Potential Energy & Conservation of Energy

In the last chapter, work transferred energy to or from kinetic energy. But when you lift a book and hold it over your head, the energy you put in does not vanish — it sits waiting, ready to reappear as kinetic energy the moment the book falls. This stored energy is called potential energy $U$. Technically, potential energy is energy that can be associated with the configuration of a system of objects that exert forces on one another.

Potential energy is only meaningful for conservative forces. A conservative force is one for which the net work done on a particle traveling any closed path is zero — equivalently, the work it does between two points depends only on those endpoints, not on the route taken. The gravitational force and the spring force are conservative. Kinetic friction is not: slide a book across a table and back, and friction has removed energy both ways; there is no stored "friction potential energy."

Defining Potential Energy: ΔU = −W

When a conservative force does work $W$ on an object, the potential energy of the system changes by:

The minus sign is the key: when the conservative force does positive work (force and displacement in the same direction), the potential energy decreases. When gravity pulls a ball downward (positive work), gravitational PE decreases — that energy moves into kinetic energy. When you lift the ball back up (gravity does negative work), gravitational PE increases — you are storing energy back in the system. It's an exact energy accounting: every joule the conservative force transfers from KE appears as potential energy, and vice versa.

For a force that varies with position, we integrate: $\Delta U = -\int_{x_i}^{x_f} F(x)\,dx$.

Gravitational Potential Energy: U(y) = mgy

Applying $\Delta U = -W_g$ for gravity ($F_g = -mg\hat{j}$, always downward) gives $\Delta U = mg(y_f - y_i)$. Setting the reference at $y_i = 0$ with $U_i = 0$:

Elastic Potential Energy: U(x) = ½kx²

Applying $\Delta U = -W_s$ for the spring force ($F_s = -kx$) gives $\Delta U = \frac{1}{2}kx_f^2 - \frac{1}{2}kx_i^2$. Setting $U = 0$ when the spring is at its relaxed length ($x = 0$):

Because $x^2 \geq 0$, elastic PE is always non-negative — it costs energy to compress or stretch a spring from its natural length in either direction. The parabolic shape $U = \frac{1}{2}kx^2$ is one of the most important curves in all of physics; it describes molecular bonds, electromagnetic resonance, and every oscillating system studied in this course.

We now have two forms of energy — kinetic $K$ and potential $U$ — that can trade with each other via conservative forces. Their sum is called the mechanical energy of the system:

Consider an isolated system in which only conservative forces do work (no friction, no drag, no external forces). When the conservative force does work $W$ on the object, $\Delta K = W$ (work–KE theorem) and $\Delta U = -W$ (potential energy definition). Adding these:

In other words, any increase in kinetic energy is exactly matched by a decrease in potential energy. The total mechanical energy does not change. This is the principle of conservation of mechanical energy:

Or equivalently: $\Delta E_\text{mec} = \Delta K + \Delta U = 0$.

The Pendulum: Energy in Continuous Conversion

A swinging pendulum is the perfect illustration of mechanical energy conservation. At the lowest point (bob moving fastest), all energy is kinetic: $U = 0$, $K = E_\text{mec}$. At the highest point (bob momentarily stopped), all energy is potential: $K = 0$, $U = E_\text{mec}$. At any intermediate point, $K + U = E_\text{mec}$ exactly. The energy sloshes continuously between the two forms, but the total never changes (assuming no air resistance or pivot friction).

Solving Problems with Conservation of Mechanical Energy

The standard approach, step by step:

1. Define the system and confirm only conservative forces do work (normal force is perpendicular to motion → no work; gravity and springs → conservative).
2. Choose a reference level for gravitational PE (often the lowest point or initial position).
3. Write $K_1 + U_1 = K_2 + U_2$, expanding each term with $K = \frac{1}{2}mv^2$, $U_g = mgy$, $U_s = \frac{1}{2}kx^2$.
4. Cancel mass if it appears on both sides (it often does for purely gravitational problems).
5. Solve for the unknown.

Classic result — free-fall equivalent: An object released from rest at height $h$ hits the ground with speed $v = \sqrt{2gh}$, regardless of whether it falls straight down, slides down a frictionless ramp, or rolls down a curved track. The shape of the path does not matter when there is no friction. This is why energy methods are so powerful.

If we know a system's potential energy function $U(x)$, we can extract everything about the force and the motion by reading a graph. This is one of the most useful analytical tools in classical mechanics.

Force from the Potential Energy Curve

Starting from $\Delta U = -F(x)\,\Delta x$ and passing to the differential limit:

The force is the negative slope of the $U(x)$ curve. Where $U$ is steeply declining (slope strongly negative), the force is large and positive (pushes in the $+x$ direction). Where $U$ is steeply rising, the force is large and negative. Where the slope is zero (a maximum or minimum of $U$), the force is zero — that's an equilibrium point.

You can verify this against known cases: for a spring, $U = \frac{1}{2}kx^2$, so $F = -d(\frac{1}{2}kx^2)/dx = -kx$ — Hooke's law. For gravity near Earth, $U = mgy$, so $F = -d(mgy)/dy = -mg$ — correct, pointing downward.

Kinetic Energy on the Graph

At any position $x$, the particle's kinetic energy is:

On a $U(x)$ graph, draw a horizontal line at the value of $E_\text{mec}$. The vertical gap between this line and the $U(x)$ curve at any position $x$ is the kinetic energy there. Where the curve is far below the $E_\text{mec}$ line, the particle moves fast. Where the curve touches or crosses the line, $K = 0$.

Turning Points

A turning point is a position $x$ where $K = 0$ — the particle momentarily stops and reverses direction. This happens where $U(x) = E_\text{mec}$, i.e., where the $U(x)$ curve intersects the horizontal $E_\text{mec}$ line. The particle cannot exist in regions where $U(x) > E_\text{mec}$ — that would require negative kinetic energy, which is impossible (since $K = \frac{1}{2}mv^2 \geq 0$).

Equilibrium: Stable, Unstable, and Neutral

At any point where $dU/dx = 0$, the force is zero — the particle is in equilibrium. But not all equilibria are alike:

• Stable equilibrium — a minimum of $U(x)$. If displaced, the force points back toward the equilibrium. Example: a marble at the bottom of a bowl. The system naturally returns.
• Unstable equilibrium — a maximum of $U(x)$. If displaced even slightly, the force pushes away from equilibrium. Example: a marble balanced on top of a ball. The smallest nudge sends it rolling away.
• Neutral equilibrium — a flat region of $U(x)$ where $dU/dx = 0$ everywhere in that region. The force is zero everywhere, and the particle stays wherever you place it. Example: a marble on a flat table.

So far we have examined systems in which only internal conservative forces act. Now we add external forces — forces from outside the system — and ask how they change the system's energy. We also face the crucial new element of friction, which converts mechanical energy to thermal energy.

Case 1: External Force, No Friction

Suppose you slowly lift a bowling ball from the floor to a shelf. The gravitational force is internal to the ball–Earth system; your lifting force is external. The work $W$ your hands do on the system equals the change in the system's mechanical energy:

If you lift slowly (so $\Delta K \approx 0$), all your work goes into gravitational PE: $W = \Delta U = mgh$. If you lift it fast, some of your work also goes into kinetic energy. The equation captures both.

Case 2: External Force with Friction — Thermal Energy

Now suppose a friction force $f_k$ acts as an object slides a distance $d$ across a surface. By experiment (and derivable from Newton's second law), the thermal energy $E_\text{th}$ generated is:

Thermal energy is the energy associated with the random microscopic motion of atoms and molecules. When two surfaces slide, the microscopic "cold welding" bonds between them are repeatedly torn and re-formed, shaking atoms into faster vibration — warming both surfaces. This energy is real energy, but it is no longer organized mechanical energy; it cannot be recovered as work without a heat engine.

With friction, the full energy equation for external work on a system becomes:

This equation is a complete energy ledger. Applied work $W$ arrives in the system; it is split between mechanical energy (organized, can do more work) and thermal energy (disorganized, escapes as heat). Not one joule is created or destroyed — it is simply redistributed. Notice that $\Delta E_\text{th} = f_k d > 0$ always: friction always generates heat, never absorbs it.

We now arrive at one of the most profound experimental facts in all of science: energy is conserved. This is not a derived theorem — it is a law based on centuries of experimental evidence, and no exception has ever been found. Every form of energy (mechanical, thermal, electrical, chemical, nuclear, electromagnetic, rest-mass) obeys it.

The Complete Energy Equation

When external work $W$ is done on a system, it accounts for all energy changes:

where $\Delta E_\text{mec} = \Delta K + \Delta U$ is the change in mechanical energy, $\Delta E_\text{th}$ is the change in thermal energy (from friction/drag), and $\Delta E_\text{int}$ accounts for any other internal energy changes (chemical energy in muscles, nuclear energy, etc.).

Isolated Systems

When no external force does work ($W = 0$), the system is isolated and the total energy cannot change:

Or equivalently, comparing two instants:

This says: the final mechanical energy equals the initial mechanical energy minus whatever was stolen away by friction and other internal processes. The mechanical energy can decrease, but only if something else increased by the same amount.

In the special case where no nonconservative forces act ($\Delta E_\text{th} = 0$, $\Delta E_\text{int} = 0$), this reduces to conservation of mechanical energy: $E_\text{mec,2} = E_\text{mec,1}$. That result from Section 8.2 is thus a special case of this more general law.

Why This Law Matters So Much

Conservation of energy is arguably the most important principle in physics because:

• It is universal. It holds for all known forces, from the subatomic to the cosmological scale.
• It constrains what is possible. Any process that would violate energy conservation is impossible — no exceptions. This is why perpetual motion machines cannot exist.
• It makes problems tractable. You can solve problems by comparing initial and final energy totals, bypassing the complicated intermediate dynamics entirely (no need to know the shape of a ramp, the forces at each instant, or the trajectory).
• It underlies all of thermodynamics. The First Law of Thermodynamics is simply energy conservation applied to heat and work at the macroscopic level.
• It is connected to a deep symmetry. Noether's theorem (1915) proves that energy conservation is a direct consequence of the fact that the laws of physics are the same today as they were yesterday — time translation symmetry. If the laws of physics never changed, energy is always conserved.