The work-energy theorem can be proven directly from Newton's 2nd law, without any reference to conservative or nonconservative forces.

The relations between conservtive forces and their potential energy (and in fact,
the existence of a scalar function satisfying these relations) is an entirely mathematical theorem. See here, here, and here.

After both of these results have been proven, one can separate the total work appearing in the work-energy theorem into the conservative and nonconservative parts. Since the conservative work is minus the difference in potential energy one can move it to the other side of the equation and get a positive difference in the total mechanical energy. In fact, the potential energy is defined so that the work equals the negative difference in the potential energy exactly because we want to get the difference in the mechanical energy (and not the difference in kinetic energy minus the difference in potential energy). That way, when the nonconservative forces do $0$ work we get conservation of energy.

It just comes from the definitions of work, kinetic energy, potential energy, and mechanical energy.

The work done by a force is defined by
$$W=\int \mathbf F\cdot\text d \mathbf x$$

The work done by all forces is given by$^*$
$$W_{net}=\int \mathbf F_{net}\cdot\text d \mathbf x=\Delta K$$

where $\Delta K$ is the change in kinetic energy. But we can also break our net force up into a sum of conservative and non-conservative forces so that
$$W_{net}=\int (\mathbf F_{c}+\mathbf F_{nc})\cdot\text d \mathbf x=W_c+W_{nc}=\Delta K$$

Furthermore, by definition of potential energy, $W_c=-\Delta U$ ,therefore$^{**}$
$$W_c+W_{nc}=-\Delta U+W_{nc}=\Delta K$$
or
$$W_{nc}=\Delta K+\Delta U$$

Mechanical energy is defined as the sum of kinetic and potential energies:
$$E=K+U$$
so that
$$\Delta E=\Delta K+\Delta U$$

Therefore we get to what we want
$$W_{nc}=\Delta E$$

This doesn't depend on assumptions of mass. If the particles' masses are changing, that mass must be coming from or going to somewhere, and so there must be interactions taking place through either conservative or non-conservative forces. Therefore, this derivation covers that scenario.

Note that this also applies to systems of particles as well, since you can apply this analysis to each particle individually, then add up everything for the entire system.

$^*$ To show why $W_{net}=\Delta K$: By Newton's second law, $\mathbf F_{net}=m\mathbf a$, and we can also treat the acceleration as constant during the interval $\text d \mathbf x$ that we integrate over. Therefore the relation $|\mathbf v+\text d\mathbf v|^2=|\mathbf v|^2+2\mathbf a\cdot\text d\mathbf x$ holds.

Now,
$$|\mathbf v+\text d\mathbf v|^2=(\mathbf v+\text d\mathbf v)\cdot(\mathbf v+\text d\mathbf v)=|\mathbf v|^2+2\mathbf v\cdot\text d\mathbf v+|\text d\mathbf v|^2$$

Since $|\text d\mathbf v|^2\approx0$, then we end up with
$$\mathbf v\cdot\text d\mathbf v=\mathbf a\cdot\text d\mathbf x$$

Therefore, our work integral becomes
$$W_{net}=\int m\mathbf a\cdot\text d \mathbf x=\int \mathbf v\cdot\text d\mathbf v=\frac12m\Delta|\mathbf v|^2=\Delta K$$

$^{**}$ Potential energy is defined through its relation to a conservative force:
$$\mathbf F_c=-\mathbf\nabla U$$
therefore,
$$W_c=-\int \mathbf\nabla U\cdot\text d\mathbf x=-\Delta U$$

## Best Answer

Mechanical energy is a term that covers kinetic energy $K$ and potential energies $U$ (gravitational $U_g$, elastic $U_{el}$ etc.).

A swinging pendulum is an example. When swinging downwards, gravitational potential energy $U_g$ is converted into kinetic energy $K$. When swinging upwards, the opposite takes place. Those energy forms simply transform into one another and never leave the system.

But doesn't gravity do work in this system?

Yes, it does. Gravity is a conservative force. The work done by conservative forces is what we callpotential energies. So gravity's work isalready includedin this consideration.The reason for this "renaming" is that when a conservative force does work, that work is

stored. The system can return to its original state again by releasing that energy. In other words, work done by a conservative forcedoesn't leave the system. Only work done by non-conservative forces disappears and leaves the system, causing the total energy to reduce.