Chapter 06: Gravitation and Newton's Synthesis

Overview

This chapter explores the fundamental concepts of gravitation, Newton's law of universal gravitation, and the synthesis of planetary motion laws. It covers the nature of gravitational forces, their mathematical formulation, and their application to planetary and satellite motion.

6-1 Newton's Law of Universal Gravitation

Definition and Historical Context

• Gravitational Force: The mutual force of attraction between any two objects in the Universe.

• Newton's Law of Universal Gravitation: Every object in the Universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Formula:

• G: Universal gravitation constant,

• m1, m2: Masses of the two objects

• r: Distance between the centers of the two masses

Example: The gravitational force between two people sitting on a bench is extremely small, but calculable using the above formula.

6-2 Vector Form of Newton's Law of Universal Gravitation

Direction and Action-Reaction Principle

• Gravitational force acts along the line joining the centers of mass of two objects.

• Newton's Third Law: The force that mass 1 exerts on mass 2 is equal and opposite to the force mass 2 exerts on mass 1.

Vector Form:

• Where is the unit vector from one mass to the other.

6-3 Gravity Near the Earth's Surface

Local Gravitational Acceleration

• Near Earth's surface, the acceleration due to gravity is approximately .

• Gravity varies slightly with altitude and location due to Earth's shape and rotation.

Formula Relating G and g:

• : Mass of the Earth

• : Radius of the Earth

Example: Calculating the value of at the top of Mt. Everest (8850 m above sea level).

6-4 Satellites and "Weightlessness"

Satellite Motion and Apparent Weightlessness

• Satellites in orbit are in continuous free fall, resulting in apparent weightlessness.

• No normal force acts on objects in orbit, which is why astronauts feel weightless.

• Geosynchronous satellites remain above the same point on Earth's equator, useful for communication and weather forecasting.

Example: Determining the height and speed required for a geosynchronous satellite.

6-5 Planets, Kepler's Laws, and Newton's Synthesis

Kepler's Laws of Planetary Motion

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

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

• Third Law (Law of Harmonies): The square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit.

Kepler's Third Law (Mathematical Form):

• : Orbital period

• : Semi-major axis

Orbital Eccentricity: Describes how much an orbit deviates from a perfect circle. Earth's orbit has low eccentricity (), while comets can have high eccentricity ().

6-7 Types of Forces in Nature

Fundamental Forces

• Gravity

• Electromagnetism

• Weak Nuclear Force (responsible for some types of radioactive decay)

• Strong Nuclear Force (binds protons and neutrons in the nucleus)

Most everyday forces (normal force, tension, friction) are manifestations of electromagnetic interactions at the atomic level.

6-8 Gravitational Field

Definition and Calculation

• The gravitational field at a point is the gravitational force per unit mass at that point.

• For a single mass :

6-9 Principle of Equivalence; Curvature of Space; Black Holes

Advanced Concepts in Gravitation

• Principle of Equivalence: Inertial mass and gravitational mass are equivalent.

• Deflection of Light: Light is deflected by massive objects due to the curvature of space, as confirmed by observations during solar eclipses.

• Black Holes: Regions of space where gravity is so strong that not even light can escape. Visualized as extreme curvature of space.

Table: Comparison of Fundamental Forces

Additional info: These notes expand on the provided slides and images, adding definitions, formulas, and examples for clarity and completeness. The table is inferred from standard physics content.