12.2 Gravitational Potential Energy

Gravitational Potential Energy Fundamentals

Gravitational potential energy (GPE) is the energy an object stores because of its position in a gravitational field. Understanding GPE lets you track how energy shifts between stored and moving forms, which is the key to solving problems involving free fall, pendulums, roller coasters, and projectile motion.

The Core Concept

GPE is always measured relative to a chosen reference point. That reference point is where you define $h = 0$, and it's up to you to pick it. Usually you'll choose the ground or the lowest point in the problem, but the physics works no matter where you set it.

The relationship between gravity's work and potential energy is:

$W_{\text{gravity}} = -\Delta U_g$

This negative sign matters. When an object falls, gravity does positive work on it, so GPE decreases. When an object rises, gravity does negative work, so GPE increases. The energy doesn't disappear; it converts between kinetic and potential forms.

Deriving the Expression

Here's how we get the standard GPE formula for objects near Earth's surface:

1. Near Earth's surface, the gravitational force on an object is $F_g = mg$, where $m$ is mass and $g \approx 9.8 \, \text{m/s}^2$.
2. If the object moves through a vertical displacement $d$, gravity does work $W = mgd$.
3. Since $\Delta U_g = -W$, we get $\Delta U_g = -mgd$.
4. Choosing a reference point where $U_g = 0$ at $h = 0$, the gravitational potential energy at height $h$ is:

$U_g = mgh$

This formula only works near Earth's surface, where $g$ is approximately constant. For objects far from Earth (like satellites), you'd need a different expression covered later in this unit.

Applications and Problem Solving

Solving Near-Earth GPE Problems

Every GPE calculation uses three variables:

• $m$ : mass of the object (kg)
• $g$ : gravitational acceleration ($9.8 \, \text{m/s}^2$)
• $h$ : height above your chosen reference point (m)

The output is energy in joules (J). A few common problem types you'll see:

• Finding GPE at a given height: A 2.0 kg book sits on a shelf 1.5 m above the floor. Its GPE relative to the floor is $U_g = (2.0)(9.8)(1.5) = 29.4 \, \text{J}$.
• Finding the change in GPE: If that book is moved to a shelf 3.0 m high, $\Delta U_g = mg\Delta h = (2.0)(9.8)(1.5) = 29.4 \, \text{J}$. The GPE increased because the book moved higher.
• Finding height from a known GPE: Rearrange to $h = \frac{U_g}{mg}$.

Watch your signs. If an object moves downward, $\Delta h$ is negative, and $\Delta U_g$ is negative too.

Mechanical Energy Conservation

When only gravity does work (no friction, no air resistance), the total mechanical energy stays constant:

$E_{\text{mech}} = KE + U_g = \text{constant}$

This means any GPE an object loses gets converted into kinetic energy, and vice versa. To use this in a problem:

1. Pick your reference point for $h = 0$.
2. Write the total mechanical energy at the starting position: $\frac{1}{2}mv_1^2 + mgh_1$.
3. Set it equal to the total mechanical energy at the final position: $\frac{1}{2}mv_2^2 + mgh_2$.
4. Solve for the unknown.

Where this shows up:

• Free fall: An object dropped from rest converts all its GPE into KE. At the bottom, $v = \sqrt{2gh}$.
• Pendulums: At the highest point, energy is all GPE. At the lowest point, energy is all KE. The pendulum swings back and forth converting between the two.
• Roller coasters: The car's speed at any point on the track can be found by comparing its height to the starting height.

When non-conservative forces like friction or air resistance are present, mechanical energy is not conserved. These forces convert some mechanical energy into thermal energy, so the object ends up with less total $KE + U_g$ than it started with. You can account for this by including a work-done-by-friction term: $E_{\text{mech,initial}} + W_{\text{non-conservative}} = E_{\text{mech,final}}$.