# UTPA STEM/CBI Courses/Physics (Calculus Based)/Universal Gravitational Force

Course Title: Calculus Based Physics I

Lecture Topic: Universal Gravitational Force

Instructor: Feng

Institution: University of Texas-Pan American

## Backwards Design

[edit | edit source]**Course Objectives**

**Primary Objectives**- By the next class period students will be able to:- Know that the universal gravitational force applies to any object that has mass
- Know how to calculate the magnitude of the universal gravitational force (G = 6.67 x 10-11 Nm2kg-2) F = Gm1m2/r122
- r can be taken to be (altitude + radius of earth) for problems considering space objects
- Weight mg comes from using the universal gravitational force formula where m2 is the mass of earth and r12 is the radius of earth, m1 is the person’s mass standing on earth’s surface, and g = Gm2/r122 = 9.8 m/s2
- Know how to discern when the gravitational force has no noticeable effect and is negligible (between two people the gravitational force is not noticeable because the masses are not heavy enough and G is so small; between planets/sun the gravitational force is noticeable)
- Know that in subatomic particle (electrons, protons) the Coulomb force (introduce formula for Coulomb force) is dominant over the gravitational force
- Know the gravitational potential energy formula and how to apply it
- Know how to find escape speed from the earth versus from the moon (also show list of escape speeds from other planets)
- Know the statements of Kepler’s three laws and how to derive and apply Kepler’s third law

**Sub Objectives**- The objectives will require that students be able to:- text
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**Difficulties**- Students may have difficulty:- Units of the universal gravitational force constant G (Nm2/kg2)
- Students tend to forget to square r when calculating gravitational force
- At an altitude h to calculate the gravitational force the denominator must be (rearth + h)2
- Negative sign in gravitational potential energy formula (negative sign indicates force is attractive)
- Assuming that rmax goes to infinity when deriving escape speed.

**Real-World Contexts**- There are many ways that students can use this material in the real-world, such as: (* = picture/video/data in Supplementary Materials in Blackboard):- The gravitational force formula applies to all bodies with mass in the universe.
- Satellites in circular motion about the earth: The relation between orbital radius and period, T2 proportional to r3 (Calculate T2/r3 using data from http://www.freemars.org/jeff/speed/index.htm, create Excel file using data)*
- Planetary motion about the sun: Kepler’s laws, Table 13.2*
- Satellite leaving solar system must be traveling at or about the escape velocity for the solar system (get some data for satellite, explain why it was able to leave solar system)*
- Solar wind leaves the sun with extreme speeds in order to escape the sun gravitational force (get some data on solar winds, compare with escape speed 617.5 km/s, sun’s escape speed, discussion on consequence of solar wind on earth magnetic field and comet tail)*
- Weight of objects on different planets, (astronauts walking on the moon)
- Tidal effects due to the gravitational pull of the moon and sun
- Identifying planets around distant stars by observing wobbles or slight deviation from calculated orbit of the distant stars*
- Saturn’s ring
- Ability to send objects to other planets: Sling shot (Voyager)*

**Model of Knowledge**

**Concept Map**- text
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**Content Priorities****Enduring Understanding**- Calculate the magnitude of the universal gravitational force (G = 6.67 x 10-11 Nm2kg-2) F = Gm1m2/r122
- Know that the universal gravitational force applies to any object that has mass
- r can be taken to be (altitude + radius of earth) for problems considering space objects
- Weight mg comes from using the universal gravitational force formula where m2 is the mass of earth and r12 is the radius of earth, m1 is the person’s mass standing on earth’s surface, and g = Gm2/r122 = 9.8 m/s2
- Contrast: discern when the gravitational force has no noticeable effect and is negligible (between two people the gravitational force is not noticeable because the masses are not heavy enough and G is so small; between planets/sun the gravitational force is noticeable)

- Gravitational potential energy formula and how to apply it
- Derive and apply escape speed for Earth
- Statements of Kepler’s three laws and derivation and application of Kepler’s third law

- Calculate the magnitude of the universal gravitational force (G = 6.67 x 10-11 Nm2kg-2) F = Gm1m2/r122
**Important to Do and Know**- Contrast: in subatomic particle (electrons, protons) the Coulomb force (introduce formula for Coulomb force) is dominant over the gravitational force
- Derive and apply escape speed for different planets or celestial objects

**Worth Being Familiar with**- Understand in some detail Keplers 1st and 2nd law (supplemental material link)

**Assessment of Learning**

**Formative Assessment**- In Class (groups)
- text
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- Homework (individual)
- Pay attention to the following new units: G Gravitational force constant(N m2/kg2), K Coulomb force constant (N m2/C2)
- Self-reading Sections Chapter 13 in the textbook

- In Class (groups)
**Summative Assessment**- text
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## Legacy Cycle

[edit | edit source]**OBJECTIVE**

By the next class period, students will be able to:

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The objectives will require that students be able to:

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**THE CHALLENGE**

Instructions: Students can refer to any resource to answer this question. Each person needs to scan and save the report as a pdf file, and email the report to Mr. Manuel Lara, Teaching Assistant on WebCT which answers the question with supporting data linking back with the relevant physics content. The report is usually due a week from the date when the question is assigned and will be kept in a WebCT folder.

Rubric for grades (on a 0-10 point scale):

- Correct algebraic set-up to begin the solution (3 points)
- Algebraic manipulation (3 points)
- Correct solution of SOHO’s distance from earth (4 points).

Format:

- Put down your name
- State challenge question and its number
- Show all your work.

Challenge Question #8: (Knight 2nd edition). In 1996, the Solar and Heliospheric Observatory (SOHO) was “parked” in an orbit slightly inside the earth’s orbit, as shown in the figure (scan and insert figure). The satellite’s period in this orbit is exactly one year, so it remains fixed relative to the earth. At this point, called a Lagrange point, the light from the sun is never blocked by the earth, yet the satellite remains “nearby” so that data are easily transmitted to earth. What is SOHO’s distance from the earth? (Useful parameters: mass of the sun = 1.991 x 10^{30} kg, mass of earth = 5.983 x 10^{24} kg, distance from the sun to earth = 1.496 x 10^{11} m, 1.000 year = 3.156 x 10^{7} s). Hint: your final algebraic equation will contain powers of the variable ranging from the fifth to the zeroth, so do not try to solve the equation algebraically – you may use computer software such as Excel, MatLab, Mathematica, Maple, graphing calculator, etc.

**GENERATE IDEAS**

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**MULTIPLE PERSPECTIVES**

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**RESEARCH & REVISE**

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**TEST YOUR METTLE**

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**GO PUBLIC**

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## Pre-Lesson Quiz

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## Test Your Mettle Quiz

[edit | edit source]- (Universal gravitational force law: Serway 7th edition page 381 #6i). A satellite originally moves in a circular orbit of radius R around the Earth. Suppose it moved into a circular orbit of radius 4R. What does the force exerted on the satellite then become?

- a. 16 times larger
- b. 8 times larger
- c. 4 times larger
- d. 2 times larger
- e. unchanged
- f. 1/2 as large
- g. 1/4 as larger
- h. 1/8 as large
- i. 1/16 as large

- (Gravitational force between two arbitrary masses: Serway 7th edition page 382 P#1). Determine the order of magnitude of the gravitational force that you exert on another person 2 m away. In your solution, state the quantities you measure or estimate and their values.
- (Gravitational force at some altitude above earth: online resource “Geostationary Satellite Problem”). A 2100 kg satellite is to be placed into a circular, geostationary orbit above the equator. How high above the equator will the satellite have to be in orbit in order for it to be stationary above one location on the equator.
- (Gravitational acceleration on a different planet or moon: Serway 7th edition page 381 #5). Suppose the gravitational acceleration at the surface of a certain satellite A of Jupiter is 2 m/s
^{2}. Satellite B has twice the mass and twice the radius of satellite A. What is the gravitational acceleration at its surface?

- a. 16 m/s
^{2} - b. 8 m/s
^{2} - c. 4 m/s
^{2} - d. 2 m/s
^{2} - e. 1 m/s
^{2} - f. 0.5 m/s
^{2} - g. 0.25 m/s
^{2}

- a. 16 m/s

- (Gravitational potential energy difference at different altitudes: Serway 7th edition page 385 #32). A 1000-kg satellite orbits the Earth at a constant altitude of 100 km. How much energy must be added to the system to move the satellite into a circular orbit with altitude 200 km? Discuss the changes in kinetic energy, potential energy, and total energy.
- (Escape velocity: Serway 7th edition page 385 #30a). What is the minimum speed, relative to the Sun, necessary for a spacecraft to escape the solar system, if it starts at the earth’s orbit?
- (Kepler’s laws: Serway 7th edition page 384 #15). Io, a satellite of Jupiter, has an orbital period of 1.77 days and an orbital radius of 4.22 x 10
^{5}km. From these data, determine the mass of Jupiter.