Universal Gravitation and Gravitational Potential Energy

Universal Gravitation

Gravitational Force Between Two Masses

The concept of universal gravitation describes the attractive force between any two masses in the universe. This force is responsible for the motion of celestial bodies and is fundamental to understanding planetary orbits and satellite motion.

• Universal Gravitation Equation: The force of gravity between two masses is given by: $F_g = G \frac{m_1 m_2}{r^2}$ where:

  • G = Universal Gravitation Constant ($6.67 \times 10^{-11} \ N\,m^2/kg^2$)

  • m_1, m_2 = masses of the two objects

  • r = distance between the centers of mass of the objects

• Example: The centripetal force acting on the Moon is the gravitational attraction from the Earth.

• Gravitational acceleration (g): Near Earth's surface, $g = 9.8 \ m/s^2$. Away from the surface, $g$ decreases with distance.

Gravitational Field and Force

The gravitational field is a region of space around a mass where another mass experiences a force. The field concept helps describe how gravity acts at a distance.

• Definitions:

  • Force: Requires two interacting objects; gets stronger as they get closer.

  • Field: Emanates from a single object; strength increases closer to the source.

• Calculating Gravitational Field: $g = \frac{F_g}{m_{obj}} = G \frac{m_{src}}{r^2}$ where $m_{src}$ is the source mass and $m_{obj}$ is the test mass.

• Force fields: All masses produce gravitational fields; all charges produce electric fields. These are vector fields.

Weight at Different Distances

The weight of an object changes with distance from the center of the Earth due to the variation in gravitational field strength.

• Example: A 2520 kg boulder thrown from 100 km above Earth:

  • Weight at surface: $2.47 \times 10^4 \ N$

  • Weight at 100 km height: $2.40 \times 10^4 \ N$

Circular Orbits and Orbital Velocity

Conditions for Circular Orbit

An object in circular orbit around a planet must have a tangential velocity such that the centripetal force equals the gravitational force.

• Equating Forces: $F_c = F_g$ $m_b \frac{v_T^2}{r} = G \frac{m_E m_b}{r^2}$ where:

  • $m_b$ = mass of the boulder

  • $m_E$ = mass of Earth

  • $r$ = distance from Earth's center

• Solving for Tangential Velocity: $v_T = \sqrt{\frac{G m_E}{r_{ES}}}$ Example calculation:

  • $G = 6.67 \times 10^{-11} \ N\,m^2/kg^2$

  • $m_E = 5.972 \times 10^{24} \ kg$

  • $r_{ES} = 6.471 \times 10^6 \ m$ (Earth radius + 100 km)

  • Result: $v_T = 7.846 \times 10^3 \ m/s$

• Too slow: The object falls to Earth (projectile motion).

• Too fast: The object escapes Earth's gravity (flies off into space).

Gravitational Potential Energy

Definition and Equation

Gravitational potential energy (Ug) is the energy associated with the position of an object in a gravitational field. It is negative because the zero reference is at infinite distance.

• Equation: $U_g = -G \frac{m_s m_o}{r}$where:

  • $m_s$ = source mass

  • $m_o$ = object mass

  • $r$ = distance between centers

• Why negative? Ug decreases as an object falls toward the source mass; energy is transformed into kinetic energy.

• Zero reference: $U_g = 0$ when $r \rightarrow \infty$.

• Change in Gravitational Potential Energy: $\Delta U_g = -G \frac{m_s m_o}{r_2} - \left(-G \frac{m_s m_o}{r_1}\right)$

• Interpretation:

  • Increasing height: Ug becomes less negative; $\Delta U_g$ is positive.

  • Decreasing height: Ug becomes more negative; $\Delta U_g$ is negative.

Summary Table: Gravitational Force and Potential Energy Equations

Quantity	Equation
Gravitational Force	$F_g = G \frac{m_s m_o}{r^2}$
Gravitational Field	$g = G \frac{m_{src}}{r^2}$
Gravitational Potential Energy	$U_g = -G \frac{m_s m_o}{r}$
Change in Potential Energy	$\Delta U_g = -G \frac{m_s m_o}{r_2} - \left(-G \frac{m_s m_o}{r_1}\right)$

Key Points and Applications

• Use "big G" equation: When not near Earth's surface or when "g" is not given.

• "Little g" shortcut: Only valid near Earth's surface.

• Force fields: All masses and charges produce fields; fields are vectors.

• Potential energy: Always negative; increases (becomes less negative) with height.

• Energy transformation: As an object falls, gravitational potential energy is converted to kinetic energy.

Additional info: The notes expand on the distinction between force and field, clarify the negative sign in gravitational potential energy, and provide context for orbital motion and energy changes in gravitational systems.