Gravitational Potential Energy Near Earth
When you lift a book off a desk and hold it above the floor, something changes that isn't immediately visible: the book gains energy by virtue of its new position. This is gravitational potential energy (GPE) — energy stored in an object because of where it sits in a gravitational field. Near Earth's surface, the gravitational field is remarkably uniform, meaning gravity pulls downward with almost exactly the same strength everywhere from ground level up to the tops of the tallest buildings. This uniformity lets us use a beautifully simple formula to calculate GPE. The formula GPE = mgh relates three measurable quantities: the object's mass, the strength of Earth's gravitational field, and the height above a chosen reference level. The key word is 'chosen' — GPE has no absolute value on its own. Only changes in GPE matter physically, so we are free to set the zero level wherever is most convenient for a given problem. This is similar to how you might measure your distance from home rather than from the centre of the Earth; the choice of reference point is arbitrary, but once chosen it must be used consistently throughout a calculation. Gravitational potential energy underpins much of classical mechanics. It converts into kinetic energy as objects fall, powers hydroelectric dams, and determines the trajectory of a rollercoaster. Mastering this concept is essential before tackling more advanced topics such as orbital mechanics, where the uniform-field approximation breaks down and a more general formula is needed.
- The uniform gravitational field assumption
- Work done against gravity defines GPE
- Choosing and using a reference level
- Conservation of energy and GPE
- Why this approximation eventually fails
The uniform gravitational field assumption
Earth's gravitational field strength g ≈ 9.81 N kg⁻¹ at the surface and decreases with altitude according to an inverse-square law. However, for heights that are small compared with Earth's radius (6,371 km), the variation in g is negligible. Climbing a 100 m building changes g by only about 0.003 %. We therefore treat g as constant, making the field 'uniform' — field lines are parallel and equally spaced, like a perfectly flat gravitational landscape.
Work done against gravity defines GPE
Energy is transferred to an object when you do work against a force. If you lift a mass m vertically by a height h at constant speed, the upward force you apply equals mg (balancing gravity), and the work done is force × distance = mgh. All of that work is stored as gravitational potential energy. This gives us the defining equation: ΔE_p = mgΔh. The Greek delta (Δ) reminds us that only changes are meaningful — GPE is always measured relative to a reference level.
Choosing and using a reference level
The reference level (where h = 0 and therefore E_p = 0) is chosen for convenience. For a ball thrown upward from a table, you might choose the table surface as h = 0. If the ball later rolls off the edge, you can redefine h = 0 as the floor. The physics — how fast the ball moves at any given point — is identical regardless of your choice. Just remember: if an object moves below your chosen zero, h is negative, and E_p is negative.
Conservation of energy and GPE
In a closed system with no friction or air resistance, total mechanical energy (kinetic + potential) is conserved. As an object falls, GPE decreases and kinetic energy (½mv²) increases by exactly the same amount. Mathematically: mgh₁ + ½mv₁² = mgh₂ + ½mv₂². This powerful equation lets you find speeds at any height without needing to know the time of flight or the exact path taken. The shape of the trajectory is irrelevant — only height and speed at two points matter.
Why this approximation eventually fails
As objects move very far from Earth's surface — rockets, satellites, spacecraft — g can no longer be treated as constant. At 6,371 km above Earth (one Earth radius up), g has already dropped to one-quarter of its surface value. In those situations, we must use the full gravitational potential energy formula E_p = −GMm/r, which you will encounter in the study of orbital mechanics. The mgh formula is the first-order approximation of this general formula for small h.
The Key Equation
- E_p — Gravitational potential energy (Joule (J)): The energy stored in the object due to its position in the gravitational field, measured relative to the chosen reference level.
- m — Mass (Kilogram (kg)): The amount of matter in the object. GPE is directly proportional to mass — a heavier object stores more energy at the same height.
- g — Gravitational field strength (Newtons per kilogram (N kg⁻¹) or m s⁻²): Near Earth's surface, g ≈ 9.81 N kg⁻¹. It represents the gravitational force experienced per unit mass.
- h — Height (Metre (m)): The vertical distance above the chosen reference level. Can be negative if the object is below the reference level.
Worked Example
Problem. A skier of mass 65 kg descends from the top of a slope at a height of 120 m above the base of the mountain to a point 45 m above the base. Assuming no friction, calculate (a) the loss in gravitational potential energy and (b) the skier's speed at the lower point, given they started from rest. Use g = 9.81 m s⁻².
Answer. The skier loses approximately 47,900 J of gravitational potential energy and reaches a speed of approximately 38.4 m s⁻¹ at the lower point.
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
Leibniz introduced the concept of 'vis viva' (living force), which he defined as mv², laying the groundwork for what we now call kinetic and potential energy. His insight that a quantity related to motion and position could be conserved throughout mechanical interactions was revolutionary. The modern concept of gravitational potential energy as stored capacity to do work developed directly from his energy framework, later refined by scientists including Lazare Carnot and William Rankine, who coined the term 'potential energy' in 1853. Leibniz and Isaac Newton famously disputed priority for the invention of calculus — a bitter controversy that divided European mathematics for decades. Ironically, both men's work was essential for formulating the equations of gravitational potential energy we use today.
Related Concepts
- Kinetic Energy — GPE and kinetic energy are interconverted as objects move under gravity. Conservation of mechanical energy requires understanding both forms simultaneously.
- Work and Energy — GPE is defined through the work done against gravity. The work-energy theorem is the foundation on which E_p = mgh is derived.
- Conservation of Energy — The principle that total mechanical energy is constant in a closed system is the key tool for solving problems involving gravitational potential energy.
- Gravitational Potential Energy in Radial Fields — The general formula E_p = −GMm/r extends the mgh approximation to situations far from Earth's surface, such as satellites and planetary orbits.
- Newton's Law of Gravitation — Understanding why g ≈ 9.81 N kg⁻¹ at Earth's surface — and why it changes with distance — comes directly from Newton's inverse-square law of gravitation.