Mr. Rogers' AP Physics C: Mechanics (With IB Physics Topics) Objectives Syllabus 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter Gravity Circular Motion Statics Rotation

Chapter 13: Universal Gravitation

F. Oscillations and gravitation 12%, cumulative 82%
4. Newton's law of gravity
5.Orbits of planets and satellites
a. Circular
b. General

 Practice Test Study Guide
 Objectives
 Essential Question: Do force fields really exist and are they similar to the force fields on Star Trek or Star Wars?
1. 1. Mathematically define gravity field.
 (gravity field) = (gravity force) (unit of mass)
1. 2. State that gravity field is a vector.
2. 3. Draw a ray diagram of a constant gravity field.
3. 4. State the meaning of the space between rays in a force field diagram.
4. 5. State the two assumptions implicit in modeling the Earth's gravity field as constant.
• The Earth is flat
• The Earth's surface is infinitely large
1. 6. List 3 types of force fields.
 Field Type Symbol Definition gravity g ( force )  /  ( unit of mass ) electrical E ( force )  /  ( unit of charge ) magnetic B ( force )  /  [ ( unit of charge )( velocity ) ]

Relevance: Air resistance is a basic characteristic that determines all kinds of important issues including fuel efficiency of vehicles and terminal velocities of falling objects.

 Activities

Lesson 1

Key Concept: Falling in Uniform Gravity Fields

Purpose: Introduce the concept of force fields using the most common force field gravity. Show how a velocity dependent force like air resistance interacts with a constant gravity field.

Interactive Discussion: Star Wars vs. real life

In Class Problem Solving:

1. Derive an expression for calculating terminal velocity.

 Mini-Lab Physics Investigation (Requires only Purpose, data, and conclusion) Title Analysis of Low velocity Air Resistance for a streamlined and non-streamlined object Purpose Determine if air resistance is directly proportional to velocity for a streamlined and non streamlined object/ Overview Air resistance is often modeled as being directly proportional to velocity when included in mathematical models using Newton's second Law. Data, Calculations Perform regression analysis on the the data for each object using Minitab and plot the residuals. Questions, Conclusions Was a linear relationship between air resistance and velocity appropriate?How did the streamlined object differ from the non streamlined one. Resources/Materials: Wind Tunnel and associated equipment.

 Essential Question: What causes non uniform force fields?

Gravity Fields Around Planets

Section 13.1, 14.2

1. 7. Correctly use the universal gravitational force equation.

• yields an action reaction pair

• r = distance between centers of mass

• G = universal gravitational constant

• the equation does not work inside a planet

• force is directly proportional to mass

• force is inversely proportional to r squared

1. 8. Draw a ray diagram of the gravity field around a planet (see red lines in figure at right). Note that the spacing of the lines is directly proportional to the g-field strength, if the drawing is made to scale.

2. 9. Calculate the gravity field strength = g (or acceleration due to gravity) using the universal gravitational force equation.

F = (G∙M∙m) / r2

g = (G∙M) / r2

1. 10. Note that the gravity field above a planet's surface acts as though it came from a point source located at the planet's center of mass. Remember, the universal gravitational force equation is only valid on a planet's exterior.

2. 11. Explain why the equation for gravity force vs. distance from a planet (a point source of gravity) is only valid above the planet's surface. (The amount of mass attracting an object toward the planet's center of mass changes as the object approaches the center of the planet. Outside the planet's surface, the amount of mass is constant)

Homefun: Read 13.1 to 13.3; Problems 1, 3, 11, 23

Chapter 14, Scenes With Real Gravity: Celebrating Disasters With Happy Hollywood Endings, pp 213 - 229

 Lesson 2Key Concept: Non-Uniform Gravity Fields Purpose: Introduce the concept of force fields and show how it can be used in problem solving. Interactive Discussion: Objectives Demo 1: Demonstrate the inverse square law with a flashlight. Video Clip: Show a video clip of Armageddon when the asteroid is split in half and travels around the Earth. Estimate the tidal forces casued by the asteroid as it travels within 300 miles of Earth's surface. (teams of 2) In Class Problem Solving: Calculate g for planet Earth. Calculate g for Zorg. Resources/Materials: Flashlight

 Essential Question: How can the force of gravity be calculated inside a planet?

The Ultimate Transportation System -- a Tunnel Through the Center of a Planet.

Relevance: Worm holes are a major yet unproven form of space travel in science fiction. In many ways, the hole-in-the-planet transportation system is analogous to a worm hole.

1. 12. Define scaling factor. The number every dimension of an object is multiplied by in order to create a different sized version of the object that looks identical to the original except for its size.
2. 13. Derive an expression for the gravity field vs. radius inside a planet, using the following:
• the gravity field inside a hollow planet is zero
• if  an object's center of mass (CM) is at a distance r from the planet's CM, only mass in the sphere of radius r will create a net gravitational force.
• the gravitational force is calculated using the universal gravity equation using the above sphere
• mass scales with the cube of the scaling factor.
1. 14. Using the displacement equation for simple harmonic motion as show below, derive the velocity and acceleration equations for simple harmonic motion.

x = (xmax)cos (ωt)

1. 15. Derive the time it would take to fall through a tunnel bored through the center of the Earth.
1. radius = 6.371 x 106 meters,
2. mass = 5.9736×1024 kg,
3. G = 6.754 × 10−11 m3/kg/s2
1.
2. 16. Plot a graph of g-field vs, distance from the center of a planet.

Homefun:  Serway

 Lesson 3Key Concept: Gravity Field Inside a Planet Purpose: Introduce scaling factors and show how scaling factors in combination with simple harmonic motion and the universal gravitational equation can be used to analyze the g-field inside a planet. Interactive Discussion: Describe the gravity field inside a hollow planet. Describe the ultimate transportation system. In Class Problem Solving: Derive an equation for the gravity field inside a planet. Give the displacement vs. time equation for simple harmonic motion, derive the velocity and acceleration vs. time equations. Calculate the time required to fall completely through a tunnel from one side of Earth to the other.

 Essential Question: How is gravitational Potential energy calculated when the g-field is not constant?

Gravitational Potential Energy From a Planet

 17. Note that that by convention the gravitational potential energy is considered to be zero at a distance of infinity from a planet. 18. Using the definition of gravitational potential energy, derive an expression for gravitational potential energy vs. distance from the center of a planet, above the planet's surface. (Note blue dashed lines at right are constant potential energy lines.)

Equations

 ΔU =  - ∫ r (force function) (dr) ∞
The force will have a negative sign because it is an attractive force.
 Ur - U∞ =  - ∫ r (-[G(Mm)] / r2) (dr) ∞
is typically used as the location of zero gravitational potential energy
Ur = - m [G(M) / r] Uf has to be negative in order to be lower than U∞.

1. 19. Explain why the equation for potential energy vs. distance from a planet (a point source of gravity) is only valid above the planet's surface. (The amount of mass attracting an object toward the planet's center of mass changes as the object approaches the center of the planet, so the force as a function of r equation is different.)

Homefun: Read 13.4 to 13.6,  Problem 27

Relevance: Gravitational potential energy is a key concept in space exploration. The energy required for getting off a planet's surface is a critical problem that could limit the exploration of planets by humans.

 Lesson 4Key Concept: Potential Energy Around a Planet Purpose: Relate gravitational potential energy to gravity force. Interactive Discussion: How are gravity field lines related to constant potential energy lines? In Class Problem Solving: Derive an expression for potential energy vs distance from the center of a planet.
 Essential Question: How can a spacecraft escape from a planet's gravity?

Gravity and Orbits

1. 20. Circular Orbit: Calculate the velocity or radius (depending on what is given) for circular orbits by combining circular motion equations with the universal gravity equation. (Note: r is the distance from the center of the planet to the spacecraft, M = mass of planet)

v = [(GM) / r]^0.5

1. 21. Elliptical Orbit (speed lowered at P): Describe what happens to a satellite in circular orbit if it's tangential velocity is decreased.
2. 22. Eliptical Orbit (speed increased at P): Describe what happens to a satellite in circular orbit if it's tangential velocity is increased.
3. 23. Escape from Orbit: Calculate escape speed from the surface of a planet. (p. 407, note: r is the distance from the center of the planet to the spacecraft, M = mass of planet)

v = [(2GM) / r]^0.5

1. 24. State the two critical speeds for orbits.
2. 25. Calculate the radius and velocity required for a geosynchronous or geostationary orbit.
1. geosynchronous--the orbit returns the satellite to the same location in the in the sky at the same time every day.
2. geostationary--a special case of geosynchronous orbit in which the satellite remains above a fixed point on the equator.

Note that an orbiting object can only remain in a fixed position above a point on the equator and then only if the radius and speed are correct.

Relevance: Orbit knowledge is key to an understanding of space travel, communication satellites, science fiction movies etc. Orbiting spacecraft are frequently incorrectly portrayed in science fiction movies.

Homefun: Read 13.4 to 13.7,  Questions 1-10 ; Problems 41

 Lesson 3Key Concept: Orbiting and Escaping from a Planet Purpose: Derive the key equations associated with orbits. Interactive Discussion: Objective s Video Clip: Show a video clip of the Space Shuttle taking off from the surface of the asteroid in Armageddon. Calculate the escape velocity needed. How fast would the ship have to be moving to take off? Work in groups of two
 Essential Question: How can the effects of air resistance be derived mathematically for a falling object?
1. Falling With Air Resistance

2.

3. 26. Explain where air resistance comes from and why it should not be called air friction. For an object to move forward, it has to push air out of the way. The change in momentum of the air creates a force that resists the motion. This is sometimes referred to as an inertial force. There is friction between the air air molecules "rubbing" past the surface of the moving object but this force is very low except at supersonic velocities.
4.
5. 27. Derive a differential equation that accounts for the air resistance of a falling object, assuming the air resistance force is directly proportional to velocity.

bv - mg = ma

bv - mg = m(dv/dt)

m(dv/dt) - bv + mg = 0

1. 28. using the above equation, derive an expression for the velocity of a falling object vs. time.
2. 29. Calculate terminal velocity for a falling object with air resistance and compare it to a falling object without air resistance. (see page 162)

Relevance: Air resistance is a basic characteristic that determines all kinds of important issues including fuel efficiency of vehicles and terminal velocities of falling objects.

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