Comprehensive Notes on Newtonian Gravity Topics

Newtonian Gravity Lecture Notes

Instructor: Omar Falou
Source: Serway / Jewett
Sections: 13.1 – 13.3
Institution: Toronto Metropolitan University

Outline / Goals

Acquire a better understanding of how long-range forces, particularly gravity, operate through a gravitational field.
Understand Newton’s Law of Universal Gravitation in detail.
Comprehend how the observed free-fall acceleration near Earth's surface is influenced by the fundamental constant G, as well as the mass and radius of the Earth.
Apply Newton's Law of Gravity and laws of motion to analyze simple circular orbits.

The Concept of Gravity and Weightlessness

Misconception about Weightlessness:
- An astronaut is often thought to be weightless due to a lack of gravity in space. This is False. Weightlessness is experienced because they are in free fall, not due to the absence of gravity.

Fundamental Questions in Gravity

Field Concept:
- How does gravity work?
- How does the Moon know its position in relation to the Earth?

Newton on “Action at a Distance”

Isaac Newton's Quote from 1692/3:
- “It is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual Contact…so that one body may act upon another at a distance thro' a vacuum, without the mediation of any thing else…is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain laws; but whether this Agent be material or immaterial, I have left to the Consideration of my readers."

Conceptual Framework of Gravity

Mass and Fields:
  - Masses create gravitational fields.
  - Masses respond to gravitational fields created by other masses.
  - This framework is useful for understanding concepts in Electricity and Magnetism as well.
  - Current understanding of gravity remains incomplete, indicating further exploration is needed.

Gravitational Field Overview

Definition of Gravitational Field:
  - A gravitational field is depicted as a vector labeling every point in space, where:
    - Magnitude: Represents the strength of the gravitational field.
    - Direction: Indicates the direction of the gravitational field, pointing toward the center of mass from which it originates.
  - A field exists around a mass even if the surrounding space is empty.

Gravitational Field Strength

Field Strength Characteristics:
  - The gravitational field strength decreases with increasing distance from the point mass.
  - Formulated as:
    - $g = - \frac{GM}{r^2}$
      - Where:
        - $g$ = gravitational field strength
        - $G$ = gravitational constant
        - $M$ = mass of the object creating the field
        - $r$ = distance from the center of mass to the observation point
Example Question:
  - Comparing gravitational field strengths:
    - Let $g_1$ be the strength 10 m above Earth's surface and $g_2$ be 20 m above Earth's surface.
    - Options:
      - A) $g_1$ is four times as large as $g_2$ .
      - B) $g_1$ is twice as large as $g_2$ .
      - C) $g_1$ is approximately the same as $g_2$ .
      - D) $g_1$ is half as large as $g_2$ .
      - E) The answer depends on the mass at those points.

Gravitational Field Near Earth's Surface

Field Characteristics:
- Near Earth's surface, gravitational field vectors point downward toward the center of the Earth.
- The magnitude of gravitational field strength remains approximately constant due to the negligible change in distance from the Earth's center (radius, $R$ ).

Adding Gravitational Fields from Multiple Masses

Concept:
  - Each mass contributes to the gravitational field present at a given point.
  - Method of addition:
    - Treat each mass individually and draw the gravitational field as if it were the only mass present.
    - To find the total gravitational field at a point, add the vector contributions from individual masses.

Forces in a Gravitational Field

Force on a Mass in a Field:
- A mass (m) placed in a gravitational field experiences a force.
- Important Note: This field $g$ created is due to other masses and is external to mass $m$ itself; thus, it does not exert a force on itself.

Newton’s Law of Universal Gravitation

Law Definition:
  - All particles in the universe exert gravitational forces on one another.
  - The magnitude of the force between two masses ( $m_1$ and $m_2$ ) is given by:
    - $F = G\frac{m_1m_2}{r^2}$
      - Where:
        - $F$ = magnitude of gravitational force
        - $G$ = universal gravitational constant
        - $r$ = distance between the center of both masses
  - This force is always attractive and aligns along the line connecting the centers of the two objects.

Gravitational Force Example Comparisons

Example Problem:
  - Compare gravitational forces acting on a person (mass = 60 kg) from two sources:
    1. Nearest Star (Alpha Centauri):
       - Distance = 4.37 light-years
       - Mass = $1.1 imes M_{sun}$ where $M_{sun} = 2 imes 10^{30} ext{ kg}$
    2. Person beside in Class:
       - Mass = 60 kg
       - Distance apart = 1 m
Focus on determining which gravitational force has a larger magnitude.

Graduating Examples in Gravity Application

Three Billiard Balls Scenario:
  - Three balls, each with mass $m = 0.3 ext{ kg}$ , positioned in a triangle:
    - Side lengths: $a = 0.4 ext{ m}, b = 0.3 ext{ m}, c = 0.5 ext{ m}$
  - Task: Calculate the force's direction and magnitude acting on $m_1$ due to the other two masses, involving:
    1. Determining the field created at $m_1$ 's position.
    2. Calculating the resultant force.

Gravity and Orbits

Understanding Orbits:
- An object in orbit is essentially in free fall; the planet curves away from it as it falls.
- If the initial velocity is sufficiently high, an object will enter a stable orbit around a planet.

Application of Newton's Laws in Circular Orbits

Circular Orbit Analysis:
  - For a planet $P$ orbiting a star $S$ :
    - Apply Newton's 2nd Law of Motion to derive:
      - $F_{ ext{centripetal}} = F_{ ext{gravity}}$
  - Representing:
    - $\frac{mv^2}{r} = \frac{GMm}{r^2}$
      - Where:
        - $v$ = orbital speed
        - $r$ = radius of orbit
Orbital Speed Derived:
- The orbital speed can be obtained as a function of the radius of orbit.

Satellites and Orbital Period Insights

Satellite Mass Independence on Orbital Period:
  - Considering two satellites in the same orbit with different masses:
    - Question: Which takes longer to complete one orbit?
      - A) The one with more mass
      - B) The one with less mass
      - C) The orbital periods are identical

Geostationary Orbit Analysis

Geostationary Orbit Requirements:
- A geostationary satellite must orbit at a specific height above Earth's surface.
- The orbit must be directly above the equator to maintain a fixed position in the sky.

Deriving Relationships in Geostationary Orbits

Centripetal vs Gravitational Forces:
  - In a geostationary orbit, the centripetal force required for uniform circular motion equals the gravitational force.
  - Leading to the equation:
    - Radius of Geostationary Orbit Relationships:
      - $r = R + h$ where $h$ is the altitude above Earth's surface.
      - Period of Orbit Relation:
        - Orbital period $T$ equates to the radius of orbit, reflecting a deeper insight into gravitational mechanics.

Summary of Key Points

All objects with mass create a gravitational field surrounding them.
The field from a point mass or spherical mass has a magnitude that points toward the mass's center.
To find a net gravitational field at any point in space, treat each object independently and add the individual fields as vectors.
The force acting on an object in a gravitational field relates to the mass sitting in the field, aligned with Newton's Law of Universal Gravitation, which also integrates circular orbits into classical mechanics understanding.