Topic 3.3 Potential Energy: define potential energy for conservative forces, relate force and potential energy by $F = -dU/dx$, and use gravitational and elastic potential energy, including the general gravitational form.

What this topic is asking

The College Board (Topic 3.3) wants you to define potential energy for conservative forces, to move between a force and its potential energy using $F = -\dfrac{dU}{dx}$, and to apply gravitational and elastic potential energy, including the general gravitational form $-GMm/r$. The calculus relationship between force and potential energy, and reading stability from a potential-energy curve, are distinctive AP Physics C expectations.

Conservative forces and potential energy

Gravity and the spring force are conservative, so each has a potential energy; friction and drag are non-conservative (path-dependent, dissipating energy as heat), so no potential energy can be defined for them. The minus sign in $\Delta U = -W$ encodes that when a conservative force does positive work (gravity pulling an object down), the potential energy decreases, and that energy reappears as kinetic energy. This packaging of conservative work into stored energy is what makes energy conservation so convenient.

Force from potential energy

The relationship between a one-dimensional conservative force and its potential energy is a calculus statement:

The force is the negative slope of the potential-energy curve. Where $U$ decreases with $x$ (negative slope), the force is positive (pushes toward increasing $x$); the force always points toward lower potential energy, "downhill" on the $U$ curve. This lets you extract the force from a given $U(x)$ by differentiating, or build $U$ from a known $F(x)$ by integrating (choosing a reference point that sets the constant). It is a two-way calculus tool the exam tests directly.

The standard potential energies

Three forms recur. Gravitational PE near a surface, where $g$ is roughly constant, is $U = mgh$ (with $h$ measured from a chosen reference). The general gravitational PE of two masses, with zero at infinite separation, is

which is negative because gravity is attractive; you must do positive work to pull the masses apart. Elastic PE of a spring is $U = \tfrac{1}{2}kx^2$, the integral of the spring force. Each follows from $U = -\int F\,dx$ applied to the corresponding force law, and each has a freely chosen zero point (only differences in $U$ are physical).

Reading a potential-energy curve

A $U(x)$ graph is a compact summary of the motion. The force at any point is the negative slope; the kinetic energy at any point is the difference between the total energy (a horizontal line) and $U(x)$; turning points are where the energy line meets the curve (where $K = 0$). Wells in the curve trap the particle in oscillation; barriers it cannot surmount limit its range. This graphical analysis, combining force, energy and stability, is a hallmark AP Physics C skill.

Try this

Q1. A conservative force gives $U(x) = 5x^2$ (SI units). Calculate the force at $x = 2.0$ m. [2 points]

• Cue. $F = -dU/dx = -10x = -10(2.0) = -20$ N (toward the origin).

Q2. State the gravitational potential energy of a satellite-Earth system and explain its sign (reference at infinity). [2 points]

• Cue. $U = -GMm/r$; it is negative because gravity is attractive and the reference (zero) is set at infinite separation.