# Unclear definition about non-conservation of Energy

We all know that the conservation of energy is written as

$$\Delta U + \Delta K = 0$$

But when we are dealing with systems in which the energy is not conserved (trivial example: the friction), we are in presence of a work done by this non conservative force, which leads us to write

$$\mathcal{L}_{nc} = \Delta E$$

Let's take a problem in which $K_f = U_i = 0$, we now have simply

$$\mathcal{L}_{nc} = U_f - K_i$$

Where "nc" stands for "non conservative".

Now I had to deal with a problem of this kind, in which I have a mass $M$ pushed by a spring which moves a bit straight (no friction here), and then there is an inclined plane with friction $\mu$.

The question is how much far, on to the inclined plane, the body travels.

Taking into account that the body has an initial velocity due to the spring release, $v_i$, and the final velocity must be zero, and the potential energies are $0$ for the initial one and $mgh = mg\ell\sin\theta$ for the final one (where $\theta$ is the angle of inclination, $\ell$ is the traveled distance), we write (according to the definition above)

$$\mathcal{L}_{nc} = \Delta E = U_f - K_i = mg\ell\sin\theta - \frac{1}{2}mv_i^2$$

Now

$$\mathcal{L}_{nc} = F\cdot \ell = mg\ell \mu\cos\theta$$

And due to this, $\ell$ can be calculated.

Now the professor instead wrote on the paper that the correct result is

$$\mathcal{L}_{nc} + mg\ell\sin\theta = \frac{1}{2}mv_i^2$$

I don't understand why the signs are messed up, and basically it's like if he says that

$$\mathcal{L}_{nc} = -\Delta E$$

Can someone please explain me the whole situation?

Thank you so much!

Details can be added, if you need any

$\mathcal{L}_{nc} = U_f+K_f- U_i - K_i$.
$\mathcal{L}_{nc} = - \Delta E$
for a positive $\Delta E$ (the absolute value of the energy loss).