Universal Gravitation and the Foundations of Orbital Motion

Universal Gravitation and the Foundations of Orbital Motion

Topics Covered in Lesson 12A

• Brief History of Astronomy

• Newton’s Law of Universal Gravitation

• Gravitational Attraction of Spherical Bodies

• Kepler’s Laws of Orbital Motion

Brief History of Astronomy

Ptolemy and the Geocentric Model

The development of astronomy began with ancient models of the universe. Ptolemy, who taught in Alexandria around A.D. 140, advocated a geocentric model, placing Earth at the center of the universe.

• Retrograde Motion: To explain the apparent backward movement of planets, Ptolemy introduced two types of circular motions:

  • Deferents: Large orbital circles along which planets move.

  • Epicycles: Small circles whose centers move along the deferents.

• This model was complex but provided a way to predict planetary positions.

Newton’s Law of Universal Gravitation

Fundamental Law

Newton’s Law of Universal Gravitation describes the attractive force between any two masses in the universe.

• Formula:

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

• F is the gravitational force between two masses.

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

• m1 and m2 are the masses of the two objects.

• r is the distance between the centers of the two masses.

Key Properties

• The force is always attractive and acts along the line joining the centers of the two masses.

• According to Newton’s third law, the forces are equal in magnitude and opposite in direction.

• If the distance between two masses is doubled, the force becomes one quarter as strong (inverse-square law).

Gravitational Attraction of Spherical Bodies

Point Mass Approximation

For spherically symmetric bodies (like planets), the gravitational force outside the body is the same as if all the mass were concentrated at the center.

• On the surface of a planet: The distance used in calculations is the radius of the planet.

• Formula for gravitational force at the surface:

$F = G \frac{M m}{R^2}$

• M is the mass of the planet, m is the mass of the object, and R is the planet’s radius.

Acceleration Due to Gravity

• The acceleration due to gravity at the surface of a planet is:

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

• g decreases with altitude above the planet’s surface.

Kepler’s Laws of Orbital Motion

Kepler’s Three Laws

Johannes Kepler formulated three empirical laws describing planetary motion:

1. First Law (Law of Ellipses): Planets move in elliptical orbits with the Sun at one focus.

2. Second Law (Law of Equal Areas): A line joining a planet and the Sun sweeps out equal areas in equal times.

3. Third Law (Law of Periods): The square of a planet’s orbital period is proportional to the cube of its average distance from the Sun.

$T^2 \propto r^3$

• Or, more precisely:

$T^2 = \left(\frac{4\pi^2}{G M_{\text{sun}}}\right) r^3$

• T is the orbital period, r is the mean distance from the Sun.

Applications

• Kepler’s laws are used to predict planetary positions and design satellite orbits (e.g., geosynchronous satellites).

• Geosynchronous satellites have an orbital period equal to one day and remain fixed relative to a point on Earth’s surface.

Summary Table: Key Concepts in Universal Gravitation

Concept	Formula	Key Points
Newton’s Law of Universal Gravitation	$F = G \frac{m_1 m_2}{r^2}$	Force between any two masses; always attractive
Acceleration due to gravity	$g = G \frac{M}{R^2}$	At planet’s surface; decreases with altitude
Kepler’s Third Law	$T^2 = \left(\frac{4\pi^2}{G M_{\text{sun}}}\right) r^3$	Relates orbital period and mean distance

Example: Calculating Gravitational Force

• Find the force between two apples of masses $m_1 = 0.110$ kg and $m_2 = 0.220$ kg separated by $r = 0.132$ m:

$F = G \frac{0.110 \times 0.220}{(0.132)^2}$

• Result: The force is extremely small, illustrating why gravity is only significant for large masses.

Additional Info

• Historical context: The transition from geocentric to heliocentric models was crucial for the development of modern physics and astronomy.

• Newton’s synthesis of celestial and terrestrial mechanics unified the understanding of motion under a single law.