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

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 $\mathrm{J}$

  • $G$ = Universal gravitational constant $\left(6.67\times {10}^{-11}\mathrm{N}·{\mathrm{m}}^2/{\mathrm{kg}}^2\right)$

  • $m_1{m}_2$ = the masses of the two objects, measured in $\mathrm{kg}$

  • $r$ = the separation distance between the center of mass of each object, measured in $\mathrm{m}$

• Notice that the equation looks very similar to Newton's law of gravitation

$\left|{\overrightarrow{F}}_g\right|=G\frac{m_1{m}_2}{r^2}$

• Consider the masses to be Earth and the Moon

• The gravitational pull of the Earth on the Moon is:

$\left|{\overrightarrow{F}}_g\right|=m_M{g}_E$

• 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:

$U_{gM}=m_M{g}_E∆y$

• Substituting in the derived expression for $g_E$:

$U_{gM}=F_g∆y$

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

$U_{gM}=\left(G\frac{m_1{m}_2}{r^2}\right)r=G\frac{m_E{m}_M}{r}$

• The value of $U_g$ is always negative

$U_{gM}=-G\frac{m_E{m}_M}{r}$

• 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:

$U_{gsys}=U_{g1}+U_{g2}$

• Where

  • $U_{gsys}$ = total gravitational potential energy of the system, measured in $\mathrm{J}$

  • $U_{g1}$ = gravitational potential energy of the central object and satellite 1, measured in $\mathrm{J}$

  • $U_{g2}$ = gravitational potential energy of the central object and satellite 2, measured in $\mathrm{J}$