Gravitational Potential Energy

This topic covers Gravitational Potential Energy.

Gravitational Potential

Gravitational Potential energy belongs to a pair of objects in a system (for instance a ball+ Earth system, galaxies of stars interacting gravitationally) and is equal to the work done against gravity. This potential energy is the energy associated within the particles inside a system and is not the same as rest or kinetic energies of the individual particles. Traditionally potential energy is represented by the symbol U and this page describes specific examples in which U is equal to the gravitational energy learned about in earlier pages to be approximately mg near the surface of the Earth or [math]\displaystyle{ F = - G \frac{m_1 m_2}{r^2}\ }[/math]
where:

• F is the force between the masses;
• G is the gravitational constant (6.674×10⁻¹¹ N · (m/kg)²);
• m₁ is the first mass;
• m₂ is the second mass;
• r is the distance between the centers of the masses.

The latter case is distance dependent and can be derived since force is the negative gradient of U. The negative indicates that potential energy decreases as particles get closer together

A Mathematical Model

In a system composed of two objects that are interacting with gravitational potential: [math]\displaystyle{ F = - G \frac{m_1 m_2}{r^2}\ }[/math]
where:

• F is the force between the masses;
• G is the gravitational constant (6.674×10⁻¹¹ N · (m/kg)²);
• m₁ is the first mass;
• m₂ is the second mass;
• r is the distance between the centers of the masses.

Close to the Surface of the Earth:

[math]\displaystyle{ U = mgh\! }[/math]

where U is the potential energy of the object assuming it is close to the surface of the Earth, m is the mass of the object, g is the acceleration of 9.8, and h is height.

Potential difference is derived to be:

[math]\displaystyle{ \,\Delta U = mg \Delta h.\ }[/math]

A Computational Model

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Examples

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