D.1.3 Gravitational potential energy and potential (HL only)

Gravitational Potential Energy and Gravitational Potential

Gravitational Potential Energy ( $E_{p}$ )

Definition

Gravitational potential energy

Gravitational potential energy is the energy stored due to the position of an object in a gravitational field.

Note

Gravitational potential energy is always negative because it is defined relative to a point at infinity, where the potential energy is zero.

The formula for gravitational potential energy between two masses $m_{1}$ and $m_{2}$ separated by a distance $r$ is:

$E_{p} = - G \frac{m_{1} m_{2}}{r}$

where:

$G$ is the gravitational constant ( $6.67 \times 10^{- 11} {N m}^{2} {kg}^{- 2}$ ).
$r$ is the distance between the centers of the masses.

Example question

Gravitational potential energy

Consider a satellite of mass $1500 kg$ orbiting Earth at a distance of $7000 km$ from its center. Calculate its gravitational potential energy.

Solution

Substituting the given values:

$E_{p} = - G \frac{M_{Earth} \cdot m_{satellite}}{r}$

$E_{p} = - \frac{6.67 \times 10^{- 11} \times 6.0 \times 10^{24} \times 1500}{7.0 \times 10^{6}}$

$\approx - 8.57 \times 10^{10} J$

Gravitational Potential ( $V_{g}$ )

Definition

Gravitational potential

Gravitational potential ( $V_{g}$ ) is the work done per unit mass to bring a small test mass from infinity to a point in a gravitational field.

It is given by:

$V_{g} = - G \frac{M}{r}$

where:

$M$ is the mass creating the gravitational field.
$r$ is the distance from the center of the mass.

Note

Gravitational potential is a scalar quantity and is measured in joules per kilogram (J kg⁻¹).

Example question

Gravitational potential

Calculate the gravitational potential at a distance of $10, 000 km$ from Earth’s center.

Solution

$V_{g} = - G \frac{M_{Earth}}{r}$

$= - \frac{6.67 \times 10^{- 11} \times 6.0 \times 10^{24}}{1.0 \times 10^{7}}$

$\approx - 4.0 \times 10^{7} {J kg}^{- 1}$

Relationship Between $E_{p}$ and $V_{g}$

The gravitational potential energy $E_{p}$ of a mass $m$ at a point in a gravitational field is related to the gravitational potential $V_{g}$ at that point by:

$E_{p} = m V_{g}$

Example

A $200 kg$ satellite is at a point where the gravitational potential is $- 5.0 \times 10^{6} {J kg}^{- 1}$ .

The gravitational potential energy is:

$E_{p} = m V_{g} = 200 \times (- 5.0 \times 10^{6}) = - 1.0 \times 10^{9} J$

Field Strength as a Potential Gradient

Gravitational Field Strength ( $g$ )

Definition

Gravitation field strength

The gravitational field strength ( $g$ ) at a point is defined as the force per unit mass experienced by a small test mass placed at that point.

It is given by:

$g = \frac{G M}{r^{2}}$

Note

Gravitational field strength is a vector quantity and is measured in newtons per kilogram (N kg⁻¹).

Potential Gradient

The gravitational field strength can also be expressed as the negative gradient of the gravitational potential:

$g = - \frac{Δ V_{g}}{Δ r}$

This means that the field strength is the rate of change of potential with respect to distance.

Gravitation potential versus distance graph.

Tip

The negative sign indicates that the gravitational field points in the direction of decreasing potential.

Example

If the gravitational potential decreases by $2.0 \times 10^{6} {J kg}^{- 1}$ over a distance of $500 m$ , the field strength is:

$g = - \frac{Δ V_{g}}{Δ r}$
$= - \frac{- 2.0 \times 10^{6}}{500} = 4000 {N kg}^{- 1}$

Work in Gravitational Fields

Work Done Moving a Mass

When a mass $m$ is moved in a gravitational field, the work done by an external force is equal to the change in gravitational potential energy.
This can be expressed as:

$W = m Δ V_{g}$

where $Δ V_{g}$ is the change in gravitational potential between the initial and final positions.

Example

A $500 kg$ satellite is moved from a point where $V_{g} = - 4.0 \times 10^{6} {J kg}^{- 1}$ to a point where $V_{g} = - 2.0 \times 10^{6} {J kg}^{- 1}$ .

The work done is:

$W = m Δ V_{g}$

$= 500 \times ((- 2.0 \times 10^{6}) - (- 4.0 \times 10^{6}))$

$= 1.0 \times 10^{9} J$

Note

Remember that work done in a gravitational field is path-independent. It depends only on the initial and final positions.

Key Takeaways

Gravitational Potential Energy( $E_{p}$ ):
- Energy stored due to the position of two masses in a gravitational field, given by $E_{p} = - G \frac{m_{1} m_{2}}{r}$ .
Gravitational Potential( $V_{g}$ ):
- Work done per unit mass to bring a test mass from infinity to a point, given by $V_{g} = - G \frac{M}{r}$ .
Field Strength as a Potential Gradient:
- Gravitational field strength is the negative gradient of potential, $g = - \frac{Δ V_{g}}{Δ r}$ .
Work in Gravitational Fields:
- Work done moving a mass is $W = m Δ V_{g}$ .

Self review

How is gravitational potential energy different from gravitational potential?
Why is gravitational potential energy negative?
How does the concept of potential gradient relate to gravitational field strength?

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D.1.3 Gravitational potential energy and potential (HL only)

Gravitational Potential Energy and Gravitational Potential

Gravitational Potential Energy (Ep)

Gravitational potential energy

Gravitational Potential (Vg)

Gravitational potential

Relationship Between Ep and Vg

Field Strength as a Potential Gradient

Gravitational Field Strength (g)

Potential Gradient

Work in Gravitational Fields

Work Done Moving a Mass

Key Takeaways

You've read 2/2 free chapters this week.

Gravitational Potential Energy

Gravitational Potential Energy ( $E_{p}$ )

Gravitational Potential ( $V_{g}$ )

Relationship Between $E_{p}$ and $V_{g}$

Gravitational Field Strength ( $g$ )