Temperature

Gravitational Potential

Learning Objective

Understand the concept of gravitational potential, how it is calculated, and its relevance to energy transformations that can affect temperature.

Key Definitions

• Gravitational potential energy (U): The work done against the gravitational field to bring a mass from infinity to a point in the field.
• Gravitational potential (V): Gravitational potential energy per unit mass. It is a scalar quantity measured in joules per kilogram (J kg⁻¹).
• Equipotential surface: A surface on which the gravitational potential is the same everywhere.

Fundamental Relationships

The gravitational potential energy of a mass \$m\$ at a distance \$r\$ from a point mass \$M\$ is

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

Dividing by \$m\$ gives the gravitational potential

\$V = -\frac{GM}{r}\$

where

• \$G = 6.674 \times 10^{-11}\ \text{N m}^2\text{kg}^{-2}\$ (gravitational constant)
• \$M\$ is the mass creating the field
• \$r\$ is the distance from the centre of \$M\$ to the point of interest

Gravitational Potential Near Earth

For points close to the Earth's surface, the field can be approximated as uniform, giving

\$V = -g h + V_0\$

where \$g = 9.81\ \text{m s}^{-2}\$, \$h\$ is the height above a reference level, and \$V_0\$ is the potential at that reference level (often taken as zero at sea level).

Equipotential Surfaces

Because \$V\$ is a scalar, the direction of the gravitational field \$\mathbf{g}\$ is always perpendicular to equipotential surfaces:

\$\mathbf{g} = -\nabla V\$

This property is useful for visualising the field and for solving problems involving work.

Relation to Temperature

When gravitational potential energy is converted into other forms, the temperature of a system can change. Two common contexts are:

1. Atmospheric heating: As air descends in the Earth's gravitational field, its potential energy decreases and is converted into kinetic energy, which can increase temperature (adiabatic heating).
2. Accretion processes: In astrophysics, material falling into a deep gravitational potential well releases energy that heats the accretion disc, raising its temperature.

Worked Example

1. Calculate the gravitational potential at a distance of \$2.0 \times 10^{7}\ \text{m}\$ from the centre of the Earth. Use \$M_{\earth}=5.97\times10^{24}\ \text{kg}\$.
2. Solution:

• Apply \$V = -\dfrac{GM}{r}\$.
• \$V = -\frac{(6.674\times10^{-11})(5.97\times10^{24})}{2.0\times10^{7}}\$
• \$V \approx -1.99\times10^{7}\ \text{J kg}^{-1}\$

3. Interpretation: A 1‑kg mass at this altitude has \$1.99\times10^{7}\ \text{J}\$ less potential energy than at infinity.

Common Mistakes

• Confusing gravitational potential (\$V\$) with gravitational field strength (\$g\$). \$V\$ is energy per unit mass (J kg⁻¹); \$g\$ is force per unit mass (m s⁻²).
• Neglecting the negative sign in \$V = -GM/r\$, which indicates that work must be done against the field to increase \$r\$.
• Assuming the uniform field approximation (\$V = -gh\$) is valid far from Earth’s surface.

Summary Table

Further Practice Questions

1. Derive the expression for the change in gravitational potential when moving from height \$h1\$ to \$h2\$ above the Earth’s surface, assuming \$h \ll R_{\earth}\$.
2. A 2 kg mass falls from a height of 500 m. Calculate the increase in its temperature if all the loss of gravitational potential energy is converted to internal energy. (Specific heat capacity of the material = \$900\ \text{J kg}^{-1}\text{K}^{-1}\$.)
3. Explain why equipotential surfaces around a non‑spherical planet are not perfect spheres.

Key Take‑aways

• Gravitational potential is a scalar field representing potential energy per unit mass.
• It is negative for attractive forces and becomes less negative as distance increases.
• Energy released from changes in gravitational potential can raise the temperature of a system.
• Equipotential surfaces provide a useful visual tool for understanding the direction of the gravitational field.