Potential Energy (Gravitational and Elastic)

Kinetic energy is the energy a body has because it is moving. Potential energy is the energy a system has because of the arrangement of its parts — a book on a high shelf, a drawn bow, two charges held apart. It is stored energy, waiting to be released, and the whole of the next chapter's machinery depends on defining it precisely here.

The definition rests on a fact established in the work chapter: some forces do path-independent work. For such a conservative force, the work done against it in moving from one configuration to another depends only on the endpoints, so we may attach a number to each configuration and call the difference the change in potential energy. For friction no such number exists, because the work depends on how you got there. Potential energy is therefore a concept available only for conservative forces — a restriction, not a technicality, and the reason the phrase "friction potential energy" is meaningless.

The general definition. The change in potential energy of a system equals the negative of the work done by the conservative force:

$$\Delta E_p = -W_{\text{conservative}}$$

Equivalently, it is the work done against that force by an external agent moving the system slowly.

1. Why the Minus Sign

The minus sign is the most misunderstood feature of the definition, and it encodes something simple. When a conservative force does positive work — gravity acting on a falling ball — the system's stored energy decreases, because that energy is being converted into kinetic energy. When you do work against the force, lifting the ball back up, the stored energy increases.

Potential energy is thus a kind of ledger of work owed. Raise a mass and you have deposited $mgh$; release it and the field withdraws exactly that amount, delivering it as kinetic energy. The bookkeeping only works because the force is conservative: the deposit and the withdrawal are guaranteed equal regardless of the route, which is exactly the property friction lacks.

Note also that potential energy belongs to a system, not to a body. Strictly it is the Earth–ball system that stores gravitational potential energy, since the force is an interaction between the two. In practice, because the Earth's motion is negligible, we speak loosely of "the potential energy of the ball" — and IB accepts this, but the underlying statement is about the pair.

2. Gravitational Potential Energy Near the Earth's Surface

Close to the ground, the gravitational field is uniform: $g$ is constant in both magnitude and direction. Lifting a body of mass $m$ through a vertical height $h$ at constant speed requires an applied force $mg$ acting through a displacement $h$, so the work done against gravity is

Three points govern its use.

Only changes are physical. The equation gives a difference, not an absolute value, so a reference level must be chosen — the ground, the table top, the lowest point of the swing. The choice is entirely free, because it cancels from every physical prediction; what matters is only that it is stated and then kept fixed throughout the problem.

Below the reference level, $E_p$ is negative. This is not an error but a consequence of the freedom above: a mine shaft $50~\text{m}$ below ground level has $E_p = -mg(50)$ relative to the surface.

Only the vertical height counts. A body carried $100~\text{m}$ horizontally gains no gravitational potential energy. On a slope of length $d$ at angle $\theta$, the relevant height is $h = d\sin\theta$ — the same path-independence established in the work chapter.

The formula $\Delta E_p = mgh$ assumes $g$ is constant, which fails over distances comparable to the Earth's radius. The general expression for the gravitational potential energy of two masses, $E_p = -\dfrac{GMm}{r}$, is developed later in this block; near the surface it reduces to $mgh$ for small height changes.

3. Elastic Potential Energy

A stretched or compressed spring stores energy too. From the work chapter, the work done in stretching a Hookean spring by an extension $x$ is the triangular area under its force–extension graph:

Equivalent forms, obtained by substituting $F = kx$, are sometimes more convenient:

Four features are worth noting.

It depends on $x^{2}$, so doubling the extension quadruples the stored energy. Stretching from $x$ to $2x$ requires three times as much additional work as the first stretch did.

It is always positive, whether the spring is stretched or compressed, since $x$ is squared. Both deformations store energy.

The reference is not arbitrary here. Unlike gravitational potential energy, the natural length of the spring provides a physically meaningful zero, so $E_p$ has an absolute value.

Beyond the limit of proportionality the formula fails, but the underlying principle does not: the stored energy is always the area under the force–extension graph, found by counting squares where no formula applies. In a material such as rubber, the loading and unloading curves differ even though the band returns to its natural length, and the area between them — the hysteresis loop — is energy dissipated as heat rather than recovered.

4. Force as the Gradient of Potential Energy

The definition $\Delta E_p = -W$ can be inverted. For a small displacement $\Delta x$ along the direction of the force, $W = F\Delta x$, so

The conservative force is the negative gradient of the potential energy. Two checks confirm the sign and the sense.

For gravity, $E_p = mgh$, so $F = -\dfrac{d(mgh)}{dh} = -mg$: a force of magnitude $mg$ directed downward, i.e. toward decreasing $E_p$. For a spring, $E_p = \tfrac{1}{2}kx^{2}$, so $F = -kx$ — Hooke's law recovered, minus sign included.

The general principle is that a body accelerates toward lower potential energy, at a rate set by how steeply the potential energy falls. This gives an immediately useful way of reading a potential energy–position graph:

• The gradient gives the force (with a sign reversal): steep means a large force.
• A minimum of $E_p$ has zero gradient, so it is an equilibrium; and since the force pushes back toward it from either side, it is a stable equilibrium — the configuration a system settles into.
• A maximum is also an equilibrium but an unstable one: any displacement produces a force pushing the body further away.
• A flat region means no force at all — neutral equilibrium.

This picture reappears throughout physics: the shape of a molecular potential well explains bonding, and the minimum of an interatomic potential curve is the equilibrium bond length.

5. Worked Examples

Example 1 — energy on a slope. A crate of mass $45~\text{kg}$ is pushed $16~\text{m}$ up a $22^\circ$ ramp. Find the gain in gravitational potential energy.

Only the vertical rise matters:

The distance along the ramp, and the presence or absence of friction, are both irrelevant to this quantity — friction affects how much work the pusher must do, not how much potential energy the crate gains.

Example 2 — comparing two stores. A spring of constant $250~\dfrac{\text{N}}{\text{m}}$ is compressed by $18~\text{cm}$ and used to launch a $0.40~\text{kg}$ ball vertically. Neglecting air resistance and the spring's own mass, how high does the ball rise above the launch point?

The elastic energy stored is

All of it becomes gravitational potential energy at the highest point, where the ball is momentarily at rest:

Notice that the two potential energies were converted directly into one another with no reference to speed, force or time. That conversion is the subject of the next chapter.

6. Common Exam Pitfalls

• Failing to state the reference level. $E_p$ has no absolute value in the uniform-field case; every answer is relative to a stated zero.
• Using the distance along a slope instead of the vertical height. Only $h = d\sin\theta$ contributes.
• Treating a negative $E_p$ as an error. Below the reference level it is negative, and that is meaningful.
• Using $E_p = \tfrac{1}{2}kx^{2}$ with $x$ as the spring's total length. $x$ is the extension or compression from the natural length.
• Forgetting that stored elastic energy goes as $x^{2}$. Doubling the compression quadruples the energy.
• Assigning potential energy to friction. Only conservative forces have a potential energy, because only they do path-independent work.
• Dropping the minus sign in $F = -\dfrac{dE_p}{dx}$, and so predicting a force pushing a system away from its stable minimum.
• Using $mgh$ over astronomical distances. It assumes a uniform field; use $-\dfrac{GMm}{r}$ when $g$ varies appreciably.

Summary

• Potential energy is energy stored in the configuration of a system, defined only for conservative forces by $\Delta E_p = -W_{\text{conservative}}$ — the work done against the force.
• The minus sign means that when the force does positive work, the stored energy falls; potential energy strictly belongs to the system, not to one body.
• Gravitational potential energy near the surface is $\Delta E_p = mgh$, depending only on the vertical height change; it requires a stated reference level, relative to which it may be negative.
• Elastic potential energy is $E_p = \tfrac{1}{2}kx^{2} = \dfrac{F^{2}}{2k}$, always positive, measured from the spring's natural length, and equal to the area under the force–extension graph even where Hooke's law fails.
• Since $E_p \propto x^{2}$, doubling a spring's extension quadruples the stored energy.
• A conservative force is the negative gradient of the potential energy, $F = -\dfrac{dE_p}{dx}$, so bodies accelerate toward lower potential energy.
• On an $E_p$–position graph, a minimum is a stable equilibrium, a maximum is unstable, and the gradient gives the force.
• $mgh$ assumes a uniform field and fails over distances comparable to the Earth's radius, where $E_p = -\dfrac{GMm}{r}$ is required.