Gravitational Potential Energy

Gravitational Potential Energy (GPE) is the energy stored in an object due to its position in a gravitational field. It depends on the object's mass, the gravitational field strength, and its height above a reference point.

Near the Earth's surface, GPE is given by: \[ U = mgh \] where:

\( U \): gravitational potential energy (in joules)
\( m \): mass of the object (in kilograms)
\( g \): gravitational field strength \( \approx 9.8 \, \text{m/s}^2 \)
\( h \): height above the reference level (in meters)

Key Notes

The reference level (where \( h = 0 \)) is arbitrary — it could be the ground, a table, or any chosen zero point.
Only changes in GPE matter in energy conservation.
GPE is a form of mechanical energy and can be converted to kinetic energy or other forms.

Gravitational Potential Energy in Universal Gravitation

When considering objects far from Earth (e.g., satellites, planets), we use the more general form: \[ U = -G \frac{m_1 m_2}{r} \] The negative sign indicates that the gravitational force is attractive, and the potential energy is zero at infinite separation.

A 5 kg object is lifted to a height of 10 m. What is its gravitational potential energy relative to the ground?

Solution:

\[ U = mgh = 5 \times 9.8 \times 10 = 490 \, \text{J} \]

The object has 490 J of gravitational potential energy relative to the ground.

A \( 1000 \, \text{kg} \) satellite is located \( 7.0 \times 10^6 \, \text{m} \) from Earth's center. What is its gravitational potential energy? Take Earth's mass as \( 5.97 \times 10^{24} \, \text{kg} \).

Solution:

\[ U = -G \frac{m_1 m_2}{r} = -\frac{6.674 \times 10^{-11} \times 5.97 \times 10^{24} \times 1000}{7.0 \times 10^6} \approx -5.69 \times 10^{10} \, \text{J} \]

The satellite has approximately \( -5.69 \times 10^{10} \, \text{J} \) of gravitational potential energy.

Written by Thenura Dilruk