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:

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

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

Quantity	Symbol	Formula	Units
Gravitational constant	$G$	$6.674 \times 10^{-11}$	N m² kg⁻²
Gravitational potential energy	$U$	$-\dfrac{GMm}{r}$	J
Gravitational potential	$V$	$-\dfrac{GM}{r}$	J kg⁻¹
Gravitational field strength (near Earth)	$g$	$9.81$ (approx.)	m s⁻²

Further Practice Questions

1. Derive the expression for the change in gravitational potential when moving from height $h_1$ to $h_2$ 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.