5.1 Absolute Gravitational Potential Energy | Work And Kinetic Energy | Physics 11

Introduction

In physics, potential energy is the energy an object possesses due to its position in a force field. For objects near the Earth's surface, the simple formula $P . E . = m g h$ is a useful approximation. However, for objects at large distances — such as satellites — a more general definition is needed. Absolute Gravitational Potential Energy provides this framework by defining the potential energy of a mass at any point in a gravitational field relative to a universal zero point.

Figure 5.1: Gravitational potential energy concept showing work done by gravity as an object moves from position r to infinity

Definition

The absolute potential energy of a body at a point in a gravitational field is defined as the work done by the gravitational force in moving the body from that point to a position of zero potential.

Zero Reference Point: By convention, the position of zero gravitational potential energy is chosen to be at an infinite distance ( $r = \infty$ ) from the source of gravity (e.g., the Earth).
Negative Value: Since gravity is an attractive force, the gravitational field does positive work when a mass moves away from the Earth. Consequently, the potential energy of a mass at any finite distance $r$ is always negative. This negative value signifies that the mass is gravitationally "bound" to the Earth and requires an input of energy to escape to infinity.

Mathematical Derivation

To find the absolute potential energy at a point, we calculate the total work done by the gravitational force when moving a mass $m$ from an initial point $r_{1}$ to a final point at infinity. The path is divided into a large number of very small intervals.

Step 1: Work Done Over a Small Interval

Let's calculate the work done, $W_{1 \to 2}$ , as the mass moves a small distance from $r_{1}$ to $r_{2}$ . The change in distance is $Δ r = r_{2} - r_{1}$ . The gravitational force is not constant over this interval, so we use an average position $r$ for the force calculation.

$F_{g} = \frac{GM m}{r ^{2}}$

Through a clever approximation for a very small interval, it can be shown that the average value of $r^{2}$ is approximately $r_{1} r_{2}$ . The work done by the gravitational force (which points inward) over the outward displacement $Δ r$ is:

$W_{1 \to 2} = F_{g} Δ r cos (18 0^{\circ}) = - \frac{GM m}{r ^{2}} Δ r$

Using the approximation $r^{2} \approx r_{1} r_{2}$ :

$W_{1 \to 2} = - \frac{GM m}{r _{1} r _{2}} (r_{2} - r_{1})$

Separating the terms gives:

$W_{1 \to 2} = - GM m (\frac{r _{2}}{r _{1} r _{2}} - \frac{r _{1}}{r _{1} r _{2}}) = - GM m (\frac{1}{r _{1}} - \frac{1}{r _{2}})$

Step 2: Total Work Done (Telescoping Sum)

The total work done ( $W_{t}$ ) in moving the mass from $r_{1}$ to a distant point $r_{n}$ is the sum of the work done in all the small intervals:

$W_{t} = W_{1 \to 2} + W_{2 \to 3} + \dots + W_{n - 1 \to n}$

$W_{t} = - GM m [(\frac{1}{r _{1}} - \frac{1}{r _{2}}) + (\frac{1}{r _{2}} - \frac{1}{r _{3}}) + \dots + (\frac{1}{r _{n - 1}} - \frac{1}{r _{n}})]$

This is a telescoping sum — all intermediate terms cancel, leaving only the first and last:

$W_{t} = - GM m (\frac{1}{r _{1}} - \frac{1}{r _{n}})$

Step 3: Moving the Mass to Infinity

To find the absolute potential energy, the final destination is infinity, so we let $r_{n} \to \infty$ . As $r_{n}$ becomes infinitely large, $\frac{1}{r _{n}} \to 0$ :

$W_{t} = - GM m (\frac{1}{r _{1}} - 0) = - \frac{GM m}{r _{1}}$

Replacing $r_{1}$ with a general distance $r$ , we obtain the final formula.

Final Formula for Absolute Potential Energy ( $U_{g}$ ):

$U_{g} (r) = - \frac{GM m}{r}$

Key Conceptual Points

Why is $U_{g}$ always negative?
The zero reference is at infinity. Since gravity is attractive, a mass at finite $r$ is in a lower energy state. To move it back to infinity, external work must be done, raising PE from a negative value up to zero.
How does $m g h$ relate to $U_{g}$ ?
The formula $m g h$ is an approximation valid only for small height changes near Earth's surface, where $g$ is approximately constant. It gives the change in PE relative to a local zero, not the absolute PE relative to infinity.
Conservative Nature of Gravity:
Gravitational PE is defined because gravity is a Conservative And Non Conservative Fields→. The work done by gravity depends only on initial and final positions, not on the path taken.

Summary

Concept	Formula / Value
Absolute Gravitational Potential Energy	$U_{g} = - \frac{GM m}{r}$
Zero reference point	$r = \infty$
Sign of $U_{g}$	Always negative (finite $r$ )
Limiting case	$U_{g} \to 0$ as $r \to \infty$

Introduction

Definition

Zero Reference Point: By convention, the position of zero gravitational potential energy is chosen to be at an infinite distance ( $r = \infty$ ) from the source of gravity (e.g., the Earth).
Negative Value: Since gravity is an attractive force, the gravitational field does positive work when a mass moves away from the Earth. Consequently, the potential energy of a mass at any finite distance $r$ is always negative. This negative value signifies that the mass is gravitationally "bound" to the Earth and requires an input of energy to escape to infinity.

Mathematical Derivation

Step 1: Work Done Over a Small Interval

$F_{g} = \frac{GM m}{r ^{2}}$

$W_{1 \to 2} = F_{g} Δ r cos (18 0^{\circ}) = - \frac{GM m}{r ^{2}} Δ r$

Using the approximation $r^{2} \approx r_{1} r_{2}$ :

$W_{1 \to 2} = - \frac{GM m}{r _{1} r _{2}} (r_{2} - r_{1})$

Separating the terms gives:

$W_{1 \to 2} = - GM m (\frac{r _{2}}{r _{1} r _{2}} - \frac{r _{1}}{r _{1} r _{2}}) = - GM m (\frac{1}{r _{1}} - \frac{1}{r _{2}})$

Step 2: Total Work Done (Telescoping Sum)

The total work done ( $W_{t}$ ) in moving the mass from $r_{1}$ to a distant point $r_{n}$ is the sum of the work done in all the small intervals:

$W_{t} = W_{1 \to 2} + W_{2 \to 3} + \dots + W_{n - 1 \to n}$

$W_{t} = - GM m [(\frac{1}{r _{1}} - \frac{1}{r _{2}}) + (\frac{1}{r _{2}} - \frac{1}{r _{3}}) + \dots + (\frac{1}{r _{n - 1}} - \frac{1}{r _{n}})]$

This is a telescoping sum — all intermediate terms cancel, leaving only the first and last:

$W_{t} = - GM m (\frac{1}{r _{1}} - \frac{1}{r _{n}})$

Step 3: Moving the Mass to Infinity

To find the absolute potential energy, the final destination is infinity, so we let $r_{n} \to \infty$ . As $r_{n}$ becomes infinitely large, $\frac{1}{r _{n}} \to 0$ :

$W_{t} = - GM m (\frac{1}{r _{1}} - 0) = - \frac{GM m}{r _{1}}$

Replacing $r_{1}$ with a general distance $r$ , we obtain the final formula.

Final Formula for Absolute Potential Energy ( $U_{g}$ ):

$U_{g} (r) = - \frac{GM m}{r}$

Key Conceptual Points

Why is $U_{g}$ always negative?
The zero reference is at infinity. Since gravity is attractive, a mass at finite $r$ is in a lower energy state. To move it back to infinity, external work must be done, raising PE from a negative value up to zero.
How does $m g h$ relate to $U_{g}$ ?
The formula $m g h$ is an approximation valid only for small height changes near Earth's surface, where $g$ is approximately constant. It gives the change in PE relative to a local zero, not the absolute PE relative to infinity.
Conservative Nature of Gravity:
Gravitational PE is defined because gravity is a Conservative And Non Conservative Fields→. The work done by gravity depends only on initial and final positions, not on the path taken.

Summary

Concept	Formula / Value
Absolute Gravitational Potential Energy	$U_{g} = - \frac{GM m}{r}$
Zero reference point	$r = \infty$
Sign of $U_{g}$	Always negative (finite $r$ )
Limiting case	$U_{g} \to 0$ as $r \to \infty$