7.1 Potential Energy and Energy Conservation

Understanding Potential Energy and Energy Conservation

Potential Energy and Work

Potential energy is energy stored in a system because of an object's position or configuration. It represents the capacity to do work or convert into kinetic energy.

When you do work against a conservative force, you increase the system's potential energy. That stored energy can later be released and converted back into work. Think of lifting a book off the floor: you do work against gravity, and the book gains gravitational potential energy. Drop it, and that potential energy converts into kinetic energy as it falls.

The total mechanical energy of a system is the sum of its potential and kinetic energy. Two key relationships connect these ideas:

• Change in potential energy: $\Delta U = -W$, where $W$ is the work done by the conservative force. The negative sign means that when a conservative force does positive work on an object (like gravity pulling it downward), the system's potential energy decreases.
• Work-energy theorem: $W_{net} = \Delta K$, meaning the net work done on an object equals its change in kinetic energy.

Conservative vs. Non-Conservative Forces

The distinction between these two types of forces determines whether you can define a potential energy and whether mechanical energy is conserved.

Conservative forces do work that depends only on the starting and ending positions, not the path taken between them. Examples include gravity, the spring force, and the electrostatic force. Because the work is path-independent, you can define a potential energy for each of these forces, and total mechanical energy is conserved.

Non-conservative forces do work that does depend on the path. Friction and air resistance are the classic examples. A longer path means more energy lost to friction. You cannot define a potential energy for these forces, and they cause the total mechanical energy of the system to decrease, typically by converting it into thermal energy (heat).

Applying Energy Conservation Principles

Conservation of Mechanical Energy

When only conservative forces act on a system, total mechanical energy stays constant:

$KE_i + PE_i = KE_f + PE_f$

This is one of the most powerful tools in introductory physics because it lets you relate conditions at two different points without needing to know every detail of the motion in between.

Problem-solving steps:

1. Choose your initial and final states (the two "snapshots" you want to relate).
2. Identify which forms of energy are present at each state (kinetic, gravitational potential, elastic potential).
3. Pick a reference point for zero potential energy (often the lowest point in the problem).
4. Write out the conservation equation with known and unknown quantities.
5. Solve algebraically for the unknown before plugging in numbers.

Common applications: free-fall problems, pendulum motion, roller coaster dynamics, and spring-mass systems. In each case, energy shifts between kinetic and potential forms, but the total remains the same (assuming friction is negligible).

For example, a 2 kg ball dropped from a height of 5 m starts with $U_g = mgh = (2)(9.8)(5) = 98 \text{ J}$ of gravitational potential energy and zero kinetic energy. Just before hitting the ground, all of that has converted to kinetic energy, so $\frac{1}{2}mv^2 = 98 \text{ J}$, giving $v \approx 9.9 \text{ m/s}$.

Calculating Potential Energy

Gravitational potential energy near Earth's surface:

$U_g = mgh$

where $m$ is mass, $g$ is gravitational acceleration ($9.8 \text{ m/s}^2$), and $h$ is height above your chosen reference point. This formula works well for problems near the surface where $g$ is approximately constant.

For objects at much larger distances (satellites, planets), use the general form:

$U_g = -\frac{GMm}{r}$

where $G$ is the gravitational constant, $M$ is the mass of the larger body, and $r$ is the distance between the centers of the two masses. The negative sign reflects the fact that gravitational potential energy is zero at infinite separation and decreases as objects move closer together.

Elastic potential energy (springs):

$U_e = \frac{1}{2}kx^2$

where $k$ is the spring constant and $x$ is the displacement from the spring's equilibrium (natural) position. Because $x$ is squared, the spring stores energy whether it's compressed or stretched.

A note on reference points: Only changes in potential energy are physically meaningful. You're free to set $U = 0$ wherever it's most convenient for the problem. For gravitational PE near Earth, that's usually the ground or the lowest point in the scenario. For springs, it's the natural (unstretched) length. Choosing a smart reference point can simplify your algebra significantly.