Gravitational Potential Energy Between Objects (College Board AP® Physics 1: Algebra-Based)

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Leander Oates

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Physics

Gravitational potential energy between objects

Gravitational potential energy between two objects

When a system consists of two approximately spherical masses such as moons, planets, or stars, the absolute gravitational potential energy of the system is given by:

$U_{g} = - G \frac{m_{1} m_{2}}{r}$

Where:
- $U_{g}$ = absolute gravitational potential energy, measured in $J$
- $G$ = Universal gravitational constant $(6.67 \times 10^{- 11} N \cdot m^{2} / {kg}^{2})$
- $m_{1} m_{2}$ = the masses of the two objects, measured in $kg$
- $r$ = the separation distance between the center of mass of each object, measured in $m$
Notice that the equation looks very similar to Newton's law of gravitation

$|{\vec{F}}_{g}| = G \frac{m_{1} m_{2}}{r^{2}}$

Consider the masses to be Earth and the Moon
The gravitational pull of the Earth on the Moon is:

$|{\vec{F}}_{g}| = m_{M} g_{E}$

The gravitational field strength is no longer constant when the object is far away from the Earth's surface
Therefore, the absolute gravitational potential energy of the Moon is:

$U_{g M} = m_{M} g_{E} ∆ y$

Substituting in the derived expression for $g_{E}$ :

$U_{g M} = F_{g} ∆ y$

Where $∆ y$ is the separation distance $r$ between the center of mass of the Moon and the center of mass of Earth, which gives:

$U_{g M} = (G \frac{m_{1} m_{2}}{r^{2}}) r = G \frac{m_{E} m_{M}}{r}$

The value of $U_{g}$ is always negative

$U_{g M} = - G \frac{m_{E} m_{M}}{r}$

When the separation distance between the objects approaches infinity, the gravitational field strength tends toward zero
As an object moves away from the Earth, work is done against the gravitational field
The object's gravitational potential energy increases with distance from the Earth to a maximum value at an infinite distance
Therefore, the point of maximum gravitational potential energy is defined to be the zero point
This means that absolute gravitational potential energy will always have a negative value

Gravitational potential energy of a system with more than two objects

Gravitational potential energy is a scalar quantity
Therefore, when a system contains more than two objects, the gravitational potential energy of the system is the sum of the gravitational potential energies of each pair of objects
For a system consisting of a central object (e.g. a planet, moon, or star) and two satellites, the total gravitational potential energy of the system is:

$U_{g s y s} = U_{g 1} + U_{g 2}$

Where
- $U_{g s y s}$ = total gravitational potential energy of the system, measured in $J$
- $U_{g 1}$ = gravitational potential energy of the central object and satellite 1, measured in $J$
- $U_{g 2}$ = gravitational potential energy of the central object and satellite 2, measured in $J$

Worked Example

A satellite of mass $4.5 \times 10^{3} kg$ is transported into an orbit $2000 km$ above the Earth's surface by a rocket of mass $8 \times 10^{3} kg$ . When the orbit is reached, the rocket releases the satellite.

The Earth has a mass of $6 \times 10^{24} kg$ and a radius of $6.4 \times 10^{6} m$ .

Calculate the total gravitational potential energy of the system just after separation.

Answer:

Step 1: Analyze the scenario and list the known quantities

Consider the system to be the Earth, the rocket, and the satellite
- Mass of Earth, $m_{E} = 6 \times 10^{24} kg$
- Mass of rocket, $m_{R} = 8 \times 10^{3} kg$
- Mass of satellite, $M_{S} = 4.5 \times 10^{3} kg$
- Radius of Earth, $r_{E} = 6.4 \times 10^{6} m$
- Distance between Earth and satellite in orbit, $d_{S} = 2000 km = 2 \times 10^{6} m$

Step 2: Write an expression for the total gravitational potential energy of the system

The absolute gravitational potential energy of the system just after separation is:

$U_{g s y s} = U_{g R} + U_{g S}$

$U_{g s y s} = (- G \frac{m_{E} m_{R}}{r_{E} + d_{S}}) + (- G \frac{m_{E} m_{S}}{r_{E} + d_{S}})$

$U_{g s y s} = - G \frac{m_{E}}{r_{E} + d_{S}} (m_{R} + m_{S})$

Step 3: Calculate the total gravitational potential energy of the system

$U_{g s y s} = - (6.67 \times 10^{- 11}) \frac{6 \times 10^{24}}{(6.4 \times 10^{6} + 2 \times 10^{6})} (8 \times 10^{3} + 4.5 \times 10^{3})$

$U_{g s y s} = - 5.96 \times 10^{11} J$

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