1. What is Gravitational Potential Energy?
Gravitational Potential Energy (GPE) is the energy an object has because of its position in a gravitational field. When you lift an object against gravity, you do work on it, and that work is stored as gravitational potential energy.
The higher the object is lifted, the greater its potential energy.
1.1. Simple Examples
- A book on a shelf has more GPE than a book on the floor.
- Water stored in a tank high above the ground has large GPE.
- A roller coaster at the top of a hill has maximum GPE.
1.1.1. Key Idea
Potential energy exists because gravity can pull the object downward at any moment.
2. Gravitational Potential Energy Near Earth's Surface
Near the Earth's surface, gravitational potential energy is given by:
\( U = mgh \)
Where:
- \( U \) is the potential energy
- \( m \) is the mass
- \( g \) is the acceleration due to gravity
- \( h \) is the height of the object above a reference point
2.1. Understanding the Formula
Lifting an object higher requires more work because gravity opposes the motion. This work gets stored as potential energy.
2.2. Dependence on Height
Potential energy increases linearly with height. Doubling the height doubles the potential energy.
3. Gravitational Potential Energy at Large Distances
At large distances (far away from Earth), the simple formula \( mgh \) does not apply. Instead, gravitational potential energy is calculated using Newton’s law of gravitation.
The general formula is:
\( U = -\dfrac{GMm}{r} \)
Where:
- \( r \) is the distance between the object and the centre of Earth.
3.1. Why the Negative Sign?
The negative sign means that gravitational potential energy is lower when the object is closer to Earth. Work must be done to move the object far away from Earth’s gravity.
3.2. Zero Potential Energy Convention
By convention, GPE is considered zero at an infinite distance from Earth. This makes mathematical calculations easier and meaningful.
4. Graphical Representation of Gravitational Potential Energy
The gravitational potential energy curve helps visualize how energy changes with distance.
4.1. GPE vs Distance
- Near Earth: GPE increases linearly with height.
- At large distances: GPE approaches zero from the negative side.
5. Relation Between GPE and Escape Velocity
To escape Earth's gravitational field, an object must gain enough energy to make its total mechanical energy zero. This means the object must overcome the negative gravitational potential energy.
5.1. Energy Interpretation
Escape velocity ensures the object has sufficient kinetic energy to cancel out its negative potential energy.
6. Importance of Gravitational Potential Energy
GPE plays a vital role in many physical and natural processes:
- Movement of rivers and waterfalls
- Hydroelectric power generation
- Motion of planets and satellites
- Behaviour of tides
- Energy storage in dams and reservoirs
Next, we study gravitational potential and the concept of gravitational field, which describe how gravity acts around a mass.