Universal Gravitation and Gravitational Potential Energy

Universal Gravitation

Gravitational Force Between Two Masses

The law of universal gravitation describes the attractive force between any two masses in the universe. This force is always attractive and acts along the line joining the centers of the two masses.

• Newton's Law of Universal Gravitation: The gravitational force between two point masses is given by:

$F_g = \frac{G m_1 m_2}{r^2}$

• G is the universal gravitational constant: $G = 6.67 \times 10^{-11} \ \mathrm{N \cdot m^2 / kg^2}$

• m1 and m2 are the masses of the two objects

• r is the distance between the centers of mass of the two objects

Gravitational Acceleration (g)

The acceleration due to gravity near the Earth's surface is approximately $9.8 \ \mathrm{m/s^2}$, but this value decreases with altitude and distance from the Earth's center.

• g is defined as the gravitational field strength:

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

• Where M is the mass of the source (e.g., Earth), and r is the distance from its center.

• At Earth's surface, $g = 9.8 \ \mathrm{m/s^2}$, but it is less at higher altitudes.

Force vs. Field

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

• Field: Emanates from a single object; the field strength increases as you approach the source. Placing a second object in the field results in a force pair.

Calculating Gravitational Field from Force

• The gravitational field can be calculated by dividing the gravitational force by the mass of the object experiencing the force:

$g = \frac{F_g}{m_{obj}} = \frac{G m_{src}}{r^2}$

• msrc is the source mass (e.g., Earth), mobj is the object in the field.

Examples and Applications

• Example: The Moon orbits the Earth due to the gravitational force between them. The acceleration due to gravity at the Moon's distance is much less than $9.8 \ \mathrm{m/s^2}$.

• Example: Calculating the weight of a 2520 kg boulder at Earth's surface and at 100 km above the surface:

  • At surface: $F_g = 2.47 \times 10^4 \ \mathrm{N}$

  • At 100 km: $F_g = 2.40 \times 10^4 \ \mathrm{N}$

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 gravitational force provides the necessary centripetal force.

• Set centripetal force equal to gravitational force:

$F_c = F_g$

$m_b \frac{v_T^2}{r} = \frac{G m_E m_b}{r^2}$

• Solve for tangential velocity $v_T$:

$v_T = \sqrt{\frac{G m_E}{r}}$

• Example: For a boulder 100 km above Earth's surface ($r = 6.471 \times 10^6$ m):

• $v_T = 7.846 \times 10^3 \ \mathrm{m/s}$

Consequences of Incorrect Velocity

• If thrown too slowly, the object falls back to Earth (projectile motion).

• If thrown too fast, the object escapes Earth's gravity and flies into space.

Gravitational Potential Energy

Definition and Equation

Gravitational potential energy (Ug) in the context of universal gravitation is the energy associated with the position of an object in a gravitational field, relative to infinity.

• The gravitational potential energy between two masses is:

$U_g = -\frac{G m_1 m_2}{r}$

• Ug is negative because the zero point is defined at infinite separation ($r \to \infty$).

• As an object falls toward Earth, Ug decreases (becomes more negative) and is converted into kinetic energy.

Change in Gravitational Potential Energy

• The change in gravitational potential energy as an object moves from $r_1$ to $r_2$ is:

$\Delta U_g = U_{g,2} - U_{g,1} = -\frac{G m_1 m_2}{r_2} + \frac{G m_1 m_2}{r_1}$

Comparison: "Little g" vs. "Big G" Equations

• Use the "big G" equation when:

  • You are not given "g" or the problem is not near Earth's surface.

  • Earth is not one of the two objects involved.

• "Little g" is a shortcut for calculations near Earth's surface.

Force Fields in Physics

Gravitational and Electric Fields

• All objects with mass produce a gravitational field.

• All objects with electric charge produce an electric field.

• Both gravitational and electric fields are vector fields, meaning they have both magnitude and direction at every point in space.

Summary Table: Key Equations in Universal Gravitation

Quantity	Equation	Description
Gravitational Force	$F_g = \frac{G m_1 m_2}{r^2}$	Force between two masses
Gravitational Field (g)	$g = \frac{G M}{r^2}$	Field strength at distance r from mass M
Orbital Velocity	$v_T = \sqrt{\frac{G M}{r}}$	Velocity for circular orbit
Gravitational Potential Energy	$U_g = -\frac{G m_1 m_2}{r}$	Potential energy relative to infinity
Change in Potential Energy	$\Delta U_g = -\frac{G m_1 m_2}{r_2} + \frac{G m_1 m_2}{r_1}$	Change as object moves from $r_1$ to $r_2$

Key Takeaways

• Universal gravitation applies everywhere, not just near Earth's surface.

• Gravitational field strength and potential energy can be calculated for any mass and distance using the universal gravitation equations.

• Gravitational potential energy is always negative, with zero defined at infinite separation.

• Force fields (gravitational, electric) are fundamental concepts in physics, describing how objects interact at a distance.