## Table of Contents

## Gravitational Potential

Gravitational potential, $\phi$, at a point in a gravitational field is the work done per unit mass, by an external force, in bringing the mass from infinity to that point.

$$\phi = – \, G \frac{M}{r}$$

, where:

- $\phi$ is the gravitational potential at distance $r$ from the mass $M$,
- $G$ is the gravitational constant ($6.674 \times 10^{-11} \text{ Nm}^{2}\text{kg}^{-2}$),
- $M$is the mass of the object creating the gravitational field,
- $r$ is the distance from the center of mass of the object ($M$) to the point in question.

Gravitational potential, $\phi$, is a scalar quantity and its units is $\text{J Kg}^{-1}$.

### Assumption Of Zero Gravitational Potential At Infinity

At infinity, $\phi$ is assumed to be zero.

- The convention of setting the gravitational potential to zero at infinity is a practical choice, simplifying calculations and allowing for a universal reference point. It signifies that as the distance from the mass creating the gravitational field approaches infinity, the gravitational potential and hence the potential energy of a mass in that field approaches zero.

### Reason For Negative Sign In Gravitational Potential Equation

The reason for the negative sign in the gravitational potential equation is twofold:

**Reference Point at Infinity**: By convention, the gravitational potential is set to zero at infinity. This means as one moves closer to the mass creating the gravitational field, the potential decreases, hence the negative sign.**Nature of Gravitational Force**: Gravitational force is inherently attractive, pulling objects together. The work done by an external force to move a mass from infinity to a point inside the gravitational field is against this attractive force. Since this external force must act in the opposite direction of the gravitational pull, the work done (and hence the gravitational potential) is considered negative.

### Gravitational Potential Difference

Gravitational potential difference is a key concept in understanding the movement of objects within a gravitational field. It represents the work done per unit mass in moving an object between two points in a gravitational field. This concept not only simplifies the calculation of energy changes but also provides a deeper understanding of gravitational interactions.

The gravitational potential difference ($\Delta \phi$) between two points in a gravitational field is defined as the difference in gravitational potential ($\phi$) between those points:

$$\Delta \phi = \phi_{2}-\phi_{1}$$

where:

- $\phi_{2}$ is the gravitational potential at the second point,
- $\phi_{1}$ is the gravitational potential at the first point.

The unit of gravitational potential difference, like gravitational potential, is joules per kilogram (J/kg).

#### Physical Significance

The gravitational potential difference is directly related to the work done by or against the gravitational force when moving a mass from one point to another within the gravitational field. If an object moves closer to the mass creating the field, the gravitational potential difference will be negative, indicating that work is done by the gravitational force. Conversely, if the object moves away, the gravitational potential difference is positive, reflecting work done against the gravitational force.

#### Applications and Calculations

This concept is particularly useful in calculating the change in gravitational potential energy of an object moving within a gravitational field without needing to integrate the gravitational force over a path. The change in gravitational potential energy ($\Delta U$) is given by multiplying the mass of the object ($m$) by the gravitational potential difference:

$$\Delta U = m \Delta \phi$$

This equation simplifies the calculation of energy changes for objects moving in gravitational fields, such as satellites orbiting Earth or planets moving around the Sun.

## Gravitational Potential Energy

Gravitational potential energy, U of a point mass m, in a gravitational field, is the work done by an external force in bringing that point mass from infinity to that point.

$$\begin{aligned} U &= G \frac{Mm}{r} \\ & \text{OR} \\ U &= m \phi \end{aligned}$$

, where:

- $U$ is the gravitational potential energy,
- $G$ is the gravitational constant ($6.674 \times 10^{-11} \text{ Nm}^{2}\text{kg}^{-2}$),
- $M$is the mass of the object creating the gravitational field,
- $m$ is the mass of the object being moved,
- $r$ is the distance from the center of mass of the object ($M$) to the point in question,
- $\phi$ is the gravitation potential at distance $r$ from the mass of the object ($M$).

Gravitational potential energy, $U$, is a scalar quantity and its units is Joules (J).

This equation shows that the gravitational potential energy of an object is directly proportional to its mass and the gravitational potential at its position.

## Understanding Gravitational Forces & Fields Through Potential Gradients

When discussing gravitational potential ($\phi$) and gravitational potential energy ($U$), two significant equations offer deeper insights into the nature of gravitational fields and forces. These equations are expressed as derivatives, highlighting the relationship between gravitational potential energy, gravitational potential, and the gravitational field strength ($g$).

### Gravitational Force From Potential Energy

The equation

$$F_{g} = \, – \frac{dU}{dr}$$

represents the gravitational force ($F_{g}$) as the negative derivative of gravitational potential energy ($U$) with respect to distance ($r$). This formulation underscores a fundamental principle: the gravitational force experienced by an object is directly related to the rate at which the potential energy changes with distance in a gravitational field. The negative sign indicates that the gravitational force is attractive, pulling objects together and pointing in the direction of decreasing potential energy.

### Gravitational Field Strength From Gravitational Potential

Similarly, the gravitational field strength ($g$) can be expressed as

$$\begin{aligned} g &= \, – \frac{d \phi}{dr} \\ & \text{OR} \\ g &= \, – \frac{\Delta \phi}{\Delta h} \end{aligned}$$

where $\phi$ is the gravitational potential. This equation, akin to the previous one, defines the gravitational field strength as the negative rate of change of gravitational potential with distance. Here, $g$ is not just a measure of force but also a vector field that points in the direction where the gravitational potential decreases the most steeply. This characteristic is pivotal in understanding how gravitational forces act over space.

#### Potential Gradient & Gravitational Field Strength

The term $\frac{d \phi}{dr}$ is known as the potential gradient. It quantifies how rapidly the gravitational potential changes with respect to distance. The potential gradient is crucial for calculating the gravitational field strength, providing a mathematical way to visualize how gravitational forces vary across different points in a field.

The direction of the gravitational field strength is inherently tied to the concept of decreasing potential. Since $g$ is defined as the negative gradient of the gravitational potential, it always points in the direction of decreasing $\phi$. This means that an object moving under the influence of gravity will naturally follow a path towards regions of lower gravitational potential, reflecting the attraction between masses.

## Equipotential Lines & Gravitational Field Strength

The concept of equipotential lines (or surfaces in three dimensions) is closely related to the discussion on gravitational field strength and potential gradient. These lines are a powerful visual tool for understanding gravitational fields, offering insights into the structure of the field and the motion of objects within it.

Equipotential lines are defined such that at every point along a line within a gravitational field, the gravitational potential $\phi$ is constant. This means that no work is done when moving an object along an equipotential line because the gravitational potential energy of the object remains unchanged. In the context of a gravitational field, these lines are always perpendicular to the gravitational field lines, which point in the direction of the field’s strength and indicate the path of steepest descent for the potential.

### Relationship Between Equipotential Lines and Gravitational Field Strength

The relationship between equipotential lines and gravitational field strength ($g$) is inherently linked through the concept of the potential gradient. Since gravitational field strength is defined as the negative gradient of the gravitational potential ($- \frac{d \phi}{dr}$), it is always directed perpendicular to equipotential lines. This perpendicular relationship is crucial because it means that the gravitational force (and thus the acceleration of any mass in the field) is directed towards regions of lower potential, guiding the motion of celestial bodies and satellites.

### Visualizing Gravitational Fields with Equipotential Lines

Equipotential lines provide a way to visualize gravitational fields. In the field surrounding a single mass, equipotential lines are concentric circles (in two dimensions) or spheres (in three dimensions) centered on the mass. The spacing between these lines indicates the strength of the gravitational field: closely spaced lines signify a steep potential gradient and a strong field, whereas widely spaced lines indicate a weak field.

### Equipotential Lines around Earth

Consider the Earth surrounded by equipotential lines representing different levels of gravitational potential. A satellite moving from one line to a closer line around Earth moves to a region of higher gravitational potential energy, indicating work done against the Earth’s gravitational field. Conversely, if the satellite moves to a line further away, it moves to a region of lower potential, indicating work done by the gravitational field.

## Worked Examples

### Example 1: Calculating Gravitational Potential

Calculate the gravitational potential at a point 10,000 km from the center of the Earth ($M = 5.972 \times 10^{24} \text{ kg}$).

**Click here to show/hide answer**

$\begin{aligned} \phi &= -\frac{GM}{r} \\ &= -\frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})}{10 \times 10^6} \\ &= -\frac{3.986 \times 10^{14}}{10 \times 10^6} \\ &= -39,860 \, \text{J/kg} \end{aligned}$$

### Example 2: Calculating Gravitational Potential Energy

Calculate the gravitational potential energy of a satellite with a mass of 1000 kg, located 10,000 km from the center of the Earth.

**Click here to show/hide answer**

First, use the gravitational potential calculated in Example 1:

$$V = -39,860 \, \text{J/kg}$$

Then, calculate the gravitational potential energy:

$$\begin{aligned} U &= m\phi \\ &= (1000)(-39,860) \\ &= -39,860,000 \, \text{J}\end{aligned}$$

### Example 3

Calculate the gravitational potential 20,000 km from the center of Earth. Take the mass of the Earth to be $5.972 \times 10^{24} \text{ kg}$.

**Click here to show/hide answer**

To calculate the gravitational potential 20,000 km from the center of the Earth, we use the formula:

$$\begin{aligned} \phi &= -\frac{GM}{r} \\ &= -\frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})}{20 \times 10^6} \\ &= -19,928,564 \, \text{J/kg} \end{aligned}$$

Therefore, the gravitational potential 20,000 km from the center of the Earth is $-19,928,564 \, \text{J/kg}$.

### Example 4

A satellite with a mass of 500 kg is in orbit 20,000 km from the center of the Earth. What is its gravitational potential energy?

**Click here to show/hide answer**

Given a satellite with a mass of 500 kg in orbit 20,000 km from the center of the Earth, we calculate its gravitational potential energy using the gravitational potential found in Example 3:

$$\begin{aligned} U &= m \phi \\ &= (500)(-19,928,564) \\ &= -9,964,282,000 \, \text{J} \end{aligned}$$

Hence, the gravitational potential energy of the satellite is $-9,964,282,000 \, \text{J}$.

### Example 5: Calculating Gravitational Potential Difference

Consider a satellite moving from an orbit where the gravitational potential is $-45,000 \, \text{J/kg}$ to another orbit with a gravitational potential of $-35,000 \, \text{J/kg}$. What is the work done against the gravitational field?

**Click here to show/hide answer**

The gravitational potential difference is:

[ \Delta V = -35,000 – (-45,000) = 10,000 \, \text{J/kg} ]

This positive difference indicates that work is done against the gravitational field to move the satellite to a higher orbit. The work done against the gravitational field is $10 \: 000 \text{J/kg}$.