Gravitational potential

The gravitational potential (V) at a point is the amount of work done per unit mass to move an object from infinity to that point against the gravitational field.

It's calculated using the equation:

$\text{V = -} \frac{\text{G M}}{\text{r}}$

Where:

• V = Gravitational potential (J kg^-1)
• G = Gravitational constant (6.67 x 10^-11 N m² kg^-2)
• M = Mass of the object causing the field (kg)
• r = Distance from the object's centre (m)

Key points:

• Gravitational potential is always negative
• The magnitude of V decreases with increasing distance from the object's center

Gravitational potential energy

The gravitational potential energy (E) of an object is the energy it has due to its position in a gravitational field. It depends on the object's mass and its height within the field. It is calculated as:

E = m V

Where:

• E = Gravitational potential energy (J)
• m = Object's mass (kg)
• V = Gravitational potential at that height (J kg^-1)

By substituting the equation for V, we get:

$ E = - \frac{\text{G m M}}{\text{r}} $

Worked example - Calculating gravitational potential energy

Calculate the gravitational potential energy of a 100 kg satellite located 400 km above the Earth's surface, assuming the Earth's mass is 5.97 x 10²⁴ kg and its radius is 6,371 km.

Step 1: Calculate distance of satellite from centre of Earth

r = radius of Earth + height above Earth’s surface

r = 6371 x 10³ + 400 x 10³ = 6771 x 10^3 m

Step 2: Calculate Gravitational Potential (V) at the Satellite's Location

$\text{V = - }\frac{\text{G M}}{\text{r}}= \frac{6.67\times10^{-11}\times5.97\times10^{24}}{6,771\times10^3}= -58.81 \text{ MJ kg}^{-1}$

Step 3: Calculate Gravitational Potential Energy (E)

E = m V = 100 x 58.81 x 10⁶ = 5,881 MJ

Gravitational potential difference

The difference in gravitational potential (ΔV) between two points in a gravitational field is equal to the work done per unit mass in moving an object between those points.

For example, moving an object from the Earth's surface to a higher point involves working against gravity, resulting in a positive ΔV.

The work done (W) moving a mass (m) through a potential difference ΔV is:

$W = mΔV$

Where:

W = work done (J)

m = mass (kg)

$\Delta$V = gravitational potential difference (J kg^-1)

Escape velocity

Escape velocity (v) is the minimum speed needed for an object to break free from the gravitational field of a celestial body without further propulsion. This velocity is where the object's kinetic energy equals its gravitational potential energy.

The escape velocity formula is derived by equating kinetic energy to gravitational potential energy:

$\frac{1}{2}\text{m v}^2=\frac{\text{G M m}}{\text{r}}$

Rearranging to find escape velocity (v) gives:

$v = \sqrt{\frac{2GM}{r}}$

Where:

• v = Escape velocity (m s^-1)
• G = Gravitational constant (6.67 x 10^-11 N m² kg^-2)
• M = Mass of the object creating the field (kg)
• r = Distance from the object's centre (m)

Worked example - Calculating the escape velocity from Earth

Calculate the escape velocity from the Earth's surface, assuming the Earth's mass is 5.97 x 10²⁴ kg and its radius is 6,371 km.

Step 1: Formula

$ \text{v = }\sqrt{\frac{2\text{ G M}}{\text{r}}} $

Step 2: Convert km to m

to convert from km to m, multiply by 1,000

6,371 km = 6,371,000 m

Step 3: Substitution and Correct Evaluation

$\text{v } = \sqrt{\frac{2 \times 6.67 \times 10^{-11} \times 5.97 \times 10^{24}}{6.371 \times 10^6}}= 11.2 \text{ km s}^{-1}$

Worked example - Calculating gravitational potential at a specific point

Calculate the gravitational potential at a point 10,000 km away from the centre of the Earth, assuming the Earth's mass is 5.97 x 10²⁴ kg.

Step 1: Formula

$\text{V = - }\frac{\text{G M}}{\text{r}}$

Step 2: Substitution and correct evaluation

$\text{V = - }\frac{6.67 \times 10^{-11} \times 5.97 \times 10^{24}}{10,000\times10^3}\text{ = -39.82 x 10}^6\text{ J kg}^{-1}$