Conservative Forces: Potential Energy

We have now seen two kinds of forces in action. One kind is like gravity. The work done by gravity doesn't depend on the path taken through the gravitational field - it only depends on the relative height of the two endpoints. The other kind is like friction. It not only depends on the path, it is (almost) always negative work, as typically friction dissipates kinetic energy as heat.

We define a conservative force to be one such that the work done by the force as you move a mass from point $x_1$ to point $x_2$ is independent of the path used to move between the points:

In this case (only), the work done going around an arbitrary closed path (starting and ending on the same point) will be identically zero!

$$W_{\rm loop} = \oint_C \vec{F} \cdot d\vec{l} = 0$$ (3.26)

(Note that the two paths from $\vec{x}_1$ to $\vec{x}_2$ combine to form a closed loop C, where the work done going forward along one path is undone coming back along the other.)

Since the work done moving a mass $m$ from an arbitrary starting point to any point in space is the same independent of the path, we can assign each point in space a numerical value: the work done by us on mass $m$, against the force, to reach it. This is the opposite of the work done by the force. We do it with this sign for reasons that will become clear in a moment. We call this field (a field is a function defined at each point in space) the potential energy of the mass $m$ in the conservative force field $\vec{F}$:

$$U(\vec{x}) = - \int_{x_0}^x \vec{F} \cdot d\vec{x} = -W$$ (3.27)

Note Well that only one limit of integration depends on $x$; the other depends on where you choose to make the potential energy zero. This is a free choice. No physics can depend on where you choose the potential energy to be zero. Note that forces only depend on potential energy differences (or gradients, actually):

$$\Delta U = - \int \vec{F}\cdot d\vec{x}$$ (3.28)

$$dU = -\vec{F} \cdot d\vec{x}$$ (3.29)

or

$$\vec{F} = \vec{\nabla}U$$ (3.30)

Why do we bother? If a mass $m$ is moving under the influence of a conservative force, then the work-energy theorem (3.11) looks like:

$$W_C = \Delta E_k$$ (3.31)

or

$$\Delta E_k - W_C = \Delta E_k + \Delta U = 0$$ (3.32)

We can now state the principle of the conservation of mechanical energy: The total mechanical energy (defined as the sum of its potential and kinetic energies) of a system moving in a conservative force field is constant.

$$E_k + U = E_{\rm tot}$$ (3.33)

(with $E_{\rm tot}$ a constant).

We can generalize this one further step by considering what happens if more than one (kind) of force is present (both conservative and nonconservative). In that case the argument above becomes:

$$W_{\rm tot}= W_C + W_{NC} = \Delta E_k$$ (3.34)

or

$$W_{NC} = \Delta E_k - W_C = \Delta E_k + \Delta U$$ (3.35)

which we state as the generalized energy conservation theorem: The work done by all the nonconservative forces acting on a system equals the change in its total mechanical energy.

Note Well that even though energy is ``lost'' to (say) friction in the generalized energy conservation theorem, this energy appears as heat. It turns out that total energy is actually conserved at all times, when one takes into account conservative and nonconservative forces, as well as the work/energy that eventually appears as heat.