The Concept of Potential Energy

The Idea of Potential Energy

When you lift a book from the floor to a shelf, you do work against gravity. That work does not vanish — it gets stored in the book-earth system as potential energy. The book now has the capacity to do work: if you let it fall, it can drive a nail or break a tile. This stored energy depends only on the position or configuration of the system, not on how it got there.

Potential energy is defined only for conservative forces — forces for which the work done in moving an object between two points is independent of the path taken. Gravity, the spring force, and electrostatic forces are conservative. Friction is not.

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Change in Potential Energy and Work Done

Consider a particle moving from point A to point B under a conservative force $\vec{F}$. The change in the particle's potential energy, $\Delta U = U_B - U_A$, is defined as the negative of the work done by the conservative force:

where $W_{AB} = \int_A^B \vec{F} \cdot d\vec{r}$ is the work done by the conservative force along the path.

Why the minus sign? If you lift a book slowly (so kinetic energy doesn't change), you apply an external force $\vec{F}_{\text{ext}}$ that exactly balances gravity: $\vec{F}_{\text{ext}} = -\vec{F}_g$. The work you do, $W_{\text{ext}}$, goes into increasing the potential energy. Since the net work on the book is zero (no change in kinetic energy), we have $W_{\text{ext}} + W_g = 0$, so $W_{\text{ext}} = -W_g = \Delta U$. The potential energy change equals the work done against the conservative force.

For an infinitesimal displacement $d\vec{r}$, the change in potential energy is

which gives the relation between force and potential energy in one dimension:

In three dimensions, this becomes the gradient:

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Gravitational Potential Energy Near the Earth's Surface

For a body of mass $m$ near the earth's surface, the gravitational force is approximately constant: $\vec{F}_g = -mg\,\hat{j}$ (taking upward as positive $y$). The work done by gravity when the body moves from height $y_A$ to $y_B$ is

The change in gravitational potential energy is therefore

If we choose the reference point $U=0$ at $y=0$ (say, the ground), then at height $h$:

This is the familiar formula. The reference point is arbitrary — only differences in potential energy matter physically.

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Elastic Potential Energy of a Spring

Consider an ideal spring obeying Hooke's law: $F_s = -kx$, where $x$ is the displacement from the natural length and $k$ is the spring constant. The negative sign indicates the force opposes the displacement.

The work done by the spring force when the spring is stretched from $x=0$ to $x$ is

The change in elastic potential energy is $\Delta U = -W_s = \frac{1}{2}kx^2$. Taking $U=0$ at the equilibrium position ($x=0$), we get

This is the energy stored in a compressed or stretched spring.

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Properties of Potential Energy (Derived from the Definition)

The textbook lists three key properties that follow directly from the definition $\Delta U = -W$. Each is proved below.

›Proof

Property (I): Potential energy is defined only for conservative forces.

If a force is non-conservative (like friction), the work done depends on the path taken. Then $W_{AB}$ is not a single-valued function of the endpoints A and B — it changes if you take a different route. Since $\Delta U$ must depend only on the initial and final positions (it is a state function), we cannot assign a unique potential energy to a configuration. Therefore, potential energy exists only for conservative forces. …