Potential Energy Formula: Calculate Gravitational Energy

Potential energy is stored energy associated with an object’s position or configuration. For an object near Earth’s surface, calculate gravitational potential energy with PE = mgh: enter its mass, height, and the local gravitational acceleration to get energy in joules.

Formula

Near Earth’s surface, gravitational potential energy relative to a chosen reference level is:

PE = m × g × h

• PE is gravitational potential energy, measured in joules (J).
• m is the object’s mass in kilograms (kg).
• g is gravitational acceleration in metres per second squared (m/s²). A commonly used approximate value near Earth’s surface is 9.8 m/s².
• h is the object’s vertical height above the chosen reference level, in metres (m).

The calculator uses those units and returns joules. One joule is one kilogram metre squared per second squared (1 J = 1 kg·m²/s²). The formula assumes that gravitational acceleration is effectively constant over the height being considered. Height means vertical separation, not the distance travelled along a sloping path.

Worked Example

Using the calculator’s default values, a 2 kg object is 5 m above the reference level and g is 9.8 m/s²:

1. Start with PE = m × g × h.
2. Substitute the values: PE = 2 kg × 9.8 m/s² × 5 m.
3. Multiply: PE = 98 kg·m²/s².
4. Since 1 kg·m²/s² equals 1 J, the result is 98.00 J.

This is the energy change associated with raising the object by 5 m, relative to the selected reference level. It is not necessarily the object’s total energy: motion, heat, and other forms of energy are not included.

What Potential Energy Means

Potential energy is energy a system has because of its arrangement or position. It is not a single kind of energy with one formula for every situation. The equation mgh applies specifically to gravitational potential energy near Earth’s surface, where g changes little across the height of interest.

Potential energy is defined relative to a reference. You may set the floor, a tabletop, or another convenient level to zero height. Changing that reference changes the numerical value assigned to the object’s potential energy, but not the energy difference between two positions. In many practical problems, the change in potential energy is what matters.

For example, if the 2 kg object in the worked example is lowered from 5 m to 1 m, its change in gravitational potential energy is ΔPE = mg(h₂ − h₁) = 2 × 9.8 × (1 − 5) = −78.4 J. The negative sign means the object’s gravitational potential energy decreases. In an ideal system, that energy can be transferred into kinetic energy as the object falls.

When to Use This Formula

Use mgh when an object’s height changes near Earth’s surface and you want its gravitational potential energy relative to a stated level, or the change between two heights. Examples include lifting a box, raising water in a tank, or estimating the energy available as an object descends.

For a change between two positions, use:

ΔPE = mg(h₂ − h₁)

This form is often more useful than calculating two separate energy values. If only the height difference is known, use that difference directly. The sign tells you whether potential energy increases or decreases; the magnitude tells you how much it changes.

Units, Gravity, and Common Mistakes

In the SI system, use kilograms, metres, and metres per second squared to obtain joules. If a mass is given in grams, divide by 1,000 to convert it to kilograms. If a height is given in centimetres, divide by 100 to convert it to metres. Mixing units without conversion can make the result wrong by factors of 10, 100, or more.

In US customary units, the equivalent common calculation uses weight force and height: gravitational potential energy equals weight in pounds-force (lbf) multiplied by height in feet (ft), giving foot-pounds of energy (ft·lbf). Do not put a mass in pounds into mgh while treating it as kilograms. A pound of mass and a pound-force are different quantities; conversions must account for that distinction.

• Use vertical height: A ramp’s length is not the height gained. Use the vertical rise.
• State the reference level: Without it, an absolute potential-energy value is ambiguous.
• Distinguish mass from weight: Mass is measured in kg; weight is a force and depends on gravity.
• Keep sensible precision: A result cannot be more reliable than the measurements and chosen value of g. Rounding the final result is usually preferable to rounding intermediate steps.
• Check the sign: The value mgh is positive for a position above the chosen zero level; a change in energy can be positive or negative.

Limits and Other Potential-Energy Formulas

Earth’s g is often approximated as 9.8 m/s², or more precisely about 9.81 m/s² for many ordinary calculations. The exact value varies slightly by location. For everyday height changes, that variation is usually small; use a specified value when a problem or measurement requires it.

For very large distances from a planet, the constant-g approximation is no longer accurate. Gravitational potential energy relative to zero at infinite distance is commonly written U = −GMm/r, where G is the gravitational constant, M and m are the masses, and r is the distance between their centres. The negative sign reflects the chosen zero and the bound gravitational system. This equation is different from mgh, which is a useful near-surface approximation.

Other forms of potential energy have their own formulas. An ideal spring has elastic potential energy PE = ½kx², where k is its spring constant and x is its displacement from the relaxed length. Electric potential energy depends on the charges and their arrangement. Do not apply mgh to a spring or an electric interaction merely because the word “potential” appears.

Frequently Asked Questions

What is the basic formula for potential energy?
For gravitational potential energy near Earth’s surface, use PE = mgh, with mass in kilograms, g in m/s², and height in metres. Other types of potential energy use different equations.

What unit is potential energy measured in?
The SI unit is the joule (J). In US customary calculations using weight and height, energy may be expressed in foot-pounds (ft·lbf).

Does potential energy depend on the zero-height point?
The numerical value does. You can choose a convenient reference level, but energy differences between the same two positions remain unchanged.

Is height the distance an object travels?
No. In mgh, height is vertical displacement above the reference level. A longer sloping route does not change the gravitational energy gained if the starting and ending heights are the same.

Can potential energy be negative?
Yes, depending on the selected reference and the type of potential-energy model. In the near-surface mgh convention, a position below the zero-height level has negative gravitational potential energy. The reference choice does not change measurable energy differences.

What is the difference between potential and kinetic energy?
Potential energy is associated with position or configuration; kinetic energy is associated with motion. As an object falls, gravitational potential energy can be converted into kinetic energy, though real systems may also transfer energy to sound, heat, or deformation.

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in m | Mass | kg | 2 | 0
in h | Height | m | 5 | 0
in g | Gravitational acceleration | m/s^2 | 9.8 | 0
out pe | Gravitational potential energy | J | m * g * h | 2