Topic 3.3: Potential Energy

What Topic 3.3 requires

Topic 3.3 has one learning objective, 3.3.A: describe the potential energy of a system. Five essential knowledge statements sit under it, and the first three are conceptual rather than computational.

• 3.3.A.1 A system composed of two or more objects has potential energy if the objects within that system only interact with each other through conservative forces.
• 3.3.A.2 Potential energy is a scalar quantity associated with the position of objects within a system.
• 3.3.A.3 The definition of zero potential energy for a given system is a decision made by the observer considering the situation to simplify or otherwise assist in analysis.
• 3.3.A.4 The potential energy of common physical systems can be described using the physical properties of that system, with sub-statements supplying the elastic and gravitational equations.
• 3.3.A.5 The total potential energy of a system containing more than two objects is the sum of the potential energy of each pair of objects within the system.

The suggested skills are 1.C, create qualitative sketches of graphs; 2.C, compare physical quantities between two or more scenarios or at different times and locations in a single scenario; 2.D, predict new values or factors of change using functional dependence; and 3.B, apply an appropriate law, definition, theoretical relationship, or model to make a claim. Skill 2.C fits this topic naturally: once the absolute value of the potential energy is yours to choose, comparing two configurations is what remains to ask about. Unit 3 is weighted at 18 to 23 percent of the multiple-choice section.

Potential energy belongs to a system (3.3.A.1)

Statement 3.3.A.1 puts two conditions on potential energy at once. The system needs two or more objects, and those objects must interact only through conservative forces. Miss either condition and there is no potential energy to speak of.

The consequence is easy to state and easy to forget. A ball held above the ground does not have gravitational potential energy; the ball-and-Earth system has it. The energy lives in the interaction between the two, in the configuration, not in the ball. The CED is unusually direct about this: its Unit 3 overview names whether a single object can have potential energy as one of the misconceptions the unit exists to correct, and Topic 3.4 states the matching positive claim in 3.4.A.1, that a system composed of only a single object can only have kinetic energy.

So write the ball-Earth system stores 20 J of gravitational potential energy, not the ball has 20 J. The arithmetic shortcut of crediting everything to the ball still gives the right numbers, because Earth's enormous mass means it takes a negligible share of the kinetic energy when the configuration changes. But learning objective 3.3.A asks you to describe the potential energy of a system, so the system language is the description being asked for.

The conservative-force condition is the reason friction never gets a potential energy. That comes straight from Topic 3.2, essential knowledge 3.2.A.1.iii: potential energies are associated only with conservative forces.

A scalar tied to position, not to speed (3.3.A.2)

Potential energy is a scalar quantity associated with the position of objects within a system. Split that into its two halves.

Scalar means it behaves like kinetic energy and work: a size with a unit and no direction. You never resolve potential energy into components, and separate contributions add as plain numbers, which is what makes 3.3.A.5 below so simple.

Associated with position is the sharper half. Kinetic energy asks how fast the objects are moving; potential energy asks where they are relative to each other. Change the configuration, meaning the relative positions of the interacting objects, and the potential energy changes. Leave the configuration alone and it does not, no matter what else happens.

This is also why 3.2.A.1.ii holds: the work done by a conservative force on a system, or the change in the potential energy of the system, will be zero if the system returns to its initial configuration. A pendulum bob back at its starting height has the same gravitational potential energy it started with, whatever route it took. Comparing configurations rather than tracking paths is the efficiency that potential energy buys you, and it is the whole reason the conservation of energy method works.

You choose where zero is (3.3.A.3)

The definition of zero potential energy for a given system is a decision made by the observer considering the situation to simplify or otherwise assist in analysis. This is not a technicality; it changes how you read every potential energy number you will ever meet.

A potential energy value on its own is not a measurable fact. A change is. That is why the equation sheet prints the near-surface gravitational relation as $\Delta U_g = mg\Delta y$, a change in potential energy tied to a change in vertical position, rather than as a bare $U = mgh$. Any two observers who pick different zeros will disagree about $U$ and agree exactly about $\Delta U$, and $\Delta U$ is what enters the physics.

Potential energy can be negative. A negative value means only that the configuration sits below the zero you chose. Nothing is broken, and nothing needs fixing.

Choose once, then hold it. Treat the zero exactly like the positive direction on an axis: declare it in your first line and never move it. Two choices tend to save the most algebra. Put $y = 0$ at the lowest point in the problem and every gravitational term stays positive, or put it at the final position and the final term vanishes. The first worked example below runs the same problem three ways to show the change coming out identical every time.

Gravitational potential energy in two forms

Essential knowledge 3.3.A.4 says the potential energy of common physical systems can be described using the physical properties of that system, and gives gravity two equations for two regimes.

The general form, 3.3.A.4.ii, covers a system of two approximately spherical distributions of mass such as moons, planets, or stars. The equation sheet prints it with an uppercase subscript:

Here $r$ is the center-to-center separation. This expression carries its own zero built in: it approaches zero as $r$ grows without limit, which is why it is negative at every finite separation and why moving the two masses farther apart raises it toward zero.

The near-surface form, 3.3.A.4.iii, is an approximation with a stated condition. Because the gravitational field near the surface of a planet is nearly constant, the change in gravitational potential energy in a system consisting of an object of mass $m$ and a planet with gravitational field of magnitude $g$, when the object is near the surface, may be approximated by

with $\Delta y$ the change in vertical position, positive upward. Use this one for ramps, drops, springs, and pendulums, where the height change is tiny next to a planetary radius. Reach for the general form only when $r$ itself changes by an amount comparable to that radius. On the value of $g$: the number printed in the CED's table of information is $g = 9.8 \, \mathrm{m/s^2}$, while a Unit 1 boundary statement reads that for all situations in which a numerical quantity is required for $g$, the value $g \approx 10 \, \mathrm{m/s^2}$ will be used, and that students will not be penalized for correctly using the more precise commonly accepted values of 9.81 or 9.8. Every calculation on this site runs on 9.8, which keeps the pages consistent with each other.

Elastic potential energy of an ideal spring (3.3.A.4.i)

The CED defines the symbol precisely: $\Delta x$ is the distance the spring has been stretched or compressed from its equilibrium length. Measure from the natural length of the spring, never from the floor, the ceiling, or wherever the block happens to sit.

Three properties follow directly from the squared term. The zero is fixed for you here rather than chosen: $U_s = 0$ at the natural length, because that is where $\Delta x = 0$. Elastic potential energy is never negative, since a square cannot be. And stretch and compression by the same distance store exactly the same energy, which is the symmetry that makes a mass on a spring oscillate evenly on both sides, as the simple harmonic motion guide develops.

For scaling questions, note the exponent. Because $U_s$ goes as $(\Delta x)^2$, a spring pulled to three times its original stretch holds nine times the energy, and cutting a compression to one third leaves one ninth of it. That is suggested skill 2.D applied to this topic.

The formula is not arbitrary. An ideal spring resists with a force of magnitude $k\Delta x$, so its force versus stretch graph is a straight line through the origin, and the area under it out to $\Delta x$ is the triangle $\frac{1}{2}(\Delta x)(k\Delta x)$. Topic 3.2's area rule produces Topic 3.3's equation exactly.

Total potential energy adds up pair by pair (3.3.A.5)

The total potential energy of a system containing more than two objects is the sum of the potential energy of each pair of objects within the system. Because potential energy is a scalar, sum means ordinary addition with signs, not vector addition.

The everyday case is a block resting on a spring that stands on the ground. Draw a boundary around block, spring, and Earth and there are two conservative interactions to account for: the gravitational one between block and Earth, worth $\Delta U_g = mg\Delta y$ when the block's height changes, and the elastic one stored in the spring, worth $U_s = \frac{1}{2}k(\Delta x)^2$. Push the block down and one term rises while the other falls. The second worked example below runs those numbers.

Two habits keep the bookkeeping honest. First, list the pairs before you write any numbers, so nothing is double counted and nothing is dropped. Second, remember which interactions get no term at all: friction and air resistance are nonconservative, so under 3.2.A.1.iii they contribute no potential energy. Energy lost to them leaves the mechanical ledger rather than moving between stores inside it, which is the distinction the work-energy theorem guide works through in detail.

Sketching potential energy graphs, and what comes next

Suggested skill 1.C asks for qualitative sketches of graphs that represent the behavior of a physical system, and this topic supplies three distinct shapes worth being able to draw from memory.

**Near-surface gravitational $U_g$ against vertical position $y$** is a straight line of slope $mg$. It crosses zero wherever you decided to put your zero, which makes the intercept a choice and the slope a fact.

**Elastic $U_s$ against $\Delta x$** is an upward parabola with its vertex at the natural length, symmetric in stretch and compression, and never dipping below the horizontal axis.

**General gravitational $U_G$ against separation $r$** is negative everywhere, deepest at small $r$, and rises steadily toward zero as $r$ increases without limit. It never crosses the axis, so a sketch that does is wrong.

From here, Topic 3.4 puts the two stores together: essential knowledge 3.4.B.1 defines mechanical energy as the sum of a system's kinetic and potential energies. Build the habit with the conservation of energy guide, check spring and height numbers against the kinetic energy calculator, which also computes both potential energies, and look up any symbol you do not recognize on the AP Physics 1 formula sheet. The Unit 3 overview shows where the remaining topics go.