

0 = 1/2 * k * D2 - mgD
mgD = mgD2 / 2d
D = D2 / 2d
2d = D