  Dan Fullerton

Gravity & Orbits

Slide Duration:

Section 1: Introduction
What is Physics?

7m 12s

Intro
0:00
Objectives
0:11
What is Physics?
0:27
Why?
0:50
0:51
Matter
1:27
Matter
1:28
Mass
1:43
Inertial Mass
1:50
Gravitational Mass
2:13
A Spacecraft's Mass
3:03
What is the Mass of the Spacecraft?
3:05
Energy
3:37
Energy
3:38
Work
3:45
Putting Energy and Work Together
3:50
Mass-Energy Equivalence
4:15
Relationship between Mass & Energy: E = mc²
4:16
Source of Energy on Earth
4:47
The Study of Everything
5:00
Physics is the Study of Everything
5:01
Mechanics
5:29
Topics Covered
5:30
Topics Not Covered
6:07
Next Steps
6:44
Three Things You'd Like to Learn About in Physics
6:45
Math Review

1h 51s

Intro
0:00
Objectives
0:10
Vectors and Scalars
1:06
Scalars
1:07
Vectors
1:27
Vector Representations
2:00
Vector Representations
2:01
2:54
2:55
Graphical Vector Subtraction
5:36
Graphical Vector Subtraction
5:37
Vector Components
7:12
Vector Components
7:13
Angle of a Vector
8:56
tan θ
9:04
sin θ
9:25
cos θ
9:46
Vector Notation
10:10
Vector Notation 1
10:11
Vector Notation 2
12:59
Example I: Magnitude of the Horizontal & Vertical Component
16:08
Example II: Magnitude of the Plane's Eastward Velocity
17:59
Example III: Magnitude of Displacement
19:33
Example IV: Total Displacement from Starting Position
21:51
Example V: Find the Angle Theta Depicted by the Diagram
26:35
Vector Notation, cont.
27:07
Unit Vector Notation
27:08
Vector Component Notation
27:25
Vector Multiplication
28:39
Dot Product
28:40
Cross Product
28:54
Dot Product
29:03
Dot Product
29:04
Defining the Dot Product
29:26
Defining the Dot Product
29:27
Calculating the Dot Product
29:42
Unit Vector Notation
29:43
Vector Component Notation
30:58
Example VI: Calculating a Dot Product
31:45
Example VI: Part 1 - Find the Dot Product of the Following Vectors
31:46
Example VI: Part 2 - What is the Angle Between A and B?
32:20
Special Dot Products
33:52
Dot Product of Perpendicular Vectors
33:53
Dot Product of Parallel Vectors
34:03
Dot Product Properties
34:51
Commutative
34:52
Associative
35:05
Derivative of A * B
35:24
Example VII: Perpendicular Vectors
35:47
Cross Product
36:42
Cross Product of Two Vectors
36:43
Direction Using the Right-hand Rule
37:32
Cross Product of Parallel Vectors
38:04
Defining the Cross Product
38:13
Defining the Cross Product
38:14
Calculating the Cross Product Unit Vector Notation
38:41
Calculating the Cross Product Unit Vector Notation
38:42
Calculating the Cross Product Matrix Notation
39:18
Calculating the Cross Product Matrix Notation
39:19
Example VII: Find the Cross Product of the Following Vectors
42:09
Cross Product Properties
45:16
Cross Product Properties
45:17
Units
46:41
Fundamental Units
46:42
Derived units
47:13
Example IX: Dimensional Analysis
47:21
Calculus
49:05
Calculus
49:06
Differential Calculus
49:49
Differentiation & Derivative
49:50
Example X: Derivatives
51:21
Integral Calculus
53:03
Integration
53:04
Integral
53:11
Integration & Derivation are Inverse Functions
53:16
Determine the Original Function
53:37
Common Integrations
54:45
Common Integrations
54:46
Example XI: Integrals
55:17
Example XII: Calculus Applications
58:32
Section 2: Kinematics
Describing Motion I

23m 47s

Intro
0:00
Objectives
0:10
Position / Displacement
0:39
Object's Position
0:40
Position Vector
0:45
Displacement
0:56
Position & Displacement are Vectors
1:05
Position & Displacement in 1 Dimension
1:11
Example I: Distance & Displacement
1:21
Average Speed
2:14
Average Speed
2:15
Average Speed is Scalar
2:27
Average Velocity
2:39
Average Velocity
2:40
Average Velocity is a Vector
2:57
Example II: Speed vs. Velocity
3:16
Example II: Deer's Average Speed
3:17
Example II: Deer's Average Velocity
3:48
Example III: Chuck the Hungry Squirrel
4:21
Example III: Chuck's Distance Traveled
4:22
Example III: Chuck's Displacement
4:43
Example III: Chuck's Average Speed
5:25
Example III: Chuck's Average Velocity
5:39
Acceleration
6:11
Acceleration: Definition & Equation
6:12
Acceleration: Units
6:19
Relationship of Acceleration to Velocity
6:52
Example IV: Acceleration Problem
7:05
The Position Vector
7:39
The Position Vector
7:40
Average Velocity
9:35
Average Velocity
9:36
Instantaneous Velocity
11:20
Instantaneous Velocity
11:21
Instantaneous Velocity is the Derivative of Position with Respect to Time
11:35
Area Under the Velocity-time Graph
12:08
Acceleration
12:36
More on Acceleration
12:37
Average Acceleration
13:11
Velocity vs. Time Graph
13:14
Graph Transformations
13:59
Graphical Analysis of Motion
14:00
Velocity and acceleration in 2D
14:35
Velocity Vector in 2D
14:39
Acceleration Vector in 2D
15:26
Polynomial Derivatives
16:10
Polynomial Derivatives
16:11
Example V: Polynomial Kinematics
16:31
Example VI: Velocity Function
17:54
Example VI: Part A - Determine the Acceleration at t=1 Second
17:55
Example VI: Part B - Determine the Displacement between t=0 and t=5 Seconds
18:33
Example VII: Tortoise and Hare
20:14
Example VIII: d-t Graphs
22:40
Describing Motion II

36m 47s

Intro
0:00
Objectives
0:09
Special Case: Constant Acceleration
0:31
Constant Acceleration & Kinematic Equations
0:32
Deriving the Kinematic Equations
1:28
V = V₀ + at
1:39
∆x = V₀t +(1/2)at²
2:03
V² = V₀² +2a∆x
4:05
Problem Solving Steps
7:02
Step 1
7:13
Step 2
7:18
Step 3
7:27
Step 4
7:30
Step 5
7:31
Example IX: Horizontal Kinematics
7:38
Example X: Vertical Kinematics
9:45
Example XI: 2 Step Problem
11:23
Example XII: Acceleration Problem
15:01
Example XIII: Particle Diagrams
15:57
Example XIV: Particle Diagrams
17:36
18:46
Free Fall
22:56
Free Fall
22:57
Air Resistance
23:24
Air Resistance
23:25
Acceleration Due to Gravity
23:48
Acceleration Due to Gravity
23:49
Objects Falling From Rest
24:18
Objects Falling From Rest
24:19
Example XVI: Falling Objects
24:55
Objects Launched Upward
26:01
Objects Launched Upward
26:02
Example XVII: Ball Thrown Upward
27:16
Example XVIII: Height of a Jump
27:48
Example XIX: Ball Thrown Downward
31:10
Example XX: Maximum Height
32:27
Example XXI: Catch-Up Problem
33:53
Example XXII: Ranking Max Height
35:52
Projectile Motion

30m 34s

Intro
0:00
Objectives
0:07
What is a Projectile?
0:28
What is a Projectile?
0:29
Path of a Projectile
0:58
Path of a Projectile
0:59
Independence of Motion
2:45
Vertical & Horizontal Motion
2:46
Example I: Horizontal Launch
3:14
Example II: Parabolic Path
7:20
Angled Projectiles
8:01
Angled Projectiles
8:02
Example III: Human Cannonball
10:05
Example IV: Motion Graphs
14:39
Graphing Projectile Motion
19:05
Horizontal Equation
19:06
Vertical Equation
19:46
Example V: Arrow Fired from Tower
21:28
Example VI: Arrow Fired from Tower
24:10
Example VII: Launch from a Height
24:40
Example VIII: Acceleration of a Projectile
29:49
Circular & Relative Motion

30m 24s

Intro
0:00
Objectives
0:08
0:32
Degrees
0:35
0:40
1:08
Example I: Part A - Convert 90 Degrees to Radians
1:09
Example I: Part B - Convert 6 Radians to Degrees
2:08
Linear vs. Angular Displacement
2:38
Linear Displacement
2:39
Angular Displacement
2:52
Linear vs. Angular Velocity
3:18
Linear Velocity
3:19
Angular Velocity
3:25
Direction of Angular Velocity
4:36
Direction of Angular Velocity
4:37
Converting Linear to Angular Velocity
5:05
Converting Linear to Angular Velocity
5:06
Example II: Earth's Angular Velocity
6:12
Linear vs. Angular Acceleration
7:26
Linear Acceleration
7:27
Angular Acceleration
7:32
Centripetal Acceleration
8:05
Expressing Position Vector in Terms of Unit Vectors
8:06
Velocity
10:00
Centripetal Acceleration
11:14
Magnitude of Centripetal Acceleration
13:24
Example III: Angular Velocity & Centripetal Acceleration
14:02
Example IV: Moon's Orbit
15:03
Reference Frames
17:44
Reference Frames
17:45
Laws of Physics
18:00
Motion at Rest vs. Motion at a Constant Velocity
18:21
Motion is Relative
19:20
Reference Frame: Sitting in a Lawn Chair
19:21
Reference Frame: Sitting on a Train
19:56
Calculating Relative Velocities
20:19
Calculating Relative Velocities
20:20
Example: Calculating Relative Velocities
20:57
Example V: Man on a Train
23:19
Example VI: Airspeed
24:56
Example VII: 2-D Relative Motion
26:12
Example VIII: Relative Velocity w/ Direction
28:32
Section 3: Dynamics
Newton's First Law & Free Body Diagrams

23m 57s

Intro
0:00
Objectives
0:11
Newton's 1st Law of Motion
0:28
Newton's 1st Law of Motion
0:29
Force
1:16
Definition of Force
1:17
Units of Force
1:20
How Much is a Newton?
1:25
Contact Forces
1:47
Field Forces
2:32
What is a Net Force?
2:53
What is a Net Force?
2:54
What Does It Mean?
4:35
What Does It Mean?
4:36
Objects at Rest
4:52
Objects at Rest
4:53
Objects in Motion
5:12
Objects in Motion
5:13
Equilibrium
6:03
Static Equilibrium
6:04
Mechanical Equilibrium
6:22
Translational Equilibrium
6:38
Inertia
6:48
Inertia
6:49
Inertial Mass
6:58
Gravitational Mass
7:11
Example I: Inertia
7:40
Example II: Inertia
8:03
Example III: Translational Equilibrium
8:25
Example IV: Net Force
9:19
Free Body Diagrams
10:34
Free Body Diagrams Overview
10:35
Falling Elephant: Free Body Diagram
10:53
Free Body Diagram Neglecting Air Resistance
10:54
Free Body Diagram Including Air Resistance
11:22
Soda on Table
11:54
Free Body Diagram for a Glass of Soda Sitting on a Table
11:55
Free Body Diagram for Box on Ramp
13:38
Free Body Diagram for Box on Ramp
13:39
Pseudo- Free Body Diagram
15:26
Example V: Translational Equilibrium
18:35
Newton's Second & Third Laws of Motion

23m 57s

Intro
0:00
Objectives
0:09
Newton's 2nd Law of Motion
0:36
Newton's 2nd Law of Motion
0:37
Applying Newton's 2nd Law
1:12
Step 1
1:13
Step 2
1:18
Step 3
1:27
Step 4
1:36
Example I: Block on a Surface
1:42
Example II: Concurrent Forces
2:42
Mass vs. Weight
4:09
Mass
4:10
Weight
4:28
Example III: Mass vs. Weight
4:45
Example IV: Translational Equilibrium
6:43
Example V: Translational Equilibrium
8:23
Example VI: Determining Acceleration
10:13
Example VII: Stopping a Baseball
12:38
Example VIII: Steel Beams
14:11
Example IX: Tension Between Blocks
17:03
Example X: Banked Curves
18:57
Example XI: Tension in Cords
24:03
Example XII: Graphical Interpretation
27:13
Example XIII: Force from Velocity
28:12
Newton's 3rd Law
29:16
Newton's 3rd Law
29:17
Examples - Newton's 3rd Law
30:01
Examples - Newton's 3rd Law
30:02
Action-Reaction Pairs
30:40
Girl Kicking Soccer Ball
30:41
Rocket Ship in Space
31:02
Gravity on You
31:23
Example XIV: Force of Gravity
32:11
Example XV: Sailboat
32:38
Example XVI: Hammer and Nail
33:18
Example XVII: Net Force
33:47
Friction

20m 41s

Intro
0:00
Objectives
0:06
Coefficient of Friction
0:21
Coefficient of Friction
0:22
Approximate Coefficients of Friction
0:44
Kinetic or Static?
1:21
Sled Sliding Down a Snowy Hill
1:22
Refrigerator at Rest that You Want to Move
1:32
Car with Tires Rolling Freely
1:49
Car Skidding Across Pavement
2:01
Example I: Car Sliding
2:21
Example II: Block on Incline
3:04
Calculating the Force of Friction
3:33
Calculating the Force of Friction
3:34
Example III: Finding the Frictional Force
4:02
Example IV: Box on Wood Surface
5:34
Example V: Static vs. Kinetic Friction
7:35
Example VI: Drag Force on Airplane
7:58
Example VII: Pulling a Sled
8:41
Example VIII: AP-C 2007 FR1
13:23
Example VIII: Part A
13:24
Example VIII: Part B
14:40
Example VIII: Part C
15:19
Example VIII: Part D
17:08
Example VIII: Part E
18:24
Retarding & Drag Forces

32m 10s

Intro
0:00
Objectives
0:07
Retarding Forces
0:41
Retarding Forces
0:42
The Skydiver
1:30
Drag Forces on a Free-falling Object
1:31
Velocity as a Function of Time
5:31
Velocity as a Function of Time
5:32
Velocity as a Function of Time, cont.
12:27
Acceleration
12:28
Velocity as a Function of Time, cont.
15:16
Graph: Acceleration vs. Time
16:06
Graph: Velocity vs. Time
16:40
Graph: Displacement vs. Time
17:04
Example I: AP-C 2005 FR1
17:43
Example I: Part A
17:44
Example I: Part B
19:17
Example I: Part C
20:17
Example I: Part D
21:09
Example I: Part E
22:42
Example II: AP-C 2013 FR2
24:26
Example II: Part A
24:27
Example II: Part B
25:25
Example II: Part C
26:22
Example II: Part D
27:04
Example II: Part E
30:50
Ramps & Inclines

20m 31s

Intro
0:00
Objectives
0:06
Drawing Free Body Diagrams for Ramps
0:32
Step 1: Choose the Object & Draw It as a Dot or Box
0:33
Step 2: Draw and Label all the External Forces
0:39
Step 3: Sketch a Coordinate System
0:42
Example: Object on a Ramp
0:52
Pseudo-Free Body Diagrams
2:06
Pseudo-Free Body Diagrams
2:07
Redraw Diagram with All Forces Parallel to Axes
2:18
Box on a Ramp
4:08
Free Body Diagram for Box on a Ramp
4:09
Pseudo-Free Body Diagram for Box on a Ramp
4:54
Example I: Box at Rest
6:13
Example II: Box Held By Force
6:35
Example III: Truck on a Hill
8:46
Example IV: Force Up a Ramp
9:29
Example V: Acceleration Down a Ramp
12:01
Example VI: Able of Repose
13:59
Example VII: Sledding
17:03
Atwood Machines

24m 58s

Intro
0:00
Objectives
0:07
What is an Atwood Machine?
0:25
What is an Atwood Machine?
0:26
Properties of Atwood Machines
1:03
Ideal Pulleys are Frictionless and Massless
1:04
Tension is Constant
1:14
Setup for Atwood Machines
1:26
Setup for Atwood Machines
1:27
Solving Atwood Machine Problems
1:52
Solving Atwood Machine Problems
1:53
Alternate Solution
5:24
Analyze the System as a Whole
5:25
Example I: Basic Atwood Machine
7:31
Example II: Moving Masses
9:59
Example III: Masses and Pulley on a Table
13:32
Example IV: Mass and Pulley on a Ramp
15:47
Example V: Ranking Atwood Machines
19:50
Section 4: Work, Energy, & Power
Work

37m 34s

Intro
0:00
Objectives
0:07
What is Work?
0:36
What is Work?
0:37
Units of Work
1:09
Work in One Dimension
1:31
Work in One Dimension
1:32
Examples of Work
2:19
Stuntman in a Jet Pack
2:20
A Girl Struggles to Push Her Stalled Car
2:50
A Child in a Ghost Costume Carries a Bag of Halloween Candy Across the Yard
3:24
Example I: Moving a Refrigerator
4:03
Example II: Liberating a Car
4:53
Example III: Lifting Box
5:30
Example IV: Pulling a Wagon
6:13
Example V: Ranking Work on Carts
7:13
Non-Constant Forces
12:21
Non-Constant Forces
12:22
Force vs. Displacement Graphs
13:49
Force vs. Displacement Graphs
13:50
Hooke's Law
14:41
Hooke's Law
14:42
Determining the Spring Constant
15:38
Slope of the Graph Gives the Spring Constant, k
15:39
Work Done in Compressing the Spring
16:34
Find the Work Done in Compressing the String
16:35
Example VI: Finding Spring Constant
17:21
Example VII: Calculating Spring Constant
19:48
Example VIII: Hooke's Law
20:30
Example IX: Non-Linear Spring
22:18
Work in Multiple Dimensions
23:52
Work in Multiple Dimensions
23:53
Work-Energy Theorem
25:25
Work-Energy Theorem
25:26
Example X: Work-Energy Theorem
28:35
Example XI: Work Done on Moving Carts
30:46
Example XII: Velocity from an F-d Graph
35:01
Energy & Conservative Forces

28m 4s

Intro
0:00
Objectives
0:08
Energy Transformations
0:31
Energy Transformations
0:32
Work-Energy Theorem
0:57
Kinetic Energy
1:12
Kinetic Energy: Definition
1:13
Kinetic Energy: Equation
1:55
Example I: Frog-O-Cycle
2:07
Potential Energy
2:46
Types of Potential Energy
2:47
A Potential Energy Requires an Interaction between Objects
3:29
Internal energy
3:50
Internal Energy
3:51
Types of Energy
4:37
Types of Potential & Kinetic Energy
4:38
Gravitational Potential Energy
5:42
Gravitational Potential Energy
5:43
Example II: Potential Energy
7:27
Example III: Kinetic and Potential Energy
8:16
Example IV: Pendulum
9:09
Conservative Forces
11:37
Conservative Forces Overview
11:38
Type of Conservative Forces
12:42
Types of Non-conservative Forces
13:02
Work Done by Conservative Forces
13:28
Work Done by Conservative Forces
13:29
Newton's Law of Universal Gravitation
14:18
Gravitational Force of Attraction between Any Two Objects with Mass
14:19
Gravitational Potential Energy
15:27
Gravitational Potential Energy
15:28
Elastic Potential Energy
17:36
Elastic Potential Energy
17:37
Force from Potential Energy
18:51
Force from Potential Energy
18:52
Gravitational Force from the Gravitational Potential Energy
20:46
Gravitational Force from the Gravitational Potential Energy
20:47
Hooke's Law from Potential Energy
22:04
Hooke's Law from Potential Energy
22:05
Summary
23:16
Summary
23:17
Example V: Kinetic Energy of a Mass
24:40
Example VI: Force from Potential Energy
25:48
Example VII: Work on a Spinning Disc
26:54
Conservation of Energy

54m 56s

Intro
0:00
Objectives
0:09
Conservation of Mechanical Energy
0:32
Consider a Single Conservative Force Doing Work on a Closed System
0:33
Non-Conservative Forces
1:40
Non-Conservative Forces
1:41
Work Done by a Non-conservative Force
1:47
Formula: Total Energy
1:54
Formula: Total Mechanical Energy
2:04
Example I: Falling Mass
2:15
Example II: Law of Conservation of Energy
4:07
Example III: The Pendulum
6:34
Example IV: Cart Compressing a Spring
10:12
Example V: Cart Compressing a Spring
11:12
Example V: Part A - Potential Energy Stored in the Compressed Spring
11:13
Example V: Part B - Maximum Vertical Height
12:01
Example VI: Car Skidding to a Stop
13:05
Example VII: Block on Ramp
14:22
Example VIII: Energy Transfers
16:15
Example IX: Roller Coaster
20:04
Example X: Bungee Jumper
23:32
Example X: Part A - Speed of the Jumper at a Height of 15 Meters Above the Ground
24:48
Example X: Part B - Speed of the Jumper at a Height of 30 Meters Above the Ground
26:53
Example X: Part C - How Close Does the Jumper Get to the Ground?
28:28
Example XI: AP-C 2002 FR3
30:28
Example XI: Part A
30:59
Example XI: Part B
31:54
Example XI: Part C
32:50
Example XI: Part D & E
33:52
Example XII: AP-C 2007 FR3
35:24
Example XII: Part A
35:52
Example XII: Part B
36:27
Example XII: Part C
37:48
Example XII: Part D
39:32
Example XIII: AP-C 2010 FR1
41:07
Example XIII: Part A
41:34
Example XIII: Part B
43:05
Example XIII: Part C
45:24
Example XIII: Part D
47:18
Example XIV: AP-C 2013 FR1
48:25
Example XIV: Part A
48:50
Example XIV: Part B
49:31
Example XIV: Part C
51:27
Example XIV: Part D
52:46
Example XIV: Part E
53:25
Power

16m 44s

Intro
0:00
Objectives
0:06
Defining Power
0:20
Definition of Power
0:21
Units of Power
0:27
Average Power
0:43
Instantaneous Power
1:03
Instantaneous Power
1:04
Example I: Horizontal Box
2:07
Example II: Accelerating Truck
4:48
Example III: Motors Delivering Power
6:00
Example IV: Power Up a Ramp
7:00
Example V: Power from Position Function
8:51
Example VI: Motorcycle Stopping
10:48
Example VII: AP-C 2003 FR1
11:52
Example VII: Part A
11:53
Example VII: Part B
12:50
Example VII: Part C
14:36
Example VII: Part D
15:52
Section 5: Momentum
Momentum & Impulse

13m 9s

Intro
0:00
Objectives
0:07
Momentum
0:39
Definition of Momentum
0:40
Total Momentum
1:00
Formula for Momentum
1:05
Units of Momentum
1:11
Example I: Changing Momentum
1:18
Impulse
2:27
Impulse
2:28
Example II: Impulse
2:41
Relationship Between Force and ∆p (Impulse)
3:36
Relationship Between Force and ∆p (Impulse)
3:37
Example III: Force from Momentum
4:37
Impulse-Momentum Theorem
5:14
Impulse-Momentum Theorem
5:15
Example IV: Impulse-Momentum
6:26
Example V: Water Gun & Horizontal Force
7:56
Impulse from F-t Graphs
8:53
Impulse from F-t Graphs
8:54
Example VI: Non-constant Forces
9:16
Example VII: F-t Graph
10:01
Example VIII: Impulse from Force
11:19
Conservation of Linear Momentum

46m 30s

Intro
0:00
Objectives
0:08
Conservation of Linear Momentum
0:28
In an Isolated System
0:29
In Any Closed System
0:37
Direct Outcome of Newton's 3rd Law of Motion
0:47
Collisions and Explosions
1:07
Collisions and Explosions
1:08
The Law of Conservation of Linear Momentum
1:25
Solving Momentum Problems
1:35
Solving Momentum Problems
1:36
Types of Collisions
2:08
Elastic Collision
2:09
Inelastic Collision
2:34
Example I: Traffic Collision
3:00
Example II: Collision of Two Moving Objects
6:55
Example III: Recoil Velocity
9:47
Example IV: Atomic Collision
12:12
Example V: Collision in Multiple Dimensions
18:11
Example VI: AP-C 2001 FR1
25:16
Example VI: Part A
25:33
Example VI: Part B
26:44
Example VI: Part C
28:17
Example VI: Part D
28:58
Example VII: AP-C 2002 FR1
30:10
Example VII: Part A
30:20
Example VII: Part B
32:14
Example VII: Part C
34:25
Example VII: Part D
36:17
Example VIII: AP-C 2014 FR1
38:55
Example VIII: Part A
39:28
Example VIII: Part B
41:00
Example VIII: Part C
42:57
Example VIII: Part D
44:20
Center of Mass

28m 26s

Intro
0:00
Objectives
0:07
Center of Mass
0:45
Center of Mass
0:46
Finding Center of Mass by Inspection
1:25
For Uniform Density Objects
1:26
For Objects with Multiple Parts
1:36
For Irregular Objects
1:44
Example I: Center of Mass by Inspection
2:06
Calculating Center of Mass for Systems of Particles
2:25
Calculating Center of Mass for Systems of Particles
2:26
Example II: Center of Mass (1D)
3:15
Example III: Center of Mass of Continuous System
4:29
Example IV: Center of Mass (2D)
6:00
Finding Center of Mass by Integration
7:38
Finding Center of Mass by Integration
7:39
Example V: Center of Mass of a Uniform Rod
8:10
Example VI: Center of Mass of a Non-Uniform Rod
11:40
Center of Mass Relationships
14:44
Center of Mass Relationships
14:45
Center of Gravity
17:36
Center of Gravity
17:37
Uniform Gravitational Field vs. Non-uniform Gravitational Field
17:53
Example VII: AP-C 2004 FR1
18:26
Example VII: Part A
18:45
Example VII: Part B
19:38
Example VII: Part C
21:03
Example VII: Part D
22:04
Example VII: Part E
24:52
Section 6: Uniform Circular Motion
Uniform Circular Motion

21m 36s

Intro
0:00
Objectives
0:08
Uniform Circular Motion
0:42
Distance Around the Circle for Objects Traveling in a Circular Path at Constant Speed
0:51
Average Speed for Objects Traveling in a Circular Path at Constant Speed
1:15
Frequency
1:42
Definition of Frequency
1:43
Symbol of Frequency
1:46
Units of Frequency
1:49
Period
2:04
Period
2:05
Frequency and Period
2:19
Frequency and Period
2:20
Example I: Race Car
2:32
Example II: Toy Train
3:22
Example III: Round-A-Bout
4:07
Example III: Part A - Period of the Motion
4:08
Example III: Part B- Frequency of the Motion
4:43
Example III: Part C- Speed at Which Alan Revolves
4:58
Uniform Circular Motion
5:28
Is an Object Undergoing Uniform Circular Motion Accelerating?
5:29
Direction of Centripetal Acceleration
6:21
Direction of Centripetal Acceleration
6:22
Magnitude of Centripetal Acceleration
8:23
Magnitude of Centripetal Acceleration
8:24
Example IV: Car on a Track
8:39
Centripetal Force
10:14
Centripetal Force
10:15
Calculating Centripetal Force
11:47
Calculating Centripetal Force
11:48
Example V: Acceleration
12:41
Example VI: Direction of Centripetal Acceleration
13:44
Example VII: Loss of Centripetal Force
14:03
Example VIII: Bucket in Horizontal Circle
14:44
Example IX: Bucket in Vertical Circle
15:24
Example X: Demon Drop
17:38
Example X: Question 1
18:02
Example X: Question 2
18:25
Example X: Question 3
19:22
Example X: Question 4
20:13
Section 7: Rotational Motion
Rotational Kinematics

32m 52s

Intro
0:00
Objectives
0:07
0:35
Once Around a Circle: In Degrees
0:36
Once Around a Circle: In Radians
0:48
0:51
1:08
Example I: Convert 90° to Radians
1:09
Example I: Convert 6 Radians to Degree
1:23
Linear vs. Angular Displacement
1:43
Linear Displacement
1:44
Angular Displacement
1:51
Linear vs. Angular Velocity
2:04
Linear Velocity
2:05
Angular Velocity
2:10
Direction of Angular Velocity
2:28
Direction of Angular Velocity
2:29
Converting Linear to Angular Velocity
2:58
Converting Linear to Angular Velocity
2:59
Example II: Angular Velocity of Earth
3:51
Linear vs. Angular Acceleration
4:35
Linear Acceleration
4:36
Angular Acceleration
4:42
Example III: Angular Acceleration
5:09
Kinematic Variable Parallels
6:30
Kinematic Variable Parallels: Translational & Angular
6:31
Variable Translations
7:00
Variable Translations: Translational & Angular
7:01
Kinematic Equation Parallels
7:38
Kinematic Equation Parallels: Translational & Rotational
7:39
Example IV: Deriving Centripetal Acceleration
8:29
Example V: Angular Velocity
13:24
Example V: Part A
13:25
Example V: Part B
14:15
Example VI: Wheel in Motion
14:39
Example VII: AP-C 2003 FR3
16:23
Example VII: Part A
16:38
Example VII: Part B
17:34
Example VII: Part C
24:02
Example VIII: AP-C 2014 FR2
25:35
Example VIII: Part A
25:47
Example VIII: Part B
26:28
Example VIII: Part C
27:48
Example VIII: Part D
28:26
Example VIII: Part E
29:16
Moment of Inertia

24m

Intro
0:00
Objectives
0:07
Types of Inertia
0:34
Inertial Mass
0:35
Moment of Inertia
0:44
Kinetic Energy of a Rotating Disc
1:25
Kinetic Energy of a Rotating Disc
1:26
Calculating Moment of Inertia (I)
5:32
Calculating Moment of Inertia (I)
5:33
Moment of Inertia for Common Objects
5:49
Moment of Inertia for Common Objects
5:50
Example I: Point Masses
6:46
Example II: Uniform Rod
9:09
Example III: Solid Cylinder
13:07
Parallel Axis Theorem (PAT)
17:33
Parallel Axis Theorem (PAT)
17:34
Example IV: Calculating I Using the Parallel Axis Theorem
18:39
Example V: Hollow Sphere
20:18
Example VI: Long Thin Rod
20:55
Example VII: Ranking Moment of Inertia
21:50
Example VIII: Adjusting Moment of Inertia
22:39
Torque

26m 9s

Intro
0:00
Objectives
0:06
Torque
0:18
Definition of Torque
0:19
Torque & Rotation
0:26
Lever Arm ( r )
0:30
Example: Wrench
0:39
Direction of the Torque Vector
1:45
Direction of the Torque Vector
1:46
Finding Direction Using the Right-hand Rule
1:53
Newton's 2nd Law: Translational vs. Rotational
2:20
Newton's 2nd Law: Translational vs. Rotational
2:21
Equilibrium
3:17
Static Equilibrium
3:18
Dynamic Equilibrium
3:30
Example I: See-Saw Problem
3:46
Example II: Beam Problem
7:12
Example III: Pulley with Mass
10:34
Example IV: Net Torque
13:46
Example V: Ranking Torque
15:29
Example VI: Ranking Angular Acceleration
16:25
Example VII: Café Sign
17:19
Example VIII: AP-C 2008 FR2
19:44
Example VIII: Part A
20:12
Example VIII: Part B
21:08
Example VIII: Part C
22:36
Example VIII: Part D
24:37
Rotational Dynamics

56m 58s

Intro
0:00
Objectives
0:08
Conservation of Energy
0:48
Translational Kinetic Energy
0:49
Rotational Kinetic Energy
0:54
Total Kinetic Energy
1:03
Example I: Disc Rolling Down an Incline
1:10
Rotational Dynamics
4:25
Rotational Dynamics
4:26
Example II: Strings with Massive Pulleys
4:37
Example III: Rolling without Slipping
9:13
Example IV: Rolling with Slipping
13:45
Example V: Amusement Park Swing
22:49
Example VI: AP-C 2002 FR2
26:27
Example VI: Part A
26:48
Example VI: Part B
27:30
Example VI: Part C
29:51
Example VI: Part D
30:50
Example VII: AP-C 2006 FR3
31:39
Example VII: Part A
31:49
Example VII: Part B
36:20
Example VII: Part C
37:14
Example VII: Part D
38:48
Example VIII: AP-C 2010 FR2
39:40
Example VIII: Part A
39:46
Example VIII: Part B
40:44
Example VIII: Part C
44:31
Example VIII: Part D
46:44
Example IX: AP-C 2013 FR3
48:27
Example IX: Part A
48:47
Example IX: Part B
50:33
Example IX: Part C
53:28
Example IX: Part D
54:15
Example IX: Part E
56:20
Angular Momentum

33m 2s

Intro
0:00
Objectives
0:09
Linear Momentum
0:44
Definition of Linear Momentum
0:45
Total Angular Momentum
0:52
p = mv
0:59
Angular Momentum
1:08
Definition of Angular Momentum
1:09
Total Angular Momentum
1:21
A Mass with Velocity v Moving at Some Position r
1:29
Calculating Angular Momentum
1:44
Calculating Angular Momentum
1:45
Spin Angular Momentum
4:17
Spin Angular Momentum
4:18
Example I: Object in Circular Orbit
4:51
Example II: Angular Momentum of a Point Particle
6:34
Angular Momentum and Net Torque
9:03
Angular Momentum and Net Torque
9:04
Conservation of Angular Momentum
11:53
Conservation of Angular Momentum
11:54
Example III: Ice Skater Problem
12:20
Example IV: Combining Spinning Discs
13:52
Example V: Catching While Rotating
15:13
Example VI: Changes in Angular Momentum
16:47
Example VII: AP-C 2005 FR3
17:37
Example VII: Part A
18:12
Example VII: Part B
18:32
Example VII: Part C
19:53
Example VII: Part D
21:52
Example VIII: AP-C 2014 FR3
24:23
Example VIII: Part A
24:31
Example VIII: Part B
25:33
Example VIII: Part C
26:58
Example VIII: Part D
28:24
Example VIII: Part E
30:42
Section 8: Oscillations
Oscillations

1h 1m 12s

Intro
0:00
Objectives
0:08
Simple Harmonic Motion
0:45
Simple Harmonic Motion
0:46
Circular Motion vs. Simple Harmonic Motion (SHM)
1:39
Circular Motion vs. Simple Harmonic Motion (SHM)
1:40
Position, Velocity, & Acceleration
4:55
Position
4:56
Velocity
5:12
Acceleration
5:49
Frequency and Period
6:37
Frequency
6:42
Period
6:49
Angular Frequency
7:05
Angular Frequency
7:06
Example I: Oscillating System
7:37
Example I: Determine the Object's Angular Frequency
7:38
Example I: What is the Object's Position at Time t = 10s?
8:16
Example I: At What Time is the Object at x = 0.1m?
9:10
Mass on a Spring
10:17
Mass on a Spring
10:18
Example II: Analysis of Spring-Block System
11:34
Example III: Spring-Block ranking
12:53
General Form of Simple Harmonic Motion
14:41
General Form of Simple Harmonic Motion
14:42
Graphing Simple Harmonic Motion (SHM)
15:22
Graphing Simple Harmonic Motion (SHM)
15:23
Energy of Simple Harmonic Motion (SHM)
15:49
Energy of Simple Harmonic Motion (SHM)
15:50
Horizontal Spring Oscillator
19:24
Horizontal Spring Oscillator
19:25
Vertical Spring Oscillator
20:58
Vertical Spring Oscillator
20:59
Springs in Series
23:30
Springs in Series
23:31
Springs in Parallel
26:08
Springs in Parallel
26:09
The Pendulum
26:59
The Pendulum
27:00
Energy and the Simple Pendulum
27:46
Energy and the Simple Pendulum
27:47
Frequency and Period of a Pendulum
30:16
Frequency and Period of a Pendulum
30:17
Example IV: Deriving Period of a Simple Pendulum
31:42
Example V: Deriving Period of a Physical Pendulum
35:20
Example VI: Summary of Spring-Block System
38:16
Example VII: Harmonic Oscillator Analysis
44:14
Example VII: Spring Constant
44:24
Example VII: Total Energy
44:45
Example VII: Speed at the Equilibrium Position
45:05
Example VII: Speed at x = 0.30 Meters
45:37
Example VII: Speed at x = -0.40 Meter
46:46
Example VII: Acceleration at the Equilibrium Position
47:21
Example VII: Magnitude of Acceleration at x = 0.50 Meters
47:35
Example VII: Net Force at the Equilibrium Position
48:04
Example VII: Net Force at x = 0.25 Meter
48:20
Example VII: Where does Kinetic Energy = Potential Energy?
48:33
Example VIII: Ranking Spring Systems
49:35
Example IX: Vertical Spring Block Oscillator
51:45
Example X: Ranking Period of Pendulum
53:50
Example XI: AP-C 2009 FR2
54:50
Example XI: Part A
54:58
Example XI: Part B
57:57
Example XI: Part C
59:11
Example XII: AP-C 2010 FR3
1:00:18
Example XII: Part A
1:00:49
Example XII: Part B
1:02:47
Example XII: Part C
1:04:30
Example XII: Part D
1:05:53
Example XII: Part E
1:08:13
Section 9: Gravity & Orbits
Gravity & Orbits

34m 59s

Intro
0:00
Objectives
0:07
Newton's Law of Universal Gravitation
0:45
Newton's Law of Universal Gravitation
0:46
Example I: Gravitational Force Between Earth and Sun
2:24
Example II: Two Satellites
3:39
Gravitational Field Strength
4:23
Gravitational Field Strength
4:24
Example III: Weight on Another Planet
6:22
Example IV: Gravitational Field of a Hollow Shell
7:31
Example V: Gravitational Field Inside a Solid Sphere
8:33
Velocity in Circular Orbit
12:05
Velocity in Circular Orbit
12:06
Period and Frequency for Circular Orbits
13:56
Period and Frequency for Circular Orbits
13:57
Mechanical Energy for Circular Orbits
16:11
Mechanical Energy for Circular Orbits
16:12
Escape Velocity
17:48
Escape Velocity
17:49
Kepler's 1st Law of Planetary Motion
19:41
Keller's 1st Law of Planetary Motion
19:42
Kepler's 2nd Law of Planetary Motion
20:05
Keller's 2nd Law of Planetary Motion
20:06
Kepler's 3rd Law of Planetary Motion
20:57
Ratio of the Squares of the Periods of Two Planets
20:58
Ratio of the Squares of the Periods to the Cubes of Their Semi-major Axes
21:41
Total Mechanical Energy for an Elliptical Orbit
21:57
Total Mechanical Energy for an Elliptical Orbit
21:58
Velocity and Radius for an Elliptical Orbit
22:35
Velocity and Radius for an Elliptical Orbit
22:36
Example VI: Rocket Launched Vertically
24:26
Example VII: AP-C 2007 FR2
28:16
Example VII: Part A
28:35
Example VII: Part B
29:51
Example VII: Part C
31:14
Example VII: Part D
32:23
Example VII: Part E
33:16
Section 10: Sample AP Exam
1998 AP Practice Exam: Multiple Choice

28m 11s

Intro
0:00
Problem 1
0:30
Problem 2
0:51
Problem 3
1:25
Problem 4
2:00
Problem 5
3:05
Problem 6
4:19
Problem 7
4:48
Problem 8
5:18
Problem 9
5:38
Problem 10
6:26
Problem 11
7:21
Problem 12
8:08
Problem 13
8:35
Problem 14
9:20
Problem 15
10:09
Problem 16
10:25
Problem 17
11:30
Problem 18
12:27
Problem 19
13:00
Problem 20
14:40
Problem 21
15:44
Problem 22
16:42
Problem 23
17:35
Problem 24
17:54
Problem 25
18:32
Problem 26
19:08
Problem 27
20:56
Problem 28
22:19
Problem 29
22:36
Problem 30
23:18
Problem 31
24:06
Problem 32
24:40
1998 AP Practice Exam: Free Response Questions (FRQ)

28m 11s

Intro
0:00
Question 1
0:15
Part A: I
0:16
Part A: II
0:46
Part A: III
1:13
Part B
1:40
Part C
2:49
Part D: I
4:46
Part D: II
5:15
Question 2
5:46
Part A: I
6:13
Part A: II
7:05
Part B: I
7:48
Part B: II
8:42
Part B: III
9:03
Part B: IV
9:26
Part B: V
11:32
Question 3
13:30
Part A: I
13:50
Part A: II
14:16
Part A: III
14:38
Part A: IV
14:56
Part A: V
15:36
Part B
16:11
Part C
17:00
Part D: I
19:56
Part D: II
21:08
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• ## Related Books 3 answersLast reply by: James GlassWed Jan 17, 2018 4:58 PMPost by Professor Dan Fullerton on January 14, 2018I would think about it as an object being free with no extra speed as 0.  If it's in the gravitational field of a large object, it would owe energy in order to get free.  Does that help? 0 answersPost by James Glass on January 14, 2018Hello,At 17:48 where you teach us about escape velocity, what is an abstract way of accepting that there is an escape velocity and total energy when there is an assumption that kinetic energy = 0 and gravitational potential energy = 0?

### Gravity & Orbits

• Any two masses exert an attractive force on each other, known as the gravitational force.
• Gravity is a non-contact force, known as a field force. Its effects are observed without the two objects coming into contact with each other.
• A gravitational field describes the gravitational force a mass feels when placed at a particular point in space.
• Inside a hollow sphere, the gravitational field is zero. Outside the sphere, you can treat the sphere as if its entire mass was concentrated at the center.
• Inside a solid sphere, treat the sphere as if the mass inside the radius is all at the center. Outside the sphere, treat the sphere as if all the mass is concentrated at the center.
• The velocity of a satellite is independent of the satellite’s mass.
• Kepler’s Laws of Planetary Motion:
• The orbits of planetary bodies are ellipses with the sun at one of the two foci of the ellipse.
• If you were to draw a line from the sun to the orbiting body, the body would sweep out equal areas along the ellipse in equal amounts of time.
• The ratio of the squares of the periods of two planets is equal to the ratio of the cubes of their semi-major axes. The ratio of the squares of the periods to the cubes of their semi-major axes is referred to as Kepler’s Constant.

### Gravity & Orbits

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Objectives 0:07
• Newton's Law of Universal Gravitation 0:45
• Newton's Law of Universal Gravitation
• Example I: Gravitational Force Between Earth and Sun 2:24
• Example II: Two Satellites 3:39
• Gravitational Field Strength 4:23
• Gravitational Field Strength
• Example III: Weight on Another Planet 6:22
• Example IV: Gravitational Field of a Hollow Shell 7:31
• Example V: Gravitational Field Inside a Solid Sphere 8:33
• Velocity in Circular Orbit 12:05
• Velocity in Circular Orbit
• Period and Frequency for Circular Orbits 13:56
• Period and Frequency for Circular Orbits
• Mechanical Energy for Circular Orbits 16:11
• Mechanical Energy for Circular Orbits
• Escape Velocity 17:48
• Escape Velocity
• Kepler's 1st Law of Planetary Motion 19:41
• Keller's 1st Law of Planetary Motion
• Kepler's 2nd Law of Planetary Motion 20:05
• Keller's 2nd Law of Planetary Motion
• Kepler's 3rd Law of Planetary Motion 20:57
• Ratio of the Squares of the Periods of Two Planets
• Ratio of the Squares of the Periods to the Cubes of Their Semi-major Axes
• Total Mechanical Energy for an Elliptical Orbit 21:57
• Total Mechanical Energy for an Elliptical Orbit
• Velocity and Radius for an Elliptical Orbit 22:35
• Velocity and Radius for an Elliptical Orbit
• Example VI: Rocket Launched Vertically 24:26
• Example VII: AP-C 2007 FR2 28:16
• Example VII: Part A
• Example VII: Part B
• Example VII: Part C
• Example VII: Part D
• Example VII: Part E

### Transcription: Gravity & Orbits

Hello, everyone, and welcome back to www.educator.com.0000

I'm Dan Fullerton and in this lesson we are going to talk about gravity and orbits.0003

Our objectives include determining the force that one, spherical asymmetric mass inserted on another.0008

Determining the strength of the gravitational field at a specified point outside a spherically symmetric mass.0013

Describing the gravitational force inside and outside the uniform sphere.0020

Calculating how to feel that the surfaced depends on the radius and the density of the sphere.0024

Stating Kepler’s laws of planetary motion and using them to describe the motion of an object in orbit.0030

Finally, applying, conservation of the angular momentum to determine the velocity and radial distance at any point in that object’s orbit.0037

Let us start by taking a look at Newton’s law of universal gravitation.0044

The force of gravity, the force of attraction between any 2 objects with mass is given by this formula -G a constant,0050

a fudge factor to make the units work out, × the mass of your first object × the mass of your second object ÷ the square of the distance between their centers.0059

R hat this means it is in the direction of r hat.0070

If we were to draw this, if we have object 1 and object 2 here, and we define from 1 to 2 as the R vector that means our unit vector in that direction, r hat is that direction.0072

The force of 1 on 2 was going to be opposite the direction of r hat, so it is going to attract 2 force of 1 on 2.0087

That is where the negative sign comes in.0098

Usually, when we are dealing with this, we will worry about magnitudes and use common sense to realize they are attracting each other0100

but the negative sign has to do with this defining the direction from 1 to 2 as in the r hat direction.0105

Now that fudge factor G, the universal gravitational constant that is equal to 6.67 × 10⁻¹¹ N m² / kg².0112

If they want to make a plot of the gravitational force between 2 objects as a function of the distance between their centers,0123

we would have this in versus square law, another place where as the distance increases between objects,0130

we see the force go down as the square of the distance.0137

Let us do a quick sample here, find the magnitude of the gravitational force exerted on the earth by the sun given the mass of the earth,0143

the mass of the sun, and the distance between them.0153

Let us start the force of gravity, it is going to be - GM 1M2 /r² in the direction of r hat, but since I'm worried about the magnitude,0157

the magnitude of the gravitational force is just going to be 6.67 × 10⁻¹¹ N m² / kg².0169

G, the universal gravitational constant × our first math, the mass of the earth about 6 × 10²⁴ kg ×0183

the mass of the sun about 2 × 10³⁸ kg ÷ the square of the distance between them 1.5 × 10¹¹ N m².0192

Put all that in my calculator and I come up with the force of about 3.5 × 10²² N.0206

Let us do another example, the diagram below represents 2 satellites of equal mass A and B and circular orbits around the planet.0218

Compared to the magnitude of the gravitational force of attraction between A in the planet,0227

the magnitude of the gravitational force of attraction between B and the planet is what?0232

Notice that B is twice as far from the center of the planet as A is.0237

Well, because it is twice as far and we have this inverse square law relationship, where the gravitational force is proportional to 1 /0242

the square of the distance between the objects.0249

If we double the distance, we get ¼ the force, answer is C because of the inverse square law relationship.0252

As we are talking about gravity, let us take a look at the gravitational field.0265

Anywhere around the earth or any other object with mass are going to have some force of gravity.0270

To help us think about that, to visualize that as we talk about the gravitational field or the force that an object would feel if placed at that position.0276

What we are going to define is if the force of gravity is GM 1M2 /r² and if we are uniform gravitational field, we can write that as MG.0286

We can do a little simplification here and cancel out one of our masses to find that the gravitational field strength is GM 1 /r².0300

That would be the force an object would feel per unit mass.0312

As you look at these gravitational field lines, as they our more sparse you have less force.0316

As they are more dense, you have a greater gravitational field strength you would have more force per unit mass.0321

Now little G, we have been talking about is the acceleration due to gravity.0328

On the surface of the earth that is 9.8 m / s².0333

If we measure it this way and use these units we are going to come up with N / kg, m/ s² unit is equivalent to 1 N / kg.0337

Gravitational field strength g is the same thing as the acceleration due to gravity.0348

Let us do a quick example here, if we say that the radius of the earth is about 6,378 km, you can actually solve given the mass of the earth0356

and the radius and find g is 9.8 m / s² or 9.8 N / kg.0367

Give it a try to confirm that reading.0377

Let us do another example, if you weigh 600 N on earth, what will you weigh on a planet with twice the mass of earth and half the radius of earth?0382

Of this is a Newton’s law of universal gravitation problem.0392

Then what happens if we double the mass?0398

If we double the masses, we are going to get twice the force, so we would double the force and you have 2 × what you had originally0402

and if you half the radius because of the inverse square law you are going to quadruple the force.0413

At the radius and 2 and the denominator and it is squared, you will get 4 × what you have.0425

Put those two together, you are going to have an 8 × 8x increase.0432

8 × 600 N is going to give us a 4,800 N as your weight on this new planet.0437

Inside a hollow sphere the gravitational field is 0 and it is important to remember, inside a hollow sphere the gravitational field is 0.0453

Outside the hollow sphere you can treat it as if all of the mass were combined into one massive point at the very center,0462

pretend its entire mass is concentrated at the center and then calculate the gravitational field. If we were to do that, we can go and graph it something like this.0471

If this is our gravitational field strength and there is our R, inside the radius of the object we have 00484

and outside we have a gravitational field strength which is equal -GM 1 /R² in the direction of r hat.0496

What if we look at a solid sphere?0511

Outside a solid sphere, we said you can treat the spheres of all the masses at the center of the sphere.0514

Inside the sphere, however, you treat the sphere as if the mass inside the radius is all at the center, the only part that counts is what is inside the radius.0519

Outside the radius of where you are in that sphere does not count.0528

Only the mass inside the radius of interest counts.0532

As an example, let us say we want to know the gravitational field strength at some radius inside this solid sphere.0536

Let us draw another sphere inside and pretend that we are standing right there, somehow inside the solid sphere.0547

To start analyzing this problem, I’m first going to define a volume mass density ρ,0554

which is the total mass ÷ the total volume or M / the volume of our entire sphere 4/3 π R³.0560

Defining the distance from the center to there is R.0573

Inside our sphere of interest here, what the volume enclosed in that sphere of interest is 4/3 π r³.0579

Our r is our radius out to the edge of our sphere of interest so then the mass enclosed by our sphere of interest0589

is going to be our volume mass density × the volume of our sphere of interest which is going to be, we have for ρ, we have capital M /4/3 π R³.0598

For our volume enclosed we have 4/3 π r³.0617

Our mass enclosed is going to be, we can factor out 4/3, π, we are going to just get M, r³ /R³.0623

Looking for the gravitational field strength, G is - G × our mass enclosed/r², where our r is the distance to the edge to the radius of our sphere of interest0640

is going to be - G /r² × our mass enclosed M r³/R³, which is just going to be - GM /R³ × r, where GM and R³ are constants.0655

What we have is a linear function of our radius and this is good for r is less than or equal to R.0677

If I wanted to make a graph of our gravitational field strength vs. our radius, there is our G and we are going to start at some 0 right at that center point.0688

We are going to have a linear increase and then when we get right to R, the edge of our total sphere are going to follow our inverse square log N.0700

Here in this region we are proportional to -r and in this region we are proportional to 1/r².0714

Let us talk about circular orbits.0725

Here we have an object M2 in a circular orbit about object M1.0728

The radius of our orbit is R, the force of gravity that is causing the centripetal force allowing it to move in the circle fg,0734

at any point it has some tangential velocity V along with a rotational or angular velocity ω.0742

As we look at this to analyze that the net centripetal force which is always MAC, M2AC for our object 2 going around object 1,0749

that has to equal the force of gravity because the force of gravity is what is causing that.0761

We know that the force of gravity by Newton's law of universal gravitation is GM 1M2 /r² and our centripetal acceleration we know is V² /r.0768

We could rewrite this as M2 V² /R must be equal to GM 1 M2 / R², which implies then that V²0784

must be equal 2 GM1 /R or if we want just the velocity, the speed at any point in time, the velocity is the √ GM 1 /r.0806

It is in a circular orbit, the radius and the mass of the planet or whether we are orbiting about determines the speed.0820

Notice there is no dependence on the mass of the object that is orbiting.0829

Taking this a bit further, we can look at the period and frequency for a circular orbit.0836

Once around the entire orbit is the circumference or 2 π r and that has to equal the distance traveled which is going to be the velocity × the period, the speed × the period,0841

which implies then that period is 2 π r / V.0856

Due to our amazing physics skills, we just found out that the velocity for something in circular orbit is the √GM 1 /r.0863

Therefore, we can write that the period is 2 π r / our velocity or speed √GM 1 /r.0873

A little bit of rearrangement we can write that as 2 π √r / √GM 1 and all those square roots are making things a little messy.0887

If we square both sides, we can write this as T² = 4 π² r² and √r² is going to be an r.0899

In the numerator we will have r³ ÷ GM 1, that is known as Kepler’s third law for circular orbits.0909

What is pretty cool about this, is if we take the ratio of T² / r³, that is equal to 4π² / GM 1, that is known as Kepler’s constant.0931

For all the planets in our solar system, we have got the same M1 so we have roughly, assuming the roughly circular orbits and they are pretty close.0944

We have the same ratio of the square of their periods to the radius so that is Kepler’s constant.0952

It has a value of about 3 × 10⁻¹⁹ s² / m³.0959

We did a pseudo derivation of Kepler’s third law.0966

We could also look at the energy for objects moving in circular orbits.0970

Our total energy is the kinetic + the potential energy and we know for an object that is moving in a circular orbit we will first offer kinetic energy is ½ M2 V².0975

The gravitational potential energy is - GM 1M2 /r, this implies then that we could write the energy as equal to ½ M2 V² - GM 1M2 /r.0991

We also know that V² = GM 1 /r so we can write this as E = ½ M2 GM1 /r - GM 1 M2 /r, which is just going to give us - GM 1M2 /2r.1014

We have an expression for the total energy for an object in the circular orbit.1046

As we are doing our analysis of things in circular orbits, we can also talk about escape velocity.1068

Now escape velocity is a concept that describes the velocity an object needs while it is in caught by a gravitational field,1073

the amount of speed it needs in order to completely escape the influence of that objects gravity.1082

We know practically, you cannot do that because you cannot get infinitely away from an object.1089

We can calculate how much energy that would require if you did everything perfectly.1095

It is a conservation of energy approach we are going to take here.1099

Let us start by looking at the total energy when we hit this point of being completely out of the influence of an object's gravity.1104

We will do that at the point where its gravitational potential energy is 0 and right at the point where it is kinetic energy is 0, where this completely stopped.1114

Anything past that, you would not need in order to escape that objects gravity that will be left over energy.1123

If we set that as our beginning condition and that means that ½ M2 V² - GM 1M2 /r must equal 0 or ½ M2 V² must equal GM 1M2 /r1129

or solving for V² that is going to be 2 GM1 /r, which implies then that the escape velocity is going to be √2GM1 /r.1155

Since we have been talking about all these orbits, let us get into Kepler’s laws.1176

Kepler’s first law of planetary motion states that the orbits of planetary bodies or ellipses with a sun at 1 of the foe side of the ellipse.1182

We have 2 foe side, the sun is one of them, there is our first planetary law by Kepler.1191

The second one gets a little bit more mathematically in depth.1200

It says if you were to draw a line from the sun to the orbiting body, the body would sweep out equal areas along the ellipse in equal amounts of time.1206

That means, let us assume that we have an object in orbit, right now it is at 0.1 and at some point it gets here to 0.2.1214

As it does that, it sweeps out this amount of area, we will call that area 1.1221

At a later point in time, it is at P3.1228

If we let it go at the same amount of time that elapsed between P1 and P2, it will move from P3 to P4 and it will sweep out that this area of the ellipse.1231

As long as the time between P1 and P2 is the same as the time between P3 and P4, the area swept out in each of these cases is the exact same.1242

That is Kepler’s second law of planetary motion.1253

If we take a look at his third law and we also did this for circular orbits, it is a bit more complex.1257

It says the ratio of the squares of the periods, squares of the periods of 2 planets is equal to the ratio of the cubes of their semi major axis.1263

Instead of r³, we are doing this for a circular path, now we are going to use the semi major axis to find here for the ellipse.1272

This would be for an elliptical path T² /a³ = 4 π² / GM 1 or you might see this written as T² = 4π² a³ /GM 1.1283

The ratio of the squares of the period for the cubes of their semi major axis that is referred to as Kepler’s constant again.1302

There is Kepler’s third law of planetary motion, this time for elliptical orbits.1309

Let us take a look at total mechanical energy for elliptical orbit.1316

We are not going to go through the entire derivation, we did it for a circular orbit.1319

This would get a little more complex but the total energy is the kinetic + the potential which would is ½ M2 V² - GM 1M2 /r, when you are talking about circular orbits.1323

We get to elliptical orbits, when we do all the math and simplify it, we come out with the total energy as - GM 1M2 /2a, that is semi major axis instead of the radius of the circle again.1338

As we look at this one, we have an object M2 and elliptical orbit about M1 and we are going to look into the couple points here.1359

As we analyze it, we can take a look starting with the derivative of the angular momentum about point P with respect to time has to equal the net torque about point B.1367

Assuming we have no external forces causing any net external torques, that has to be 0.1380

Therefore, our angular momentum about point P, I should say equal must be concert.1386

That means that the magnitude of the angular momentum about point P which will be at different points, if we look at that point A for example,1401

that will be M2 velocity of A, radius at A, which would be the semi major axis × the sin θ A has to equal the same thing when it is at point B, M2 VB RB sin θ B.1410

And because the mass is the same, the masses cancel out, VA RA sin θ = VB RB sin θ B and that is true for any of the points on the ellipse.1429

Specifically, when you are at points A and B, the perigee point and what is known as the apogee point, at those points θ A = θ B those are 90°,1440

which implies then that VA RA must equal VB RB, my simplification at those specific points.1453

Let us finish up with a couple problems here.1466

A rocket is launched vertically from the surface of the earth with an initial velocity of 10 km /s, what maximum height does it reached neglecting air resistance?1469

Some things that might be helpful, the mass of the earth, the radius of the earth, and we may not assume that the acceleration due to gravity is constant,1479

it is going to be moving so far away from the surface of the earth.1486

The g is going to be changing in this problem.1490

We can solve this with the conservation of energy approach.1493

The initial kinetic energy + the initial gravitational potential energy must equal the final kinetic + final gravitational potential energies.1497

This implies then that ½ × M2, the mass of our object × the square of its velocity - GM 1M2 / the radius of the earth.1508

Its gravitational potential energy when it is on the surface of the earth because now our sending arbitrary 0 point is infinity has to equal to,1522

its highest point kinetic is 0 and its gravitational potential energy will be - GM 1M2 /r.1531

So then this implies, if we solve for the square of the velocity V² - 2 GM 1 /E = -2 GM1 /R.1542

Notice that M2 has completely have been eliminated from the equation but we kind of expected it.1558

A little bit more rearrangement, 1 /r = -V² / 2 GM1 + 2 GM 1/2 GM 1 RE, radius of the earth which implies that1565

1 /r = - V² / 2 G M1 + 1 / the radius of the earth,1584

which implies that 1 /r = we have our negative square the initial velocity 10,000 m / s² ÷ 2 × G 6.67 × 10⁻¹¹ N m² / kg² × the mass of the earth1594

6 × 10²⁴ kg + 1 /the radius of the earth 6.37 × 10⁶ m, which implies then that r is approximately equal to 3.12 × 10 ⁺7m.1615

Now it is important here to realize that, that is the total distance from the center of the earth.1635

If we want to know the maximum height it reaches, how far above the surface of the earth, what we have calculated here,1640

if there is earth, there is radius of the earth, we have calculated how high it goes there.1648

We are really want to know this distance so that means we have to go a little bit further, the height above the surface of the earth,1655

let us call that H = the radius - the radius of the earth which is going to be our 3.12 × 10⁷ m - the radius of the earth1664

6.37 × 10⁶ m or 2.48 × 10 ⁺7m above the surface of the earth.1679

Let us finish out by doing an old AP free response problem.1692

We are going to start with a 2007 exam free response question number 2.1697

We are looking at the Mars global surveyor, entered its final orbit about Mars and gives us some data about that,1705

we are asked to calculate the radius of the global surveyor's orbit.1711

For part A, it is going in some orbit where we have a radius of Mars of 3.43 × 10⁶ m /s, I should probably should put that in red for Mars.1716

The period it tells us is 118 min or 7080 s, our mass is 930 kg and our velocity is 3400 m/s, calculate the radius then.1734

Period is the distance traveled ÷ the velocity so that is going to be 2 π × the radius ÷ the speed of which implies then1754

that r is going to be equal the VT / 2 π which is 3400 m /s × our period 7080 s / 2 π or I come up with a radius of our orbit of about 3.83 × 10 ⁺6m.1764

Looking at part B, calculate the mass of Mars.1792

We can go back to our Newton’s second law to write that the net centripetal force which is going to be MV² /r must equal the gravitational force,1797

Newton’s law of universal gravitation because the gravitational force is providing that centripetal force GM 1M2 /r²,1811

which implies then that V² is going to be equal 2 G × the mass of Mars ÷ R,1821

which implies then that the mass of Mars must equal the radius × to square root of velocity ÷1831

the gravitational constant which is 3.83 × 10 ⁺6m, we just found that out up there.1838

The square of our velocity is going to be 3400 m /s² ÷ the universal gravitational constant 6.67 × 10⁻¹¹ N m² / kg².1848

It comes out to be about 6.64 × 10²³ kg.1863

For C, it is asking us to find the total mechanical energy of the global surveyor in that orbit.1876

Our total energy is the kinetic energy + the potential energy which is going to be ½ MV² – G mass of the satellite,1885

Mass of Mars over the radius which will be ½ × our mass 930 kg × our speed 3400² – G 6.67 × 10⁻¹¹ ×1897

the mass of our satellite 930 × the mass of Mars 6.64 × 10²³ ÷ 3.83 × 10⁶.1915

I come up with a total energy then, when I put that in my calculator of about -5.38 × 10⁹ joules.1930

Let us take a look at D, if the global surveyor was replaced in the lower circular orbit, will the new orbital period be greater than or less than a given period?1947

That is definitely going to be less than.1957

The velocity must go up, the shorter circumference to travel so the distance it is traveling is going down.1961

And if velocity is displacement /time, distance traveled / time, and time is displacement/ velocity.1968

We have a distance going down and the velocity going up, both those factors lead to a shorter time period /orbit.1977

I might explain that out in a little more detail and sentences but that is the idea,1992

And for part E, we have a question about the global surveyor entered an elliptical orbit, if the speed of its closest approach was 3400 m/s,1997

find the speed at the furthest point of its orbit.2012

We can do that by conservation of spin angular momentum and state that the mass of our satellite × the velocity position 1 × the radius of position 1 =2015

the mass of the satellite × the velocity of position 2 × the radius of position 2 or V1 r1 = V2 r2.2026

If we want the velocity at that new point, V2 is just V1 r1/r2 which would be our 3400 m/s × our radius 3.71 + 34.3 × 10⁵ m.2037

I'm just going to use a short hand, we do not need to put all of the thousands in there since they are all going to factor out here, over 4.36 + 34.3 × 10⁵ m.2057

We will have this ratio × 3400, gives us a velocity of about 3343 m /s.2077

Hopefully, that gets you a great start on gravity and orbits.2090

Thank you for joining us today at www.educator.com.2093

I look forward to seeing you again real soon and make it a great day everybody.2095

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