KAHIBARO
Table of Contents
Meaning and Physical Idea
Gravitational potential energy is the energy associated with the position of an object in a gravitational field. It tells us how much energy is stored because of where the object is, not because of how fast it is moving.
Near Earth, when you lift an object upward, you must do work against gravity. That work is stored as gravitational potential energy. If the object is later allowed to fall, that stored energy can change into kinetic energy.
This idea is specific to gravity as a conservative force. The broader meaning of potential energy and conservation ideas belong to nearby chapters. Here we focus on the gravitational case.
Gravitational Potential Energy Near Earth's Surface
Close to Earth's surface, the gravitational force on an object of mass $m$ is approximately constant and equal to its weight:
$$F_g = mg$$
where $g \approx 9.8\ \text{m/s}^2$.
If an object is moved vertically upward by a height $\Delta h$, the increase in gravitational potential energy is
$$\Delta U = mg\Delta h$$
If we choose the potential energy to be zero at some reference height, then the gravitational potential energy at height $h$ is commonly written as
$$U = mgh$$
This formula works well when the height change is small compared with Earth's radius.
Important near Earth's surface:
$$\Delta U = mg\Delta h$$
and, with a chosen zero level,
$$U = mgh$$
This formula is valid when $g$ can be treated as constant.
Interpreting the Sign
The sign of $\Delta U$ matters.
If an object moves upward, then $\Delta h > 0$, so
$$\Delta U > 0$$
Its gravitational potential energy increases.
If an object moves downward, then $\Delta h < 0$, so
$$\Delta U < 0$$
Its gravitational potential energy decreases.
Gravity naturally pulls objects downward, toward lower gravitational potential energy.
Choice of Reference Level
Potential energy does not need an absolute zero fixed by nature in this near Earth model. What matters physically is the change in potential energy, not the numerical value by itself.
You may choose the ground, a table, or any convenient level as $U = 0$. The same object can then have different numerical values of $U$, but all correct calculations give the same $\Delta U$.
For example, if a book is on a shelf $2\ \text{m}$ above the floor, then relative to the floor
$$U = mg(2)$$
But if the shelf itself is chosen as the zero level, then the same book has
$$U = 0$$
Both choices are valid.
Only differences in gravitational potential energy are physically important in most mechanics problems:
$$\Delta U = U_f - U_i$$
Why the Formula Makes Sense
Suppose you slowly lift an object straight upward at constant speed. Since the speed is constant, the upward force you apply balances the downward gravitational force. The applied force has magnitude $mg$.
The work you do over height $\Delta h$ is
$$W = F\Delta h = mg\Delta h$$
That work is stored as gravitational potential energy:
$$\Delta U = mg\Delta h$$
So the higher the object is lifted, and the more massive it is, the more gravitational potential energy it gains.
Gravitational Potential Energy for Large Distances
When distances from Earth become very large, or when dealing with planets and satellites, gravity is not constant with height. Then the simple formula $U = mgh$ is no longer accurate.
In that case, the gravitational potential energy between two masses $M$ and $m$ separated by distance $r$ is
$$U(r) = -\frac{GMm}{r}$$
where $G$ is the gravitational constant.
This formula is especially useful in astronomy and orbital motion. Its full use belongs more naturally with gravitation and planetary motion, but it is important to know that the near Earth formula is only an approximation.
The negative sign means that the zero of gravitational potential energy is chosen at infinite separation. Bound objects have negative gravitational potential energy.
General gravitational potential energy for two masses:
$$U(r) = -\frac{GMm}{r}$$
Near Earth's surface, this becomes approximately
$$\Delta U = mg\Delta h$$
Comparing the Two Formulas
The two common forms are related but used in different situations.
| Situation | Formula | Main assumption |
|---|---|---|
| Near Earth's surface | $U = mgh$ or $\Delta U = mg\Delta h$ | $g$ is approximately constant |
| Large distances from a planet or star | $U(r) = -\dfrac{GMm}{r}$ | Gravitational force changes with distance |
Everyday Examples
A backpack on the floor has less gravitational potential energy than the same backpack on a shelf. A roller coaster car at the top of a hill has more gravitational potential energy than at the bottom. A water tank high above the ground stores gravitational potential energy that can later be used when the water flows downward.
In each case, height matters. More mass and more height mean more gravitational potential energy in the near Earth model.
Example Calculation Near Earth
Take a $3.0\ \text{kg}$ object lifted by $2.5\ \text{m}$. The increase in gravitational potential energy is
$$\Delta U = mg\Delta h$$
$$\Delta U = (3.0)(9.8)(2.5) = 73.5\ \text{J}$$
So the object gains
$$73.5\ \text{J}$$
of gravitational potential energy.
Visualizing Height and Energy
Relation to Work Done by Gravity
If the object moves downward, gravity does positive work, and gravitational potential energy decreases. If the object moves upward, gravity does negative work, and gravitational potential energy increases.
This relationship is
$$W_{\text{gravity}} = -\Delta U$$
So if gravitational potential energy goes up by $20\ \text{J}$, then the work done by gravity is $-20\ \text{J}$.
For gravity as a conservative force:
$$W_{\text{gravity}} = -\Delta U$$
An increase in gravitational potential energy means negative work by gravity.
Common Mistakes
A common mistake is to use $U = mgh$ without choosing or understanding the reference level. Another is to confuse height with total distance traveled. In gravitational potential energy near Earth, only the change in vertical height matters.
For example, if an object is carried along a long ramp to a platform, the increase in gravitational potential energy depends only on the vertical rise, not on the length of the path.
Another common mistake is forgetting the sign of $\Delta h$. Moving down lowers gravitational potential energy.
Summary
Gravitational potential energy is energy due to position in a gravitational field. Near Earth's surface, the change is
$$\Delta U = mg\Delta h$$
and with a chosen zero level,
$$U = mgh$$
For larger distances, the more general formula is
$$U(r) = -\frac{GMm}{r}$$
The numerical value of gravitational potential energy depends on the chosen reference level, but changes in gravitational potential energy are what matter physically.