Conservative Forces

67 Conservative Forces

In this chapter we will take a closer look at the conservation of energy equation. To start with, let’s generalize the derivations we did last chapter.

Exercise 52.1: Conservative Forces

Consider a force $F(x)$ that depends only on position, and choose an arbitrary reference point $x_0$ . We define the potential energy $U(x)$ via

$U(x) \equiv - \int_{x_0}^x F(x')dx'.$

A. Show that the work done by the function $F$ in taking a particle from $x_{\rm i}$ to $x_{\rm f}$ can be written as $W=-\Delta U = - (U_{\rm f}-U_{\rm i})$ .

B. Plug the above expression into the work–energy theorem to prove that energy is conserved for objects subject to this force $F(x)$ .

There are two take aways here. First, in 1D, if a force depends only on position, then this force conserved energy. Second, the potential energy of this force is given by

$U(x) = - \int_{x_0}^x F(x')dx'.$

Notice that the potential energy depends on the reference point $x_0$ : if we choose a different reference point, the potential energy is different. But- that’s something we already know. For instance, for gravity, we can set $y=0$ wherever we want: that’s our reference point.

67 Conservative Forces

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