7.3 Gravitational Potential Energy

Gravitational potential energy is the energy an object has because of its position in a gravitational field. Understanding it is essential for solving problems involving falling objects, projectile motion, and energy conservation, which are core topics throughout introductory physics.

Gravitational Potential Energy

Definition of gravitational potential energy

Gravitational potential energy ($PE_g$) is energy stored in an object due to its height above some reference level. The higher an object sits relative to that reference point, the more gravitational potential energy it has.

• Depends on the object's mass and its height above a chosen reference level (often the ground)
• Measured in joules (J)
• It's a scalar quantity, so it has magnitude but no direction
• It's one type of potential energy, which broadly means energy due to an object's position or configuration

The reference level is something you choose. You could set it at the floor, the tabletop, or the bottom of a cliff. What matters physically is the change in height, not the absolute height itself. This means $PE_g$ can even be negative if the object is below your chosen reference point.

Calculation of gravitational potential energy

Near Earth's surface, gravitational potential energy is calculated with:

$PE_g = mgh$

• $m$ = mass of the object (kg)
• $g$ = acceleration due to gravity, approximately $9.8 \text{ m/s}^2$ on Earth
• $h$ = height above the reference level (m)

Example: A 5 kg object sitting 2 m above the ground has:

$PE_g = 5 \text{ kg} \times 9.8 \text{ m/s}^2 \times 2 \text{ m} = 98 \text{ J}$

This equation works well near Earth's surface, where $g$ is roughly constant. For situations far from Earth (like satellites in orbit), you'd need a different formula, but that's beyond the scope of this unit.

Connection to work: When you lift an object, you do work against gravity. That work gets stored as gravitational potential energy. The change in $PE_g$ equals the work done against gravity:

$\Delta PE_g = mg \Delta h$

where $\Delta h$ is the change in height. Notice that only the vertical displacement matters. If you carry a box horizontally across a room at constant height, $\Delta h = 0$ and the gravitational potential energy doesn't change.

Applications of gravitational potential energy

The most powerful application of $PE_g$ is through conservation of mechanical energy. In a system with no non-conservative forces (like friction or air resistance), the total mechanical energy stays constant:

$KE_1 + PE_{g1} = KE_2 + PE_{g2}$

This means gravitational potential energy and kinetic energy trade back and forth:

• A falling object converts $PE_g$ into $KE$, speeding up as it drops
• An object thrown upward converts $KE$ into $PE_g$, slowing down as it rises

How to solve conservation of energy problems:

1. Choose a reference level for height (often the lowest point in the problem).
2. Identify the initial and final states. Write out $KE$ and $PE_g$ for each.
3. Set total energy at the initial state equal to total energy at the final state.
4. Solve for the unknown variable.

Example: An object is dropped from rest at a height of 5 m. Find its speed just before hitting the ground (ignore air resistance).

• Initial state: The object is at rest, so $KE_1 = 0$. Its potential energy is $PE_{g1} = mgh_1$.
• Final state: At ground level, $h = 0$, so $PE_{g2} = 0$. All energy is now kinetic: $KE_2 = \frac{1}{2}mv_2^2$.
• Apply conservation of energy:

$mgh_1 = \frac{1}{2}mv_2^2$

Notice that mass cancels from both sides. Solving for $v_2$:

$v_2 = \sqrt{2gh_1} = \sqrt{2 \times 9.8 \text{ m/s}^2 \times 5 \text{ m}} \approx 9.9 \text{ m/s}$

The fact that mass cancels is worth remembering. It tells you that all objects dropped from the same height (with no air resistance) hit the ground at the same speed, regardless of mass.

Gravitational potential energy and related concepts

• Gravitational force: The force that gives rise to gravitational potential energy. Near Earth's surface, $F_g = mg$.
• Mechanical energy: The sum of kinetic energy and all forms of potential energy in a system. For problems in this unit, that's $KE + PE_g$.
• Conservative force: Gravity is a conservative force, meaning the work it does depends only on the starting and ending positions, not the path taken between them. This is exactly why conservation of energy works for gravitational problems.
• Reference level: Your chosen "zero height." Pick whichever level makes the math simplest for a given problem.