SPHY 215 General Physics Spring 2022 Notes

Course Information

• Instructor/Presenter: Laurence L. Henry, PhD

• Course Description: The syllabus and course outline are posted on MOODLE in the Announcements folder. Students must read it carefully and completely.

• Assignment Information:

  • Assignment #1 will be posted shortly.

  • Submission details are provided on MOODLE.

• Reading Requirement: Students are instructed to read relevant references online regarding the topic of Gravitation as indicated in the Course Outline/Syllabus.

Lecture Overview: Gravitation

• Main topics of the lecture include:

  • Concept of a gravitational field

  • Newton's law of gravitation

  • Principle of Superposition and applications to multiple point mass gravitation problems

  • Two shell theorems related to gravitation

  • Definition and explanation of conservative fields

  • Gravitational potential

  • Gravitational potential energy

Newton’s Law of Gravitation

• Definition of Point Mass: A point mass is defined as a mass where all its mass is concentrated at a single point, or when the distances involved are large compared to the size of the masses being considered.

• Mass Representation: Mass is symbolized as M or m and can be distinguished using subscripts (e.g., m1, m2).

• SI Units:

  • Mass must be expressed in kilograms (kg) or their multiples.

  • Distances must be measured in meters (m).

• Newton's Law of Gravitation: The gravitational force between two point masses is given by the formula: $F = \frac{GMm}{r^2}$

  • Where:

  • F = Gravitational force

  • G = Gravitational constant = $6.67 \times 10^{-11} \text{N} \cdot \text{m}^2/\text{kg}^2$

  • M and m = masses of the objects

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

• Force Order of Subscripts: It’s important to note that the order of the subscripts matters, e.g.:

  • For m1 attracting m2: $F<em>{12} = \frac{G m</em>1 m_2}{r^2}$

  • For m2 attracting m1: $F<em>{21} = \frac{G m</em>1 m_2}{r^2}$

Example of Calculating Gravitational Force

• Problem: Calculate the gravitational force between two masses, where m = 25.6 kg and M = 156.8 kg at a distance of r = 15.8 cm.

• Solution: Convert cm to m: r = 0.158 m.

• Force calculation:
  $F = \frac{ (6.67 \times 10^{-11} \text{N} \cdot \text{m}^2/\text{kg}^2)(156.8 \text{kg})(25.6 \text{kg})}{(0.158)^2}$

Principle of Superposition

• Definition: The Principle of Superposition allows us to calculate the net force acting on a point mass due to multiple other point masses.

• Application: The net force is the sum of the forces exerted by all other masses individually acting on the point mass.

• Example: Refer to examples posted on Moodle in announcements regarding this principle.

  • For multiple masses:
    $F<em>5 = F</em>{51} + F<em>{52} + F</em>{53} + F<em>{54} + F</em>{56}$

• Assignment Connection: Problem #5 of Assignment #1 requires the use of the Principle of Superposition.

Two Shell Theorems Dealing with Gravitation

1. Theorem #1: The gravitational force between a point mass m and mass M is given by:
   $F_{mM} = \frac{G m M}{r^2}$

2. Theorem #2: For a point mass m inside a spherical shell of mass M:
   $F_{mM} = 0$

• The implications of these theorems will vary depending on the configuration of the mass distribution.

Conservative Fields

• Definition: A gravitational field is considered a conservative field when the work done moving between two points A and B is independent of the path taken.

• Graphical Representation:

  • Work done in different paths from point A to point B will yield the same result:

  • $W<em>{path1} = W</em>{path2} = W_{path3}$

Gravitational Field Strength (GFS)

• Definition: GFS at a point in a gravitational field caused by mass M is the force acting on a test mass m placed in this field, divided by the mass of the test mass m.

• Mathematically Expressed:
  $GFS = \frac{F}{m} = \frac{GMm/r^2}{m} = \frac{GM}{r^2}$

• Units: The unit for GFS is N/kg.

• Example of GFS:

  • At the surface of Earth:
    $GFS = \frac{(6.67 \times 10^{-11} \text{N} \cdot \text{m}^2/\text{kg}^2)(5.98 \times 10^{24} \text{kg})}{(6.37 \times 10^6 ext{m})^2} = 9.83 ext{N/kg}$

  • This value represents the acceleration due to gravity (g) at the surface of the Earth.

Gravitational Potential (Vg)

• Definition: The gravitational potential Vg at a location at a specific distance from mass M is defined as:
  $V_g = \frac{GM}{r}$

• Units: Measured in Joules/kg (J/kg).

• Examples:

  • (i) Compute the gravitational potential at a distance of 6.89 m from an 8.9 kg mass:
    $V_g = \frac{(6.67 \times 10^{-11} \text{N} \cdot \text{m}^2/\text{kg}^2)(8.9 ext{kg})}{6.89 ext{m}} = 8.62 \times 10^{-11} ext{J/kg}$

Gravitational Potential Energy (Ug)

• Definition: The gravitational potential energy U that a mass m has when located in a gravitational field created by mass M is given by: $U_{gravity} = \frac{GMm}{r}$

  • Reference: Potential energy is considered zero at infinity (when two masses are infinitely far apart).

• Example Calculation:

  • For a mass of 1.6 kg at a distance of 6.89 m from an 8.9 kg mass:
    $U = \frac{(6.67 \times 10^{-11} \text{N} \cdot \text{m}^2/\text{kg}^2)(8.9 ext{kg})(1.6 ext{kg})}{6.89 ext{m}} = 1.38 \times 10^{-10} ext{J}$

Important Notes on Potential and Potential Energy

• Reference Context: Unless a reference is provided, gravitational potential and gravitational potential energy are meaningless.

• Default Reference: The zero of energy is assumed to be at an infinite distance from the mass causing the gravitational field.

• Gravitational Influence: Both masses set up gravitational fields, but the examples often focus on the field present before the second mass is added.

Summary of Key Gravitation Concepts

• Newton’s Law of Gravitation: $F = \frac{GMm}{r^2}$ (magnitude), where the force is attractive.

• Usage of Superposition Principle: To find net gravitational force from multiple point masses.

• Shell Theorems: Highlight behavior of gravitational fields given certain uniform mass distributions.

• Conservative Field Nature: Work in conservative fields is path-independent.

• Gravitational Field Strength: $GFS = \frac{GM}{r^2}$

• Gravitational Potential and Energy:

  • $V_g = \frac{GM}{r}$

  • $U_g = \frac{GMm}{r}$

Closing Remarks on Gravitational Physics

• Gravitation Completion: Problems involving potential energy, example given includes:

  • What is the binding energy of the earth-moon system?
    $U = \frac{GMm<em>{Me}}{R</em>{em}}$

The symbols used in the notes, particularly in the context of gravitation, represent various physical quantities and units:

• F: Represents Gravitational force, measured in Newtons (N).

• G: Denotes the Gravitational constant, with a value of $6.67 \times 10^{-11} \text{N} \cdot \text{m}^2/\text{kg}^2$.

• M and m: Symbolize the masses of objects, typically expressed in kilograms (kg).

• r: Represents the distance between the centers of two masses, measured in meters (m).

• GFS: Stands for Gravitational Field Strength, with units of N/kg.

• Vg: Represents Gravitational Potential, measured in Joules/kg (J/kg).

• U or Ug: Denotes Gravitational Potential Energy, measured in Joules (J).

Additionally, units like kg (kilogram), m (meter), N (Newton), and J (Joule) are used for mass, distance, force, and energy, respectively.