Gravitational Potential Energy-Derivations, Formulas, Examples | JEE NEET Notes

Introduction

Gravitational Potential Energy (GPE) is a form of energy stored in an object due to its position in a gravitational field. It is one of the most fundamental concepts in mechanics and plays a crucial role in various applications such as free-fall motion, planetary motion, and roller coasters.

Define Gravitational Potential Energy(GPE)

The energy possessed by objects due to changes in their position in a gravitational field is called Gravitational Potential Energy.

It is the energy of the object due to the gravitational forces. The work done per unit mass to bring the body from infinity to a location inside the gravitational field of any object is known as gravitational potential and the energy change here is called Gravitational Potential Energy.

What Is Gravitational Potential Energy?

When a body of mass (m) is moved from infinity to a point inside the gravitational influence of a source mass (M) without accelerating it, the amount of work done in displacing it into the source field is stored in the form of potential energy. This is known as gravitational potential energy. It is represented by the symbol U_g.

Explanation: 

We know that the potential energy of a body at a given position is defined as the energy stored in the body at that position. If the position of the body changes due to the application of external forces, the change in potential energy is equal to the amount of work done on the body by the forces.

Under the action of gravitational force, the work done is independent of the path taken for a change in position, so the force is a conservative force. Besides, all such forces have some potential in them.

The gravitational influence on a body at infinity is zero; therefore, potential energy is zero, which is called a reference point.

Mathematically, Gravitational Potential Energy is given by the product of the mass (m) of the object, acceleration due to gravity (g), and height (h) above the ground as 

U = mgh

where:

• U = Gravitational Potential Energy (Joules, J)
• m = Mass of the object (kilograms, kg)
• g = Acceleration due to gravity (9.8 m/s² on Earth)
• h = Height above the reference level (meters, m)

Derivation of Gravitational Potential Energy Equation

Consider a source mass ‘M’ is placed at a point along the x-axis; initially, a test mass ‘m’ is at infinity. A small amount of work done in bringing it without acceleration through a very small distance (dx) is given by

dw = Fdx

Here, F is an attractive force, and the displacement is towards the negative x-axis direction, so F and dx are in the same direction. Then,

dw = (GMm/x²)dx

Integrating on both sides

\(\begin{array}{l}w = \int_{\infty }^{r} \frac{GMm}{x^{2}}dx\end{array} \)

\(\begin{array}{l}w = -[\frac{GMm}{x}]_{\infty }^{r}\end{array} \)

\(\begin{array}{l}w = -[\frac{GMm}{r}] – (\frac{-GMm}{\infty })\end{array} \)

\(\begin{array}{l}w = \frac{-GMm}{r}\end{array} \)

Since the work done is stored as its potential energy U, the gravitational potential energy at a point which is at a distance ‘r’ from the source mass is given by;

U = -GMm/r

If a test mass moves from a point inside the gravitational field to the other point inside the same gravitational field of source mass, then the change in potential energy of the test mass is given by;

ΔU = GMm (1/r_i – 1/r_f)

If r_i > r_f then ΔU is negative.

Expression for Gravitational Potential Energy at Height (h) – Derive ΔU = mgh

If a body is taken from the surface of the earth to a point at a height ‘h’ above the surface of the earth, then r_i = R and r_f = R + h, then,

ΔU = GMm [1/R – 1/(R+h)]

ΔU = GMmh/R(R + h)

When h<<R, then (R + h) ≈ R and g = GM/R²

On substituting this in the above equation, we get,

Gravitational Potential Energy is

ΔU = mgh

⇒ Note: 

• The weight of a body at the centre of the earth is zero due to the fact that the value of g at the centre of the earth is zero.
• At a point in the gravitational field where the gravitational potential energy is zero, the gravitational field is zero.

📌 Key Features of Gravitational Potential Energy:

1. Height Dependent: Gravitational Potential Energy increases with an increase in height.
2. Mass Proportional: Heavier objects possess more Gravitational Potential Energy.
3. Work-Energy Relation: Work done to raise an object is stored as Gravitational Potential Energy.
4. Energy Transformation: Converts into kinetic energy during free fall.
5. Negative Potential Energy: In astrophysics, Gravitational Potential Energy is often negative due to convention.

Solved Problems

Example 1. Calculate the gravitational potential energy of a body of mass 10 kg and is 25 m above the ground.

Solution:

Given, Mass m = 10 Kg and Height h = 25 m

G.P.E is given as,

U = m × g × h = 10 Kg 9.8 m/s² × 25 m = 2450 J.

Example 2. If the mass of the earth is 5.98 ×10²⁴ kg and the mass of the sun is 1.99 × 10³⁰ kg, and the earth is 160 million km away from the sun, calculate the GPE of the earth.

Solution:

Given, the mass of the Earth (m) = 5.98 × 10²⁴ Kg and mass of the Sun (M) = 1.99 × 10³⁰ Kg

The gravitational potential energy is given by:

U = -GMm/r

U = (6.673 ∗ 10^-11 ∗ 5.98 ∗ 10²⁴ ∗1.99∗10³⁰)/(160∗10⁹) = 4963 x 10³⁰ J

Example 3. A basketball weighing 2.2 kg falls off a building to the ground 50 m below. Calculate the gravitational potential energy of the ball when it arrives below.

Solution:

GPE = (2.2 kg)(9.8 m/s²)(50 m) = 1078 J.

Example 4: A 2 kg body free falls from rest from a height of 12 m. Determine the work done by the force of gravity and the change in gravitational potential energy. Consider the acceleration due to gravity to be 10 m/s².

Solution:

Since, W = mgh

Substituting the values in the above equation, we get

W = 2 × 12 × 10 = 240 N

The change in gravitational potential energy is equal to the work done by gravity.

Therefore, Gravitational Potential Energy= 240 Joule.

📌 Real-Life Applications of Gravitational Potential Energy

1. Hydropower Plants – Water stored at a height has high Gravitational Potential Energy, which converts into kinetic energy to generate electricity.
2. Roller Coasters – At the highest point, a roller coaster has maximum Gravitational Potential Energy, which later transforms into kinetic energy.
3. Dams and Reservoirs – Water stored at an elevated height holds significant gravitational potential energy.
4. Satellites and Space Exploration – The gravitational potential energy of a satellite determines its orbit around a planet.
5. Bungee Jumping – Before the fall, a person at a height has maximum Gravitational Potential Energy, which is converted into kinetic energy.

📌 Frequently Asked Questions (FAQs)

🔹 Q1: What is the SI unit of Gravitational Potential Energy?

A: The SI unit of gravitational potential energy is Joule (J).

🔹 Q2: How does gravitational potential energy change with height?

A: Gravitational Potential Energy increases as height increases since it is directly proportional to height (U = mgh).

🔹 Q3: Does Gravitational Potential Energy depend on mass?

A: Yes, Gravitational Potential Energy is directly proportional to mass; more massive objects have higher gravitational potential energy.

🔹 Q4: What happens to gravitational potential energy when an object falls?

A: As an object falls, its Gravitational Potential Energy converts into kinetic energy, keeping total mechanical energy conserved.

🔹 Q5: Is gravitational potential energy always positive?

A: No, in physics (especially astrophysics), it is often considered negative because work is required to bring an object to zero potential energy.

📌 Multiple Choice Questions (MCQs) with Answers and Explanations

🔸 Q1: Which factor does NOT affect gravitational potential energy?

A) Mass of the object
B) Acceleration due to gravity
C) Height above the ground
D) Speed of the object

Answer: ✅ D) Speed of the object
Explanation: Gravitational Potential Energy depends on mass, height, and gravitational acceleration, but not on speed as speed is related to kinetic energy.

🔸 Q2: If the mass of an object is doubled, how will its gravitational potential energy change?

A) It will remain the same
B) It will become half
C) It will double
D) It will become four times

Answer: ✅ C) It will double
Explanation: Since Gravitational Potential Energy is given by U = mgh, if m doubles, U also doubles.

🔸 Q3: What happens to gravitational potential energy when an object is taken to a greater height?

A) It decreases
B) It remains the same
C) It increases
D) It first increases then decreases

Answer: ✅ C) It increases
Explanation: Since Gravitational Potential Energy is directly proportional to height (h), an increase in height leads to an increase in gravitational potential energy.

🔸 Q4: Why is gravitational potential energy considered negative in astrophysics?

A) Because energy is lost when moving upwards
B) Because work is needed to bring an object to zero potential energy
C) Because gravity is a repulsive force
D) Because the mass of an object is negative in space

Answer: ✅ B) Because work is needed to bring an object to zero potential energy
Explanation: In space physics, potential energy is considered negative since work must be done to move an object to a reference level where potential energy is zero.

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