  Dan Fullerton

Circular & Relative Motion

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 1 answer Last reply by: Professor Dan FullertonMon Aug 23, 2021 3:49 PMPost by Carissa Shuxteau on August 22, 2021If you have a ball that rolls down a loop roller coaster, where you know the diameter of the ball and the amount of time it takes to pass through photogate A, through photogate B, and the time elapsed as it passes between photogate A and B, and the acceleration is not constant, how would you determine the average and instantaneous acceleration of the ball? 1 answer Last reply by: Professor Dan FullertonSun Apr 26, 2020 6:14 PMPost by Samantha Zhan on April 26, 2020For the parabolic form  y(X) = (vy0/vx)X - (g/2vx^2)X^2could the (vy0/vx) be expanded to v0sin?/v0cos? and simply to tan??making the equation y(X) = Xtan? - (g/2vx^2)X^2 3 answers Last reply by: Professor Dan FullertonTue Jun 6, 2017 2:13 PMPost by K Lee on June 6, 2017Where should I go to get example problems that are relevant to the topics? I have Giancoli's Physics: Principles with Applications book, but it is algebra-based, not calculus-based. Would it be good if I do example problems out of that book after each lesson even though it's not calculus based? Thanks! 1 answer Last reply by: Professor Dan FullertonFri Apr 15, 2016 10:57 AMPost by Sunkyung Jung on April 15, 2016Can you help me with this question, please?Plane A is flying at 400 mph in the northeast direction relative to the earth. Plane B is flying at 500 mph in the north direction relative to the earth. What is the direction of motion of Plane B as observed from Plane A? 1 answer Last reply by: Professor Dan FullertonTue Dec 1, 2015 8:21 AMPost by nathan lau on November 29, 2015at 11:11 how did you take the derivative of (rcos(wt)i hat+rsin(wt)j hat) to get the velocity? i'm not sure where the wr's came from, or most of the rest of the answer. 1 answer Last reply by: Professor Dan FullertonThu Apr 23, 2015 8:51 PMPost by Miras Karazhigitov on April 23, 2015In example 4, does acceleration goes toward center of the Earth 0 answersPost by Daniel Fullerton on December 7, 2014That would be a great answer! 0 answersPost by Shaina M on December 6, 2014If I said the answer for that last problem was 5m/s at an angle of 53.13deg NE would that be right?

### Circular & Relative Motion

• Once around a circle is 360 degrees, or 2*Pi radians, where a radian is the distance around an arc equal to the length of the arc’s radius.
• Angular velocity is the time rate of change of the angular displacement, or the first derivative of the angular displacement with respect to time.
• Angular acceleration is the time rate of change of the angular velocity, or the first derivative of the angular velocity with respect to time, or the second derivative of the angular displacement with respect to time.
• The direction of the angular velocity and angular acceleration vectors is given by the right hand rule.
• There is no way to distinguish between motion at rest and motion at a constant velocity in an inertial reference frame.

### Circular & Relative Motion

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 0:08
• Degrees
• Example I: Radians and Degrees 1:08
• Example I: Part A - Convert 90 Degrees to Radians
• Example I: Part B - Convert 6 Radians to Degrees
• Linear vs. Angular Displacement 2:38
• Linear Displacement
• Angular Displacement
• Linear vs. Angular Velocity 3:18
• Linear Velocity
• Angular Velocity
• Direction of Angular Velocity 4:36
• Direction of Angular Velocity
• Converting Linear to Angular Velocity 5:05
• Converting Linear to Angular Velocity
• Example II: Earth's Angular Velocity 6:12
• Linear vs. Angular Acceleration 7:26
• Linear Acceleration
• Angular Acceleration
• Centripetal Acceleration 8:05
• Expressing Position Vector in Terms of Unit Vectors
• Velocity
• Centripetal Acceleration
• Magnitude of Centripetal Acceleration
• Example III: Angular Velocity & Centripetal Acceleration 14:02
• Example IV: Moon's Orbit 15:03
• Reference Frames 17:44
• Reference Frames
• Laws of Physics
• Motion at Rest vs. Motion at a Constant Velocity
• Motion is Relative 19:20
• Reference Frame: Sitting in a Lawn Chair
• Reference Frame: Sitting on a Train
• Calculating Relative Velocities 20:19
• Calculating Relative Velocities
• Example: Calculating Relative Velocities
• 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

### Transcription: Circular & Relative Motion

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

I am Dan Fullerton and in this lesson we are going to start to explore circular and relative motion.0004

Our objectives include understanding and applying relationships between translational and rotational kinematics which we are going to continue in future lessons.0009

Use the right hand rule to determine the direction of the angular velocity vector.0017

Describe the meaning of the phrase motion is relative and calculating the velocity of an object relative to various reference frames.0023

Let us start by talking about how we measure circular motion in degrees and once around the circle is 360°.0031

In radians however once around the circle is 2 π.0040

A radians measures the distance around an arc equal to the length of the arcs radius.0044

If there is a circle there is our radius it takes 2 π × this value to go once around the circle the circumference.0049

We call that distance around the circle δ s or C circumference = 2 π × the radius of that circle.0058

It is going to be useful to convert from radians to degrees and back again.0068

Especially, in this APC course, we are using both types of measurements depending on the problem.0072

If you are using your calculators, be very careful you got them in the right mode that when you are inputting angles0077

you know whether you are using degrees or radians in your calculator is also on the same page as you are.0082

First let us convert 90° to radians.0088

If we start with 90° and we want radians from 90° / 1 we will multiply that there are 2 π radians in one time around the circle or 360°.0091

Therefore I can say that we have π / 2 radians is equal to 90° or 1.57 radians if you to put it in decimal form.0105

Interesting to know radians is not an actual unit of measure.0119

It is helpful for us to use mentally but radians are officially unit less.0123

Let us convert 6 radians to degrees going the other way.0129

We have 6 radians and we want to convert to degrees.0132

I know there are 2 π radians and 360° therefore our radians will make a ratio of 1 and 6 × 360 degrees / 2 π is about 344°.0137

Nice and straightforward.0156

How do we use these?0158

Linear position their displacement we have talked about is being given by δ r or we are also going to see it now especially as we talk about things in circles as δ s.0163

Angular position or displacement is given by δ θ.0172

How many degrees you have is you are going around the circle.0176

S = r × θ if you know your radius and the angle through which you pass you can find how far you have traveled0180

this distance along the circumference s = r θ or δ s = r δ θ.0191

We can also take a look at that in terms of velocity.0198

Linear speed and velocity we have been talking about is being given by V.0201

Angular speed or velocity is given by ω that is the curvy w symbol.0206

If we look V our average velocity is dr dt or ds dt the time rate of change of position.0214

Angular velocity is the time rate of change of θ your angular displacement.0222

Know that any point here your velocity vector is constantly changing the pointing in this direction.0228

Ω is around the circle your angular velocity as you go around there.0234

Now the direction of angular velocity is a little bit tricky because if you look here depending0240

on where you are in the circle your linear velocity is constantly changing.0244

Trying to describe a constant velocity for something around the circle is very tough in the angular world.0250

When we talk about angular velocity to define its direction we are going to use the right hand rule.0256

For something that is moving counterclockwise around the circle take the fingers of your right hand and wrap them in that counterclockwise direction.0262

Your thumb points in what we call the direction of the angular velocity vector.0271

To illustrate that I got a diagram on our next slide here.0275

If something is moving in this direction wrap your hands around it and your thumb points in the direction of the angular velocity vector.0279

It is very interesting to note that the angular velocity vector does not tell you the direction the object is moving at that specific point in time.0286

It is of very nebulous concept the first couple of times you see that.0294

We will also see this with angular acceleration and some other things angular momentum coming up as well.0298

How do we convert linear to angular velocity?0307

Linear velocity was ds dt or dr dt and using those interchangeably at this point.0312

S = r θ how we found our linear displacement along the curve from an angular displacement.0319

We can write this then it as velocity =dr θ dt.0328

But as we go around a circle the radius is a constant that is what makes a circle.0336

We can pull that out of the derivation.0341

We have V =r d θ dt but we just to find d θ dt as ω.0344

ω is d θ dt and we can write that our velocity vector is equal to r × ω vector V= rω.0354

We have a conversion method between the 2.0367

Let us take an example.0374

If we look at Earth's angular velocity find the magnitude of Earth's angular velocity and radians per second.0375

Ω angular velocity is angular displacement over some time interval.0384

Once around is 2 π radians and how long does it take the Earth once all the way around.0392

That is a day or 24 hours so that is going to be π /12 radians / hr.0400

radians / hr is kind of goofy metric.0411

Let us make that into radians / s to more standard unit.0414

I know that 1hr is 3600s.0417

My hours make a ratio of 1 and I come up with about 7.27 × 10⁻⁵ radians /s.0423

We will look at angular displacement angular velocity what come next? angular acceleration.0442

Linear acceleration we said was given by the A vector.0449

Angular acceleration is given by the α vector.0452

Angular acceleration was a derivative of velocity with respect to time.0457

Angular acceleration is the derivative of angular velocity with respect to time.0462

How fast something is accelerating as it goes around that circle?0468

By accelerating, we are not talking about the centripetal acceleration towards the center of a circle0473

that allows it to keep moving a circle or talk about whether it speeding up or slowing down in its path around the circle.0477

Looking at centripetal acceleration in much more detail and putting this together.0484

We can express the position vector in terms of unit vectors.0490

If we say that the position vector r(t) = function x is sum function of time in the i hat direction0494

+ y position as function of time in the j hat direction to give us r vector.0505

When we do this is we are going around the circle we should be fairly easy to see.0514

If we broke this up into components our x position is going to be r cos θ.0521

R y is going to be r sin θ.0527

x(t) is given by r cos θ or θ is going to be a function of time.0531

Ry function, y is a function (t) is r sin θ.0540

We could then write that r(t) Position vector is our cos θ in the x direction × I hat + r sin θ in the y direction or × j hat.0546

We also know θ = ωt angular displacement is angular velocity × time.0565

It is like in the linear world linear displacement is linear velocity × time.0576

R(t) = r cos ωt i hat + r sin ωt j hat.0584

We have gone that far let us take a look and say what if we want velocity though?0602

Velocity was dr dt which is going to be the derivative with respect to time of all the stuff we have up here.0607

R cos ωt i hat + r sin ωt j hat.0619

Let us take the derivative there then that is going to be equal to the derivative of r cos ω t0635

is going to be - ωr sin ωt i hat + derivative of r sin ωt is going to be ωr cos ωt j hat.0642

There is our velocity function.0666

Just write rv in here so we do not forget.0668

There is velocity, let us go to acceleration.0673

Let us keep going.0675

We know the acceleration is dv dt so that is going to be the derivative with respect to time of r - ωr sin ωt i hat + ωr cos ωt j hat.0677

Acceleration, as we take that derivative am going to have -ω² r cos ωt I hat - ω² r sin ωt j hat.0703

We can factor an ω² out of that so I could write that as acceleration is - ω² will have r cos ωt i hat + r sin ωt j hat.0730

That looks mighty familiar this term right here because we defined that way up here.0756

If we look up above we said that r cos ωt I hat + r sin ωt j hat that was our r vector.0763

We could then say that acceleration is - ω² r.0779

Why that negative sign?0786

This is because the acceleration points in the opposite direction of the radius.0789

If our radius vector r is going out to that position this implies that acceleration is in the opposite direction towards the center of the circle.0793

That is centripetal acceleration.0802

We can make it look a little more familiar if you like.0805

If you want to look at the magnitude of that centripetal acceleration that is ω² r still not familiar0808

but what if they put in our substitutions V = ωr or ω = V / r.0818

We can say that the magnitude of a = V² / r² × r or V² / r.0825

Centripetal acceleration V² / r it all works out.0835

Let us do an amusement park example.0841

Riders on a merry go round move in a circle of radius 4m executing 4 revolutions every minute or 1 revolution every 15s.0845

Find the angular velocity and the centripetal acceleration of a rider on the merry go round.0854

Let us start with the angular velocity that will be angular displacement divided by the time interval0860

which is going to be 4 × around in each time around is 2 π radians that takes 60s for all of that which will come out to be 0.419 radians / s.0867

If we want centripetal acceleration that is going to be ω² r or 0.419² × r radius 4m or 0.702 m/s².0882

Another example, let us look at the moon.0904

The Moon revolves around the earth every 27.3 days and the radius of the orbit is 382 mega meters.0906

What is the magnitude and direction of the acceleration of the moon relative to earth?0914

Couple ways we can do this let us start with the old fashioned way.0920

We know that the speed is distance /time.0924

It travels one time around 2π r one circumference in the period of t which is going to be 2π × r radius 382,000,000 m0928

in our time period is going to be 27.3 days but we want that in seconds.0944

Let us do the conversion.0949

27.3 days × we know there are 24 hr in one day and we know that there are 3600s in 1hr so why do all that and I come up with about 1018 m/s.0950

The centripetal acceleration ac =V² / r which is going to be r 1018 m / s² divided by our radius again 382,000,000m or about 0.00271 m/s².0972

That is one way to do it.0996

Let us take what we just learned during the different way.0998

Our angular velocity is δ θ divided by the time interval below 2π radians in again 27.3 days × 24 hours in the day × 3600 s/hr.1001

That comes up with about 2.66 × 10 -6 radians / s.1024

If we want the centripetal acceleration that is going to be ω² r which is 2.66 × 10⁻⁶² × r radius 382,000,000m.1032

I get the same thing 0.00271 m/s².1050

Let us move on and talk about reference frames for couple moments.1063

A reference frame describes the motion of an observer watching something else moving.1067

Our most common reference frame is earth that is the one we deal with every day.1071

We typically measure the motion of things compare to the earth because that is what we typically see is its surroundings.1075

Now the laws of physics, we study in this course assume we are in an inertial not accelerating reference frame.1081

That is not quite true.1087

As we are spinning on the earth we are constantly accelerating1089

and have other motion from the universe but it is negligible compared to what we are typically dealing with.1092

Let us assume we are going to call the earth an inertial reference frame.1098

There is no way to distinguish between motion at a rest and motion at the constant velocity in an inertial reference frame.1102

What that means is, let us think about what is going on if you are in an airplane as an example.1109

Let us take the airplane and put a wing in here.1114

Here we are nice and happy in our airplane.1119

Assuming this is an extremely smooth airplane, no turbulence, no bumps, whatsoever.1123

You cannot tell the difference whether you are sitting on the runway or whether you are in the air moving 500 miles/hr if all the window shades are down.1130

As long as you are moving it that constant speed at the inertial reference frame there is no way to distinguish the difference between the 2.1140

That is what we are talking about when we are talking about an inertial reference frame.1147

How can you tell if you cannot look out the window?1151

There is no experiment you can do to tell the difference between those 2.1153

Once we have that settled what we have talked about that reference frames we will talk about motion being relative.1161

For example imagine you are sitting in a lawn chair watching a train travel past you.1166

It is going to the right at 15m/s.1170

From your reference frame if you saw a cup of water through the train’s window it would look to you at your lawn chair like it is moving at 50 m/s.1175

If you are sitting on the train right in front of a cup of water though the couple of water would appear to be at rest not moving at all.1184

It all depends on your frame of reference.1192

Now imagine, instead you are on the trains staring out of the window you are watching some students sitting on a lawn chair for some unknown reason.1197

From your reference frame, the cup of water on the train remains still but you see that student as though they are moving to the left that 50 m/s.1204

Again, it all depends on your frame of reference.1213

What is going on as far as motion and how you describe that?1216

When we want to calculate velocities between 2 things, relative velocities, let us consider 2 objects a and b.1221

Taking the velocity of a with respect to some reference framed b helps us understand exactly what we are talking about.1228

It is pretty straightforward.1235

For example you might want to know the speed of a car with respect to the ground.1237

Or if you are walking on the train you might want to know the speed of a person with respect to the train.1241

It does not matter that the train is speeding along at 40 m/s.1246

What you are really concerned about is how fast that person is moving with respect to the rest of the train.1250

Let us do some examples here as we talk about strategies.1256

When we are calculating relative velocities here is a little trick I use to help keep things straight.1260

What we are going to do is say that the velocity with some object a with respect to c is defined as the velocity of a1265

with respect to some intermediate object b + the velocity of b with respect to some objects c.1275

As long as you change things like this a to b, b to c, c to d.1281

As long as you have them all daisy chained you can say that the total is the velocity of the first letter with respect to the last letter.1288

In the example we were doing for example.1296

The example we are doing for example a redundant again.1299

The example we were doing, we will call velocity of b with respect to c, the velocity of the train with respect to the ground.1302

We will call object a, that is a is our cup, b is our train, and c is the ground.1314

Velocity of a with respect to b then would be the velocity of the cup with respect to the train.1324

The velocity of a with respect to c then would be the velocity of the cup with respect to the ground.1335

If you want to know the velocity of the cup with respect to the ground you take the velocity of the cup1344

with respect to the train and add it to the velocity of the train with respect to the ground.1348

And the math will all work out.1351

As you look at that pattern here we could have any number of these that we daisy chain1354

for example the velocity of some a with respect to e would be the velocity of a with respect to b + the velocity of some b1359

with respect to c + the velocity of c with respect to d + the velocity of d with respect to e.1365

As we do all these all we have to do is take a look and go with we got a b and b they match.1375

C and c they match.1381

D and d they match.1382

Our total is going to be the first with respect to the last.1384

Velocity of a with respect to e.1389

It will be a little easier to show you how that is done with an example or 2 that is by just talking about it.1392

Let us do a couple.1398

Let us say we have a train we will call that a traveling at 60 m/s to the east with respect to the ground.1400

We will call the ground c.1408

A businessman b on the train runs it 5 m/s to the west with respect to the train.1410

Find the velocity of the man b with respect to the ground c.1415

We are looking for the velocity of b with respect to c.1422

One way I could do that is I can find the velocity of b with respect to a + the velocity of a with respect to c.1427

That would work because we have the a's in the middle so we end up with b and c for a total.1436

Based on what we are given I think we will be ok there.1441

The velocity of b with respect to a would be the velocity of the man with respect to the train.1445

It says the man on the train runs 5 m/s to the west with respect to the train.1449

Velocity of b with respect to a is 5 to the west.1455

Let us call it to the east positive.1458

That will be -5 m/s because the man is running to the west with respect to the train + V ac the velocity of the train1461

with respect to the ground that is 60 m/s to the east.1470

That will be + 60 m/s I will end up with 55 m/s east.1474

The velocity of the businessman with respect to the ground is 55 m/s east.1481

You probably could have done that in your head on a simple problem like this but1487

as we get more involved having a way to keep track like this can be very valuable.1491

Let us take a look at an airspeed example.1497

An airplane let us call that P flies at 250 m/s to the east with respect to the air.1500

The air we will give that an a is moving at 15 m/s to the east with respect to the ground.1505

We will call to the east our positive direction again.1512

Find the velocity of the plane with respect to the ground.1516

We are looking for the velocity of the plane with respect to the ground.1521

We are given the velocity of the plane with respect to the air that is 250 m/s and we are given1525

the velocity of the air with respect to the ground which is 15 m/s.1533

If we want the velocity of the plane with respect to the ground that will be the velocity of the plane1541

with respect to the air + the velocity of air with respect to the ground.1545

Those matches in the middle are left with P and g that should work.1550

So that is going to be 250 m/s + 15 m/s or the velocity of the plane with respect to the ground is 265 m/s.1555

This works in multiple dimensions as well.1570

An airplane P flies a 250 m/s to the east with respect to the air.1574

The air is moving at 35 m/s to the north with respect to the ground.1581

Air with respect to the ground is 35 m/s north.1586

Find the velocity of the plane with respect to the grounds.1593

We want plane with respect to the ground.1596

The velocity of the plane with respect to the ground is going to be the velocity of the plane1600

with respect to the air + the velocity of the air with respect to the ground.1604

It will be Pg Pg that should work.1610

What I am going to do is draw this out.1613

The velocity of the plane with respect to the air that is 250 m/s to the east.1615

The velocity with of air with respect to the ground is 35 m/s to the north and we are going to add that1623

so I will line them up to tail to remember as we talked about vectors.1628

There is our velocity of air with respect to the ground which is 35 m/s.1632

If I want the sum of those the velocity of the plane with respect to the ground I will draw a line1640

from the starting point of my first factor to the ending point of my last once they are lined up to tail.1647

To add these now in vector form let us find a magnitude and find the speed here first.1651

We can do that using the Pythagorean Theorem.1657

The magnitude of the velocity of the plane with respect to the ground is going to be √250² + 35² or about 252 m/s.1659

If we wanted to know this angle θ we can find that as well.1676

Θ equals the inverse tan.1681

We know the opposite and the adjacent side.1683

Inverse tan of opposite / adjacent which will be the inverse tan of 35 m/s / 250 m/s or about 7.97 degrees.1685

My answer the velocity of the plane with respect to the ground is 252 m/s and angle of roughly 8° NE.1700

One last 2D problem here with relative velocities.1711

An oil tanker let us call it T travels east at 3 m/s with respect to the ground while a tugboat moves north at 4 m/s with respect to the tanker.1717

Our tug boat we will call b with respect to our tanker T.1728

What is the velocity of our tug boat with respect to the ground?1735

We want the velocity of our tug boat with respect to the ground and that is going to be the velocity of our tugboat1740

with respect to the tanker + the velocity of the taker with respect to the ground.1746

All right drawing these out, we know that we have as I look at the problem tanker travels east that 3 m/s with respect to the ground.1753

The tanker with respect to the ground VT g is 3 m/s.1761

We also have the tugboat pushes it north or moves forward moves north at 4 m/s with respect to the tanker.1771

The velocity of our boat with respect to that tanker is going to be 4 m/s.1778

We can add these in any way we want I am just going to drop this way so tip to tail again.1784

Velocity of tugboat with respect to the tanker is 4 m/s.1788

Find the velocity of the tugboat with respect to the ground.1793

To do that I will add these up starting at the starting point of my first going to the ending point of my last.1796

And I can do that when my head that is a 345 right triangle.1803

The velocity of our tugboat with respect to the ground must be 5 m/s.1807

And we can use trigonometry if they wanted to if we needed to know that angle exactly.1812

Hopefully that gets you a good start on circular and relative motion.1817

Thank you so much for watching www.educator.com and make it a great day everybody.1821

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