I'm not sure I quite get this Conservation of Energy Problem (simple physics)

Factual Questions
1

This is for homework, I’m fairly certain I have the answer, but not entirely certain why.

There’s a block pushed back against a spring n meters (not using the actual numbers since they’re irrelevant), with spring constant k. Then after release, the block travels x meters before stopping. I’m supposed to figure out the coefficient of kinetic friction.

  1. I get that the potential energy of the spring is 1/2 k n^2

  2. Then fx = 1/2 * k * n^2 (where f is the force of friction)

  3. f = (1/2 * k * n^2)/x

  4. (1/2 * k * n^2)/x = mg z (where z is the coefficient, since I can’t make a mu symbol)

  5. Therefore, z = (1/2 * k * n^2)/(mgx)

Okay, I think I did it right, but no matter how hard I try I can’t get from step one to two without just flat out saying “the overall energy is clearly spent over the course of the problem, so I can set them equal since the energy spent by friction is equal to the energy stored/imparted by the spring.”

I don’t get, mathematically, how to get there. It’s clearly conservation of energy, but

E = U + K
0 = 1/2 * k * n ^2 + fx
fx = - 1/2 * k * n^2

Using

E = U + non-conservative work

Yields the same result.

Which is the wrong sign, what am I missing here? How do I get from step 1 to 2? I don’t need to show this, it’s just that I really want to make sure I understand what’s going on before it bites me.

2

“the overall energy is clearly spent over the course of the problem, so I can set them equal since the energy spent by friction is equal to the energy stored/imparted by the spring.” is pretty much equal to 0 = 1/2 * k * n ^2 + fx

The difference is that in the sentence you just compare the absolute values of the energies involved, in the equation you set up a more rigorous mathematical description, one that tells you whether the force f is doing positive or negative work on the system.

3

I’m not totally sure where the confusion is. Is it just about the sign?

I’ll post this in case it helps: (apologies if it doesn’t)

In your step 2 above, you’ve applied the Work-Kinetic Energy Theorem, which says that the net work done on an object equals its change in kinetic energy, or Wnet=ΔKE.

The work done by friction should be equivalent to the change in the block’s kinetic energy, which should be negative:

When the block leaves the spring the initial KE = 1/2kn^2, and the final KE = 0 when it comes to rest.

So the ΔKE = 0 - 1/2kn^2 = -1/2kn^2

And the Wnet = fx

This leads to fx = -1/2kn^2.
Now, the fx side of the equation is also going to be negative when you plug values in because the frictional force is opposite in direction to the displacement x. (in other words, if the displacement is in the positive direction, then the frictional force is acting in the negative direction)

4

What is X - origin, direction ?
In the spring equation, 0=center of ideal spring motion. Positive seems to be toward compression. The weight released will travel in a -x direction.
In the motion/friction equation, x seems to be where the motion starts, in the direction of motion - which would be -x compared to the spring equation?

5

That’s what it is, when I was trying to figure out why my equation seemed to work I was fixating on n^2 not being able to be negative and neglected that f = -zmg, rather than zmg.

Thanks.