Gravitational Potential Energy: Meaning, Formula and Real-Life Applications

Gravitational Potential Energy is the energy stored in an object because of its position in Earth's gravitational field. Whenever an object is lifted above the ground, work is done against gravity, and this work is stored as gravitational potential energy. The higher the object is placed or the greater its mass, the more energy it stores.

From a book placed on a shelf to water stored in a dam, gravitational potential energy is present in many everyday situations. In this article, you will learn what is gravitational potential energy, the gravitational potential energy formula, its derivation, factors affecting it, and its real-life applications.

Table of Contents

• What is Gravitational Potential Energy
• Gravitational Potential Energy Formula
• Gravitational Potential Energy Derivation
• Mathematical Expression for Gravitational Potential Energy at Height 
• Factors Affecting Gravitational Potential Energy
• Real-Life Applications of Gravitational Potential Energy
• Frequently Asked Questions on Gravitational Potential Energy

What is Gravitational Potential Energy

Imagine you are holding a cricket ball above the ground. Even though the ball is not moving, it has stored energy because it is at a certain height. If you release it, the ball falls towards the ground due to gravity. The stored energy changes into kinetic energy as the ball moves.

In simple words, 

Gravitational potential energy is the energy stored in an object because of its position in Earth's gravitational field.

The higher an object is placed, the greater its gravitational potential energy. Likewise, a heavier object also stores more gravitational potential energy than a lighter one at the same height.

Do you know? If an object is resting on the ground, its gravitational potential energy is usually taken as zero because its height is considered zero with respect to the reference level.

Gravitational Potential Energy Formula

At an infinite distance from a mass, the gravitational effect becomes negligible, and the gravitational potential energy is taken as zero. This point is called the reference point. Near the Earth's surface, gravitational potential energy is commonly expressed as,

${U = mgh}$

Where,

• U is gravitational Potential Energy (Joule)
• m is the mass of the object (kg)
• g is the acceleration due to gravity (9.8 m/s² on Earth)
• h is the height of the object above the reference point (m)

This formula shows that gravitational potential energy depends on three things:

• Mass of the object
• Height above the ground
• Value of gravity

Since gravitational potential energy is a form of energy, its SI unit is the joule (J). The dimensional formula is ML²T⁻².

Gravitational Potential Energy Derivation

The gravitational potential energy derivation explains how the expression for potential energy is obtained by calculating the work done against gravity. 

Let us consider a source mass M fixed at a point and a test mass m located at infinity. Since the gravitational force at infinity is considered zero, the potential energy is also taken as zero at this reference point.

As the test mass is brought slowly towards the source mass without changing its speed, a small amount of work is done over a tiny displacement dx.

The gravitational force between the two masses is:

$F=\frac{GMm}{x^2}$

The small amount of work done is:

$dW=F\,dx$

Substituting the value of F:

$dW=\frac{GMm}{x^2}\,dx$

To find the total work done in moving the test mass from infinity to a distance r, integrate both sides:

$W=\int_{\infty}^{r}\frac{GMm}{x^2}\,dx$

On solving the integral:

$W=-\frac{GMm}{r}$

Since the work done is stored as gravitational potential energy,

${U=-\frac{GMm}{r}}$

where,

• U is gravitational potential energy (J)
• G is the universal gravitational constant
• M is the mass of the source body
• m is the mass of the object
• r is the distance between the centres of the two masses

The negative sign indicates that gravity is an attractive force. As the object moves farther away from the source mass, its gravitational potential energy increases and approaches zero at infinity.

Mathematical Expression for Gravitational Potential Energy at Height 

Near the Earth's surface, let

• Initial distance from Earth's centre:  $r_i = R$
• Final distance: $r_f = R + h$

The change in gravitational potential energy is,

$\Delta U = GMm\left(\frac{1}{R}-\frac{1}{R+h}\right)$

Simplifying,

$\Delta U=\frac{GMmh}{R(R+h)}$

For objects close to the Earth's surface, the height h is very small compared to the Earth's radius R (h≪R). Therefore,

$R+h \approx R$

Using the relation,

$g=\frac{GM}{R^2}$

the equation becomes:

${\Delta U=mgh}$

This is the gravitational potential energy formula commonly used for objects near the Earth's surface.

At the centre of the Earth, the value of gravitational acceleration g is zero, so the weight of an object becomes zero. At an infinite distance from the Earth, gravitational potential energy is taken as zero, which serves as the reference level.

Factors Affecting Gravitational Potential Energy

The gravitational potential energy (GPE) of an object depends on three main factors: its mass, its height above a reference point and the strength of the gravitational field. A change in any of these factors changes the amount of gravitational potential energy stored in the object.

1. Mass of the Object

The mass of an object is directly proportional to its gravitational potential energy. A heavier object has more gravitational potential energy than a lighter object when both are at the same height.

Example: A 10 kg suitcase placed on a shelf stores more gravitational potential energy than a 2 kg backpack placed on the same shelf.

2. Height Above the Reference Point

The higher an object is lifted above the chosen reference level (usually the ground), the greater its gravitational potential energy. More work is required to lift an object to a greater height, increasing the energy stored in it.

Example: A flower pot on the third floor has more gravitational potential energy than the same pot kept on the ground.

3. Gravitational Field Strength (g)

Gravitational potential energy also depends on the strength of the gravitational field. On Earth, the value of g is approximately 9.8 m/s², while it is about 1.62 m/s² on the Moon. Therefore, the same object at the same height has less gravitational potential energy on the Moon than on Earth.

Example: An astronaut lifting a tool on the Moon stores less gravitational potential energy in it than when lifting the same tool to the same height on Earth.

Real-Life Applications of Gravitational Potential Energy

Gravitational potential energy has many practical applications in science, engineering and everyday life. It helps explain how energy is stored due to an object's position and how that stored energy changes into other forms.

• Projectile Motion: When a ball is thrown upward, its kinetic energy gradually changes into gravitational potential energy. As it falls, the stored potential energy converts back into kinetic energy.
• Hydroelectric Power Plants: Water stored at a height in dams has high gravitational potential energy. When released, this energy changes into kinetic energy, which turns turbines to generate electricity.
• Roller Coasters: At the highest point of the track, a roller coaster has maximum gravitational potential energy. As it moves downward, this energy changes into kinetic energy, increasing the coaster's speed.
• Space Science: Scientists use gravitational potential energy to calculate the energy needed to launch rockets, place satellites into orbit and plan space missions.
• Engineering and Construction: Engineers consider gravitational potential energy while designing bridges, elevators, cranes, dams, and other structures to ensure safe and efficient operation.
• Energy Conservation: In many physical systems, gravitational potential energy continuously converts into kinetic energy and back again, while the total mechanical energy remains constant when no external forces, such as friction, act on the system.

Gravitational potential energy is the energy stored in an object because of its position in a gravitational field. It depends on the object's mass, height and gravity. From everyday activities to engineering and space science, it plays a vital role in explaining how stored energy changes into kinetic energy.