Home » Physics » AP Physics 1 & 2 » Rotational Dynamics

Dan Fullerton

Dan Fullerton

Rotational Dynamics

Slide Duration:

Table of Contents

Section 1: Introduction
What is Physics?

7m 38s

Intro
0:00
Objectives
0:12
What is Physics?
0:31
What is Matter, Energy, and How to They Interact
0:55
Why?
0:58
Physics Answers the 'Why' Questions.
1:05
Matter
1:23
Matter
1:29
Mass
1:33
Inertial Mass
1:53
Gravitational Mass
2:12
A Spacecraft's Mass
2:58
Energy
3:37
Energy: The Ability or Capacity to Do Work
3:39
Work: The Process of Moving an Object
3:45
The Ability or Capacity to Move an Object
3:54
Mass-Energy Equivalence
4:51
Relationship Between Mass and Energy E=mc2
5:01
The Mass of An Object is Really a Measure of Its Energy
5:05
The Study of Everything
5:42
Introductory Course
6:19
Next Steps
7:15
Math Review

24m 12s

Intro
0:00
Outline
0:10
Objectives
0:28
Why Do We Need Units?
0:52
Need to Set Specific Standards for Our Measurements
1:01
Physicists Have Agreed to Use the Systeme International
1:24
The Systeme International
1:50
Based on Powers of 10
1:52
7 Fundamental Units: Meter, Kilogram, Second, Ampere, Candela, Kelvin, Mole
2:02
The Meter
2:18
Meter is a Measure of Length
2:20
Measurements Smaller than a Meter, Use: Centimeter, Millimeter, Micrometer, Nanometer
2:25
Measurements Larger Than a Meter, Use Kilometer
2:38
The Kilogram
2:46
Roughly Equivalent to 2.2 English Pounds
2:49
Grams, Milligrams
2:53
Megagram
2:59
Seconds
3:10
Base Unit of Time
3:12
Minute, Hour, Day
3:20
Milliseconds, Microseconds
3:33
Derived Units
3:41
Velocity
3:45
Acceleration
3:57
Force
4:04
Prefixes for Powers of 10
4:21
Converting Fundamental Units, Example 1
4:53
Converting Fundamental Units, Example 2
7:18
Two-Step Conversions, Example 1
8:24
Two-Step Conversions, Example 2
10:06
Derived Unit Conversions
11:29
Multi-Step Conversions
13:25
Metric Estimations
15:04
What are Significant Figures?
16:01
Represent a Manner of Showing Which Digits In a Number Are Known to Some Level of Certainty
16:03
Example
16:09
Measuring with Sig Figs
16:36
Rule 1
16:40
Rule 2
16:44
Rule 3
16:52
Reading Significant Figures
16:57
All Non-Zero Digits Are Significant
17:04
All Digits Between Non-Zero Digits Are Significant
17:07
Zeros to the Left of the Significant Digits
17:11
Zeros to the Right of the Significant Digits
17:16
Non-Zero Digits
17:21
Digits Between Non-Zeros Are Significant
17:45
Zeroes to the Right of the Sig Figs Are Significant
18:17
Why Scientific Notation?
18:36
Physical Measurements Vary Tremendously in Magnitude
18:38
Example
18:47
Scientific Notation in Practice
19:23
Example 1
19:28
Example 2
19:44
Using Scientific Notation
20:02
Show Your Value Using Correct Number of Significant Figures
20:05
Move the Decimal Point
20:09
Show Your Number Being Multiplied by 10 Raised to the Appropriate Power
20:14
Accuracy and Precision
20:23
Accuracy
20:36
Precision
20:41
Example 1: Scientific Notation w/ Sig Figs
21:48
Example 2: Scientific Notation - Compress
22:25
Example 3: Scientific Notation - Compress
23:07
Example 4: Scientific Notation - Expand
23:31
Vectors & Scalars

25m 5s

Intro
0:00
Objectives
0:05
Scalars
0:29
Definition of Scalar
0:39
Temperature, Mass, Time
0:45
Vectors
1:12
Vectors are Quantities That Have Magnitude and Direction
1:13
Represented by Arrows
1:31
Vector Representations
1:47
Graphical Vector Addition
2:42
Graphical Vector Subtraction
4:58
Vector Components
6:08
Angle of a Vector
8:22
Vector Notation
9:52
Example 1: Vector Components
14:30
Example 2: Vector Components
16:05
Example 3: Vector Magnitude
17:26
Example 4: Vector Addition
19:38
Example 5: Angle of a Vector
24:06
Section 2: Mechanics
Defining & Graphing Motion

30m 11s

Intro
0:00
Objectives
0:07
Position
0:40
An Object's Position Cab Be Assigned to a Variable on a Number Scale
0:43
Symbol for Position
1:07
Distance
1:13
When Position Changes, An Object Has Traveled Some Distance
1:14
Distance is Scalar and Measured in Meters
1:21
Example 1: Distance
1:34
Displacement
2:17
Displacement is a Vector Which Describes the Straight Line From Start to End Point
2:18
Measured in Meters
2:27
Example 2: Displacement
2:39
Average Speed
3:32
The Distance Traveled Divided by the Time Interval
3:33
Speed is a Scalar
3:47
Example 3: Average Speed
3:57
Average Velocity
4:37
The Displacement Divided by the Time Interval
4:38
Velocity is a Vector
4:53
Example 4: Average Velocity
5:06
Example 5: Chuck the Hungry Squirrel
5:55
Acceleration
8:02
Rate At Which Velocity Changes
8:13
Acceleration is a Vector
8:26
Example 6: Acceleration Problem
8:52
Average vs. Instantaneous
9:44
Average Values Take Into Account an Entire Time Interval
9:50
Instantaneous Value Tells the Rate of Change of a Quantity at a Specific Instant in Time
9:54
Example 7: Average Velocity
10:06
Particle Diagrams
11:57
Similar to the Effect of Oil Leak from a Car on the Pavement
11:59
Accelerating
13:03
Position-Time Graphs
14:17
Shows Position as a Function of Time
14:24
Slope of x-t Graph
15:08
Slope Gives You the Velocity
15:09
Negative Indicates Direction
16:27
Velocity-Time Graphs
16:45
Shows Velocity as a Function of Time
16:49
Area Under v-t Graphs
17:47
Area Under the V-T Graph Gives You Change in Displacement
17:48
Example 8: Slope of a v-t Graph
19:45
Acceleration-Time Graphs
21:44
Slope of the v-t Graph Gives You Acceleration
21:45
Area Under the a-t Graph Gives You an Object's Change in Velocity
22:24
Example 10: Motion Graphing
24:03
Example 11: v-t Graph
27:14
Example 12: Displacement From v-t Graph
28:14
Kinematic Equations

36m 13s

Intro
0:00
Objectives
0:07
Problem-Solving Toolbox
0:42
Graphs Are Not Always the Most Effective
0:47
Kinematic Equations Helps us Solve for Five Key Variables
0:56
Deriving the Kinematic Equations
1:29
Kinematic Equations
7:40
Problem Solving Steps
8:13
Label Your Horizontal or Vertical Motion
8:20
Choose a Direction as Positive
8:24
Create a Motion Analysis Table
8:33
Fill in Your Givens
8:42
Solve for Unknowns
8:45
Example 1: Horizontal Kinematics
8:51
Example 2: Vertical Kinematics
11:13
Example 3: 2 Step Problem
13:25
Example 4: Acceleration Problem
16:44
Example 5: Particle Diagrams
17:56
Example 6: Quadratic Solution
20:13
Free Fall
24:24
When the Only Force Acting on an Object is the Force of Gravity, the Motion is Free Fall
24:27
Air Resistance
24:51
Drop a Ball
24:56
Remove the Air from the Room
25:02
Analyze the Motion of Objects by Neglecting Air Resistance
25:06
Acceleration Due to Gravity
25:22
g = 9.8 m/s2
25:25
Approximate g as 10 m/s2 on the AP Exam
25:37
G is Referred to as the Gravitational Field Strength
25:48
Objects Falling From Rest
26:15
Objects Starting from Rest Have an Initial velocity of 0
26:19
Acceleration is +g
26:34
Example 7: Falling Objects
26:47
Objects Launched Upward
27:59
Acceleration is -g
28:04
At Highest Point, the Object has a Velocity of 0
28:19
Symmetry of Motion
28:27
Example 8: Ball Thrown Upward
28:47
Example 9: Height of a Jump
29:23
Example 10: Ball Thrown Downward
33:08
Example 11: Maximum Height
34:16
Projectiles

20m 32s

Intro
0:00
Objectives
0:06
What is a Projectile?
0:26
An Object That is Acted Upon Only By Gravity
0:29
Typically Launched at an Angle
0:43
Path of a Projectile
1:03
Projectiles Launched at an Angle Move in Parabolic Arcs
1:06
Symmetric and Parabolic
1:32
Horizontal Range and Max Height
1:49
Independence of Motion
2:17
Vertical
2:49
Horizontal
2:52
Example 1: Horizontal Launch
3:49
Example 2: Parabolic Path
7:41
Angled Projectiles
8:30
Must First Break Up the Object's Initial Velocity Into x- and y- Components of Initial Velocity
8:32
An Object Will Travel the Maximum Horizontal Distance with a Launch Angle of 45 Degrees
8:43
Example 3: Human Cannonball
8:55
Example 4: Motion Graphs
12:55
Example 5: Launch From a Height
15:33
Example 6: Acceleration of a Projectile
19:56
Relative Motion

10m 52s

Intro
0:00
Objectives
0:06
Reference Frames
0:18
Motion of an Observer
0:21
No Way to Distinguish Between Motion at Rest and Motion at a Constant Velocity
0:44
Motion is Relative
1:35
Example 1
1:39
Example 2
2:09
Calculating Relative Velocities
2:31
Example 1
2:43
Example 2
2:48
Example 3
2:52
Example 1
4:58
Example 2: Airspeed
6:19
Example 3: 2-D Relative Motion
7:39
Example 4: Relative Velocity with Direction
9:40
Newton's 1st Law of Motion

10m 16s

Intro
0:00
Objective
0:05
Newton's 1st Law of Motion
0:16
An Object At Rest Will Remain At Rest
0:21
An Object In Motion Will Remain in Motion
0:26
Net Force
0:39
Also Known As the Law of Inertia
0:46
Force
1:02
Push or Pull
1:04
Newtons
1:08
Contact and Field Forces
1:31
Contact Forces
1:50
Field Forces
2:11
What is a Net Force?
2:30
Vector Sum of All the Forces Acting on an Object
2:33
Translational Equilibrium
2:37
Unbalanced Force Is a Net Force
2:46
What Does It Mean?
3:49
An Object Will Continue in Its Current State of Motion Unless an Unbalanced Force Acts Upon It
3:50
Example of Newton's First Law
4:20
Objects in Motion
5:05
Will Remain in Motion At Constant Velocity
5:06
Hard to Find a Frictionless Environment on Earth
5:10
Static Equilibrium
5:40
Net Force on an Object is 0
5:44
Inertia
6:21
Tendency of an Object to Resist a Change in Velocity
6:23
Inertial Mass
6:35
Gravitational Mass
6:40
Example 1: Inertia
7:10
Example 2: Inertia
7:37
Example 3: Translational Equilibrium
8:03
Example 4: Net Force
8:40
Newton's 2nd Law of Motion

34m 55s

Intro
0:00
Objective
0:07
Free Body Diagrams
0:37
Tools Used to Analyze Physical Situations
0:40
Show All the Forces Acting on a Single Object
0:45
Drawing FBDs
0:58
Draw Object of Interest as a Dot
1:00
Sketch a Coordinate System
1:10
Example 1: Falling Elephant
1:18
Example 2: Falling Elephant with Air Resistance
2:07
Example 3: Soda on Table
3:00
Example 4: Box in Equilibrium
4:25
Example 5: Block on a Ramp
5:01
Pseudo-FBDs
5:53
Draw When Forces Don't Line Up with Axes
5:56
Break Forces That Don’t Line Up with Axes into Components That Do
6:00
Example 6: Objects on a Ramp
6:32
Example 7: Car on a Banked Turn
10:23
Newton's 2nd Law of Motion
12:56
The Acceleration of an Object is in the Direction of the Directly Proportional to the Net Force Applied
13:06
Newton's 1st Two Laws Compared
13:45
Newton's 1st Law
13:51
Newton's 2nd Law
14:10
Applying Newton's 2nd Law
14:50
Example 8: Applying Newton's 2nd Law
15:23
Example 9: Stopping a Baseball
16:52
Example 10: Block on a Surface
19:51
Example 11: Concurrent Forces
21:16
Mass vs. Weight
22:28
Mass
22:29
Weight
22:47
Example 12: Mass vs. Weight
23:16
Translational Equilibrium
24:47
Occurs When There Is No Net Force on an Object
24:49
Equilibrant
24:57
Example 13: Translational Equilibrium
25:29
Example 14: Translational Equilibrium
26:56
Example 15: Determining Acceleration
28:05
Example 16: Suspended Mass
31:03
Newton's 3rd Law of Motion

5m 58s

Intro
0:00
Objectives
0:06
Newton's 3rd Law of Motion
0:20
All Forces Come in Pairs
0:24
Examples
1:22
Action-Reaction Pairs
2:07
Girl Kicking Soccer Ball
2:11
Rocket Ship in Space
2:29
Gravity on You
2:53
Example 1: Force of Gravity
3:34
Example 2: Sailboat
4:00
Example 3: Hammer and Nail
4:49
Example 4: Net Force
5:06
Friction

17m 49s

Intro
0:00
Objectives
0:06
Examples
0:23
Friction Opposes Motion
0:24
Kinetic Friction
0:27
Static Friction
0:36
Magnitude of Frictional Force Is Determined By Two Things
0:41
Coefficient Friction
2:27
Ratio of the Frictional Force and the Normal Force
2:28
Chart of Different Values of Friction
2:48
Kinetic or Static?
3:31
Example 1: Car Sliding
4:18
Example 2: Block on Incline
5:03
Calculating the Force of Friction
5:48
Depends Only Upon the Nature of the Surfaces in Contact and the Magnitude of the Force
5:50
Terminal Velocity
6:14
Air Resistance
6:18
Terminal Velocity of the Falling Object
6:33
Example 3: Finding the Frictional Force
7:36
Example 4: Box on Wood Surface
9:13
Example 5: Static vs. Kinetic Friction
11:49
Example 6: Drag Force on Airplane
12:15
Example 7: Pulling a Sled
13:21
Dynamics Applications

35m 27s

Intro
0:00
Objectives
0:08
Free Body Diagrams
0:49
Drawing FBDs
1:09
Draw Object of Interest as a Dot
1:12
Sketch a Coordinate System
1:18
Example 1: FBD of Block on Ramp
1:39
Pseudo-FBDs
1:59
Draw Object of Interest as a Dot
2:00
Break Up the Forces
2:07
Box on a Ramp
2:12
Example 2: Box at Rest
4:28
Example 3: Box Held by Force
5:00
What is an Atwood Machine?
6:46
Two Objects are Connected by a Light String Over a Mass-less Pulley
6:49
Properties of Atwood Machines
7:13
Ideal Pulleys are Frictionless and Mass-less
7:16
Tension is Constant in a Light String Passing Over an Ideal Pulley
7:23
Solving Atwood Machine Problems
8:02
Alternate Solution
12:07
Analyze the System as a Whole
12:12
Elevators
14:24
Scales Read the Force They Exert on an Object Placed Upon Them
14:42
Can be Used to Analyze Using Newton's 2nd Law and Free body Diagrams
15:23
Example 4: Elevator Accelerates Upward
15:36
Example 5: Truck on a Hill
18:30
Example 6: Force Up a Ramp
19:28
Example 7: Acceleration Down a Ramp
21:56
Example 8: Basic Atwood Machine
24:05
Example 9: Masses and Pulley on a Table
26:47
Example 10: Mass and Pulley on a Ramp
29:15
Example 11: Elevator Accelerating Downward
33:00
Impulse & Momentum

26m 6s

Intro
0:00
Objectives
0:06
Momentum
0:31
Example
0:35
Momentum measures How Hard It Is to Stop a Moving Object
0:47
Vector Quantity
0:58
Example 1: Comparing Momenta
1:48
Example 2: Calculating Momentum
3:08
Example 3: Changing Momentum
3:50
Impulse
5:02
Change In Momentum
5:05
Example 4: Impulse
5:26
Example 5: Impulse-Momentum
6:41
Deriving the Impulse-Momentum Theorem
9:04
Impulse-Momentum Theorem
12:02
Example 6: Impulse-Momentum Theorem
12:15
Non-Constant Forces
13:55
Impulse or Change in Momentum
13:56
Determine the Impulse by Calculating the Area of the Triangle Under the Curve
14:07
Center of Mass
14:56
Real Objects Are More Complex Than Theoretical Particles
14:59
Treat Entire Object as if Its Entire Mass Were Contained at the Object's Center of Mass
15:09
To Calculate the Center of Mass
15:17
Example 7: Force on a Moving Object
15:49
Example 8: Motorcycle Accident
17:49
Example 9: Auto Collision
19:32
Example 10: Center of Mass (1D)
21:29
Example 11: Center of Mass (2D)
23:28
Collisions

21m 59s

Intro
0:00
Objectives
0:09
Conservation of Momentum
0:18
Linear Momentum is Conserved in an Isolated System
0:21
Useful for Analyzing Collisions and Explosions
0:27
Momentum Tables
0:58
Identify Objects in the System
1:05
Determine the Momenta of the Objects Before and After the Event
1:10
Add All the Momenta From Before the Event and Set Them Equal to Momenta After the Event
1:15
Solve Your Resulting Equation for Unknowns
1:20
Types of Collisions
1:31
Elastic Collision
1:36
Inelastic Collision
1:56
Example 1: Conservation of Momentum (1D)
2:02
Example 2: Inelastic Collision
5:12
Example 3: Recoil Velocity
7:16
Example 4: Conservation of Momentum (2D)
9:29
Example 5: Atomic Collision
16:02
Describing Circular Motion

7m 18s

Intro
0:00
Objectives
0:07
Uniform Circular Motion
0:20
Circumference
0:32
Average Speed Formula Still Applies
0:46
Frequency
1:03
Number of Revolutions or Cycles Which Occur Each Second
1:04
Hertz
1:24
Formula for Frequency
1:28
Period
1:36
Time It Takes for One Complete Revolution or Cycle
1:37
Frequency and Period
1:54
Example 1: Car on a Track
2:08
Example 2: Race Car
3:55
Example 3: Toy Train
4:45
Example 4: Round-A-Bout
5:39
Centripetal Acceleration & Force

26m 37s

Intro
0:00
Objectives
0:08
Uniform Circular Motion
0:38
Direction of ac
1:41
Magnitude of ac
3:50
Centripetal Force
4:08
For an Object to Accelerate, There Must Be a Net Force
4:18
Centripetal Force
4:26
Calculating Centripetal Force
6:14
Example 1: Acceleration
7:31
Example 2: Direction of ac
8:53
Example 3: Loss of Centripetal Force
9:19
Example 4: Velocity and Centripetal Force
10:08
Example 5: Demon Drop
10:55
Example 6: Centripetal Acceleration vs. Speed
14:11
Example 7: Calculating ac
15:03
Example 8: Running Back
15:45
Example 9: Car at an Intersection
17:15
Example 10: Bucket in Horizontal Circle
18:40
Example 11: Bucket in Vertical Circle
19:20
Example 12: Frictionless Banked Curve
21:55
Gravitation

32m 56s

Intro
0:00
Objectives
0:08
Universal Gravitation
0:29
The Bigger the Mass the Closer the Attraction
0:48
Formula for Gravitational Force
1:16
Calculating g
2:43
Mass of Earth
2:51
Radius of Earth
2:55
Inverse Square Relationship
4:32
Problem Solving Hints
7:21
Substitute Values in For Variables at the End of the Problem Only
7:26
Estimate the Order of Magnitude of the Answer Before Using Your Calculator
7:38
Make Sure Your Answer Makes Sense
7:55
Example 1: Asteroids
8:20
Example 2: Meteor and the Earth
10:17
Example 3: Satellite
13:13
Gravitational Fields
13:50
Gravity is a Non-Contact Force
13:54
Closer Objects
14:14
Denser Force Vectors
14:19
Gravitational Field Strength
15:09
Example 4: Astronaut
16:19
Gravitational Potential Energy
18:07
Two Masses Separated by Distance Exhibit an Attractive Force
18:11
Formula for Gravitational Field
19:21
How Do Orbits Work?
19:36
Example5: Gravitational Field Strength for Space Shuttle in Orbit
21:35
Example 6: Earth's Orbit
25:13
Example 7: Bowling Balls
27:25
Example 8: Freely Falling Object
28:07
Example 9: Finding g
28:40
Example 10: Space Vehicle on Mars
29:10
Example 11: Fg vs. Mass Graph
30:24
Example 12: Mass on Mars
31:14
Example 13: Two Satellites
31:51
Rotational Kinematics

15m 33s

Intro
0:00
Objectives
0:07
Radians and Degrees
0:26
In Degrees, Once Around a Circle is 360 Degrees
0:29
In Radians, Once Around a Circle is 2π
0:34
Example 1: Degrees to Radians
0:57
Example 2: Radians to Degrees
1:31
Linear vs. Angular Displacement
2:00
Linear Position
2:05
Angular Position
2:10
Linear vs. Angular Velocity
2:35
Linear Speed
2:39
Angular Speed
2:42
Direction of Angular Velocity
3:05
Converting Linear to Angular Velocity
4:22
Example 3: Angular Velocity Example
4:41
Linear vs. Angular Acceleration
5:36
Example 4: Angular Acceleration
6:15
Kinematic Variable Parallels
7:47
Displacement
7:52
Velocity
8:10
Acceleration
8:16
Time
8:22
Kinematic Variable Translations
8:30
Displacement
8:34
Velocity
8:42
Acceleration
8:50
Time
8:58
Kinematic Equation Parallels
9:09
Kinematic Equations
9:12
Delta
9:33
Final Velocity Squared and Angular Velocity Squared
9:54
Example 5: Medieval Flail
10:24
Example 6: CD Player
10:57
Example 7: Carousel
12:13
Example 8: Circular Saw
13:35
Torque

11m 21s

Intro
0:00
Objectives
0:05
Torque
0:18
Force That Causes an Object to Turn
0:22
Must be Perpendicular to the Displacement to Cause a Rotation
0:27
Lever Arm: The Stronger the Force, The More Torque
0:45
Direction of the Torque Vector
1:53
Perpendicular to the Position Vector and the Force Vector
1:54
Right-Hand Rule
2:08
Newton's 2nd Law: Translational vs. Rotational
2:46
Equilibrium
3:58
Static Equilibrium
4:01
Dynamic Equilibrium
4:09
Rotational Equilibrium
4:22
Example 1: Pirate Captain
4:32
Example 2: Auto Mechanic
5:25
Example 3: Sign Post
6:44
Example 4: See-Saw
9:01
Rotational Dynamics

36m 6s

Intro
0:00
Objectives
0:08
Types of Inertia
0:39
Inertial Mass (Translational Inertia)
0:42
Moment of Inertia (Rotational Inertia)
0:53
Moment of Inertia for Common Objects
1:48
Example 1: Calculating Moment of Inertia
2:53
Newton's 2nd Law - Revisited
5:09
Acceleration of an Object
5:15
Angular Acceleration of an Object
5:24
Example 2: Rotating Top
5:47
Example 3: Spinning Disc
7:54
Angular Momentum
9:41
Linear Momentum
9:43
Angular Momentum
10:00
Calculating Angular Momentum
10:51
Direction of the Angular Momentum Vector
11:26
Total Angular Momentum
12:29
Example 4: Angular Momentum of Particles
14:15
Example 5: Rotating Pedestal
16:51
Example 6: Rotating Discs
18:39
Angular Momentum and Heavenly Bodies
20:13
Types of Kinetic Energy
23:41
Objects Traveling with a Translational Velocity
23:45
Objects Traveling with Angular Velocity
24:00
Translational vs. Rotational Variables
24:33
Example 7: Kinetic Energy of a Basketball
25:45
Example 8: Playground Round-A-Bout
28:17
Example 9: The Ice Skater
30:54
Example 10: The Bowler
33:15
Work & Power

31m 20s

Intro
0:00
Objectives
0:09
What Is Work?
0:31
Power Output
0:35
Transfer Energy
0:39
Work is the Process of Moving an Object by Applying a Force
0:46
Examples of Work
0:56
Calculating Work
2:16
Only the Force in the Direction of the Displacement Counts
2:33
Formula for Work
2:48
Example 1: Moving a Refrigerator
3:16
Example 2: Liberating a Car
3:59
Example 3: Crate on a Ramp
5:20
Example 4: Lifting a Box
7:11
Example 5: Pulling a Wagon
8:38
Force vs. Displacement Graphs
9:33
The Area Under a Force vs. Displacement Graph is the Work Done by the Force
9:37
Find the Work Done
9:49
Example 6: Work From a Varying Force
11:00
Hooke's Law
12:42
The More You Stretch or Compress a Spring, The Greater the Force of the Spring
12:46
The Spring's Force is Opposite the Direction of Its Displacement from Equilibrium
13:00
Determining the Spring Constant
14:21
Work Done in Compressing the Spring
15:27
Example 7: Finding Spring Constant
16:21
Example 8: Calculating Spring Constant
17:58
Power
18:43
Work
18:46
Power
18:50
Example 9: Moving a Sofa
19:26
Calculating Power
20:41
Example 10: Motors Delivering Power
21:27
Example 11: Force on a Cyclist
22:40
Example 12: Work on a Spinning Mass
23:52
Example 13: Work Done by Friction
25:05
Example 14: Units of Power
28:38
Example 15: Frictional Force on a Sled
29:43
Energy

20m 15s

Intro
0:00
Objectives
0:07
What is Energy?
0:24
The Ability or Capacity to do Work
0:26
The Ability or Capacity to Move an Object
0:34
Types of Energy
0:39
Energy Transformations
2:07
Transfer Energy by Doing Work
2:12
Work-Energy Theorem
2:20
Units of Energy
2:51
Kinetic Energy
3:08
Energy of Motion
3:13
Ability or Capacity of a Moving Object to Move Another Object
3:17
A Single Object Can Only Have Kinetic Energy
3:46
Example 1: Kinetic Energy of a Motorcycle
5:08
Potential Energy
5:59
Energy An Object Possesses
6:10
Gravitational Potential Energy
7:21
Elastic Potential Energy
9:58
Internal Energy
10:16
Includes the Kinetic Energy of the Objects That Make Up the System and the Potential Energy of the Configuration
10:20
Calculating Gravitational Potential Energy in a Constant Gravitational Field
10:57
Sources of Energy on Earth
12:41
Example 2: Potential Energy
13:41
Example 3: Energy of a System
14:40
Example 4: Kinetic and Potential Energy
15:36
Example 5: Pendulum
16:55
Conservation of Energy

23m 20s

Intro
0:00
Objectives
0:08
Law of Conservation of Energy
0:22
Energy Cannot Be Created or Destroyed.. It Can Only Be Changed
0:27
Mechanical Energy
0:34
Conservation Laws
0:40
Examples
0:49
Kinematics vs. Energy
4:34
Energy Approach
4:56
Kinematics Approach
6:04
The Pendulum
8:07
Example 1: Cart Compressing a Spring
13:09
Example 2
14:23
Example 3: Car Skidding to a Stop
16:15
Example 4: Accelerating an Object
17:27
Example 5: Block on Ramp
18:06
Example 6: Energy Transfers
19:21
Simple Harmonic Motion

58m 30s

Intro
0:00
Objectives
0:08
What Is Simple Harmonic Motion?
0:57
Nature's Typical Reaction to a Disturbance
1:00
A Displacement Which Results in a Linear Restoring Force Results in SHM
1:25
Review of Springs
1:43
When a Force is Applied to a Spring, the Spring Applies a Restoring Force
1:46
When the Spring is in Equilibrium, It Is 'Unstrained'
1:54
Factors Affecting the Force of A Spring
2:00
Oscillations
3:42
Repeated Motions
3:45
Cycle 1
3:52
Period
3:58
Frequency
4:07
Spring-Block Oscillator
4:47
Mass of the Block
4:59
Spring Constant
5:05
Example 1: Spring-Block Oscillator
6:30
Diagrams
8:07
Displacement
8:42
Velocity
8:57
Force
9:36
Acceleration
10:09
U
10:24
K
10:47
Example 2: Harmonic Oscillator Analysis
16:22
Circular Motion vs. SHM
23:26
Graphing SHM
25:52
Example 3: Position of an Oscillator
28:31
Vertical Spring-Block Oscillator
31:13
Example 4: Vertical Spring-Block Oscillator
34:26
Example 5: Bungee
36:39
The Pendulum
43:55
Mass Is Attached to a Light String That Swings Without Friction About the Vertical Equilibrium
44:04
Energy and the Simple Pendulum
44:58
Frequency and Period of a Pendulum
48:25
Period of an Ideal Pendulum
48:31
Assume Theta is Small
48:54
Example 6: The Pendulum
50:15
Example 7: Pendulum Clock
53:38
Example 8: Pendulum on the Moon
55:14
Example 9: Mass on a Spring
56:01
Section 3: Fluids
Density & Buoyancy

19m 48s

Intro
0:00
Objectives
0:09
Fluids
0:27
Fluid is Matter That Flows Under Pressure
0:31
Fluid Mechanics is the Study of Fluids
0:44
Density
0:57
Density is the Ratio of an Object's Mass to the Volume It Occupies
0:58
Less Dense Fluids
1:06
Less Dense Solids
1:09
Example 1: Density of Water
1:27
Example 2: Volume of Gold
2:19
Example 3: Floating
3:06
Buoyancy
3:54
Force Exerted by a Fluid on an Object, Opposing the Object's Weight
3:56
Buoyant Force Determined Using Archimedes Principle
4:03
Example 4: Buoyant Force
5:12
Example 5: Shark Tank
5:56
Example 6: Concrete Boat
7:47
Example 7: Apparent Mass
10:08
Example 8: Volume of a Submerged Cube
13:21
Example 9: Determining Density
15:37
Pressure & Pascal's Principle

18m 7s

Intro
0:00
Objectives
0:09
Pressure
0:25
Pressure is the Effect of a Force Acting Upon a Surface
0:27
Formula for Pressure
0:41
Force is Always Perpendicular to the Surface
0:50
Exerting Pressure
1:03
Fluids Exert Outward Pressure in All Directions on the Sides of Any Container Holding the Fluid
1:36
Earth's Atmosphere Exerts Pressure
1:42
Example 1: Pressure on Keyboard
2:17
Example 2: Sleepy Fisherman
3:03
Example 3: Scale on Planet Physica
4:12
Example 4: Ranking Pressures
5:00
Pressure on a Submerged Object
6:45
Pressure a Fluid Exerts on an Object Submerged in That Fluid
6:46
If There Is Atmosphere Above the Fluid
7:03
Example 5: Gauge Pressure Scuba Diving
7:27
Example 6: Absolute Pressure Scuba Diving
8:13
Pascal's Principle
8:51
Force Multiplication Using Pascal's Principle
9:24
Example 7: Barber's Chair
11:38
Example 8: Hydraulic Auto Lift
13:26
Example 9: Pressure on a Penny
14:41
Example 10: Depth in Fresh Water
16:39
Example 11: Absolute vs. Gauge Pressure
17:23
Continuity Equation for Fluids

7m

Intro
0:00
Objectives
0:08
Conservation of Mass for Fluid Flow
0:18
Law of Conservation of Mass for Fluids
0:21
Volume Flow Rate Remains Constant Throughout the Pipe
0:35
Volume Flow Rate
0:59
Quantified In Terms Of Volume Flow Rate
1:01
Area of Pipe x Velocity of Fluid
1:05
Must Be Constant Throughout Pipe
1:10
Example 1: Tapered Pipe
1:44
Example 2: Garden Hose
2:37
Example 3: Oil Pipeline
4:49
Example 4: Roots of Continuity Equation
6:16
Bernoulli's Principle

20m

Intro
0:00
Objectives
0:08
Bernoulli's Principle
0:21
Airplane Wings
0:35
Venturi Pump
1:56
Bernoulli's Equation
3:32
Example 1: Torricelli's Theorem
4:38
Example 2: Gauge Pressure
7:26
Example 3: Shower Pressure
8:16
Example 4: Water Fountain
12:29
Example 5: Elevated Cistern
15:26
Section 4: Thermal Physics
Temperature, Heat, & Thermal Expansion

24m 17s

Intro
0:00
Objectives
0:12
Thermal Physics
0:42
Explores the Internal Energy of Objects Due to the Motion of the Atoms and Molecules Comprising the Objects
0:46
Explores the Transfer of This Energy From Object to Object
0:53
Temperature
1:00
Thermal Energy Is Related to the Kinetic Energy of All the Particles Comprising the Object
1:03
The More Kinetic Energy of the Constituent Particles Have, The Greater the Object's Thermal Energy
1:12
Temperature and Phases of Matter
1:44
Solids
1:48
Liquids
1:56
Gases
2:02
Average Kinetic Energy and Temperature
2:16
Average Kinetic Energy
2:24
Boltzmann's Constant
2:29
Temperature Scales
3:06
Converting Temperatures
4:37
Heat
5:03
Transfer of Thermal Energy
5:06
Accomplished Through Collisions Which is Conduction
5:13
Methods of Heat Transfer
5:52
Conduction
5:59
Convection
6:19
Radiation
6:31
Quantifying Heat Transfer in Conduction
6:37
Rate of Heat Transfer is Measured in Watts
6:42
Thermal Conductivity
7:12
Example 1: Average Kinetic Energy
7:35
Example 2: Body Temperature
8:22
Example 3: Temperature of Space
9:30
Example 4: Temperature of the Sun
10:44
Example 5: Heat Transfer Through Window
11:38
Example 6: Heat Transfer Across a Rod
12:40
Thermal Expansion
14:18
When Objects Are Heated, They Tend to Expand
14:19
At Higher Temperatures, Objects Have Higher Average Kinetic Energies
14:24
At Higher Levels of Vibration, The Particles Are Not Bound As Tightly to Each Other
14:30
Linear Expansion
15:11
Amount a Material Expands is Characterized by the Material's Coefficient of Expansion
15:14
One-Dimensional Expansion -> Linear Coefficient of Expansion
15:20
Volumetric Expansion
15:38
Three-Dimensional Expansion -> Volumetric Coefficient of Expansion
15:45
Volumetric Coefficient of Expansion is Roughly Three Times the Linear Coefficient of Expansion
16:03
Coefficients of Thermal Expansion
16:24
Example 7: Contracting Railroad Tie
16:59
Example 8: Expansion of an Aluminum Rod
18:37
Example 9: Water Spilling Out of a Glass
20:18
Example 10: Average Kinetic Energy vs. Temperature
22:18
Example 11: Expansion of a Ring
23:07
Ideal Gases

24m 15s

Intro
0:00
Objectives
0:10
Ideal Gases
0:25
Gas Is Comprised of Many Particles Moving Randomly in a Container
0:34
Particles Are Far Apart From One Another
0:46
Particles Do Not Exert Forces Upon One Another Unless They Come In Contact in an Elastic Collision
0:53
Ideal Gas Law
1:18
Atoms, Molecules, and Moles
2:56
Protons
2:59
Neutrons
3:15
Electrons
3:18
Examples
3:25
Example 1: Counting Moles
4:58
Example 2: Moles of CO2 in a Bottle
6:00
Example 3: Pressurized CO2
6:54
Example 4: Helium Balloon
8:53
Internal Energy of an Ideal Gas
10:17
The Average Kinetic Energy of the Particles of an Ideal Gas
10:21
Total Internal Energy of the Ideal Gas Can Be Found by Multiplying the Average Kinetic Energy of the Gas's Particles by the Numbers of Particles in the Gas
10:32
Example 5: Internal Energy of Oxygen
12:00
Example 6: Temperature of Argon
12:41
Root-Mean-Square Velocity
13:40
This is the Square Root of the Average Velocity Squared For All the Molecules in the System
13:43
Derived from the Maxwell-Boltzmann Distribution Function
13:56
Calculating vrms
14:56
Example 7: Average Velocity of a Gas
18:32
Example 8: Average Velocity of a Gas
19:44
Example 9: vrms of Molecules in Equilibrium
20:59
Example 10: Moles to Molecules
22:25
Example 11: Relating Temperature and Internal Energy
23:22
Thermodynamics

22m 29s

Intro
0:00
Objectives
0:06
Zeroth Law of Thermodynamics
0:26
First Law of Thermodynamics
1:00
The Change in the Internal Energy of a Closed System is Equal to the Heat Added to the System Plus the Work Done on the System
1:04
It is a Restatement of the Law of Conservation of Energy
1:19
Sign Conventions Are Important
1:25
Work Done on a Gas
1:44
Example 1: Adding Heat to a System
3:25
Example 2: Expanding a Gas
4:07
P-V Diagrams
5:11
Pressure-Volume Diagrams are Useful Tools for Visualizing Thermodynamic Processes of Gases
5:13
Use Ideal Gas Law to Determine Temperature of Gas
5:25
P-V Diagrams II
5:55
Volume Increases, Pressure Decreases
6:00
As Volume Expands, Gas Does Work
6:19
Temperature Rises as You Travel Up and Right on a PV Diagram
6:29
Example 3: PV Diagram Analysis
6:40
Types of PV Processes
7:52
Adiabatic
8:03
Isobaric
8:19
Isochoric
8:28
Isothermal
8:35
Adiabatic Processes
8:47
Heat Is not Transferred Into or Out of The System
8:50
Heat = 0
8:55
Isobaric Processes
9:19
Pressure Remains Constant
9:21
PV Diagram Shows a Horizontal Line
9:27
Isochoric Processes
9:51
Volume Remains Constant
9:52
PV Diagram Shows a Vertical Line
9:58
Work Done on the Gas is Zero
10:01
Isothermal Processes
10:27
Temperature Remains Constant
10:29
Lines on a PV Diagram Are Isotherms
10:31
PV Remains Constant
10:38
Internal Energy of Gas Remains Constant
10:40
Example 4: Adiabatic Expansion
10:46
Example 5: Removing Heat
11:25
Example 6: Ranking Processes
13:08
Second Law of Thermodynamics
13:59
Heat Flows Naturally From a Warmer Object to a Colder Object
14:02
Heat Energy Cannot be Completely Transformed Into Mechanical Work
14:11
All Natural Systems Tend Toward a Higher Level of Disorder
14:19
Heat Engines
14:52
Heat Engines Convert Heat Into Mechanical Work
14:56
Efficiency of a Heat Engine is the Ratio of the Engine You Get Out to the Energy You Put In
14:59
Power in Heat Engines
16:09
Heat Engines and PV Diagrams
17:38
Carnot Engine
17:54
It Is a Theoretical Heat Engine That Operates at Maximum Possible Efficiency
18:02
It Uses Only Isothermal and Adiabatic Processes
18:08
Carnot's Theorem
18:11
Example 7: Carnot Engine
18:49
Example 8: Maximum Efficiency
21:02
Example 9: PV Processes
21:51
Section 5: Electricity & Magnetism
Electric Fields & Forces

38m 24s

Intro
0:00
Objectives
0:10
Electric Charges
0:34
Matter is Made Up of Atoms
0:37
Protons Have a Charge of +1
0:45
Electrons Have a Charge of -1
1:00
Most Atoms Are Neutral
1:04
Ions
1:15
Fundamental Unit of Charge is the Coulomb
1:29
Like Charges Repel, While Opposites Attract
1:50
Example 1: Charge on an Object
2:22
Example 2: Charge of an Alpha Particle
3:36
Conductors and Insulators
4:27
Conductors Allow Electric Charges to Move Freely
4:30
Insulators Do Not Allow Electric Charges to Move Freely
4:39
Resistivity is a Material Property
4:45
Charging by Conduction
5:05
Materials May Be Charged by Contact, Known as Conduction
5:07
Conductors May Be Charged by Contact
5:24
Example 3: Charging by Conduction
5:38
The Electroscope
6:44
Charging by Induction
8:00
Example 4: Electrostatic Attraction
9:23
Coulomb's Law
11:46
Charged Objects Apply a Force Upon Each Other = Coulombic Force
11:52
Force of Attraction or Repulsion is Determined by the Amount of Charge and the Distance Between the Charges
12:04
Example 5: Determine Electrostatic Force
13:09
Example 6: Deflecting an Electron Beam
15:35
Electric Fields
16:28
The Property of Space That Allows a Charged Object to Feel a Force
16:44
Electric Field Strength Vector is the Amount of Electrostatic Force Observed by a Charge Per Unit of Charge
17:01
The Direction of the Electric Field Vector is the Direction a Positive Charge Would Feel a Force
17:24
Example 7: Field Between Metal Plates
17:58
Visualizing the Electric Field
19:27
Electric Field Lines Point Away from Positive Charges and Toward Negative Charges
19:40
Electric Field Lines Intersect Conductors at Right Angles to the Surface
19:50
Field Strength and Line Density Decreases as You Move Away From the Charges
19:58
Electric Field Lines
20:09
E Field Due to a Point Charge
22:32
Electric Fields Are Caused by Charges
22:35
Electric Field Due to a Point Charge Can Be Derived From the Definition of the Electric Field and Coulomb's Law
22:38
To Find the Electric Field Due to Multiple Charges
23:09
Comparing Electricity to Gravity
23:56
Force
24:02
Field Strength
24:16
Constant
24:37
Charge/ Mass Units
25:01
Example 8: E Field From 3 Point Charges
25:07
Example 9: Where is the E Field Zero?
31:43
Example 10: Gravity and Electricity
36:38
Example 11: Field Due to Point Charge
37:34
Electric Potential Difference

35m 58s

Intro
0:00
Objectives
0:09
Electric Potential Energy
0:32
When an Object Was Lifted Against Gravity By Applying a Force for Some Distance, Work Was Done
0:35
When a Charged Object is Moved Against an Electric Field by Applying a Force for Some Distance, Work is Done
0:43
Electric Potential Difference
1:30
Example 1: Charge From Work
2:06
Example 2: Electric Energy
3:09
The Electron-Volt
4:02
Electronvolt (eV)
4:15
1eV is the Amount of Work Done in Moving an Elementary Charge Through a Potential Difference of 1 Volt
4:28
Example 3: Energy in eV
5:33
Equipotential Lines
6:32
Topographic Maps Show Lines of Equal Altitude, or Equal Gravitational Potential
6:36
Lines Connecting Points of Equal Electrical Potential are Known as Equipotential Lines
6:57
Drawing Equipotential Lines
8:15
Potential Due to a Point Charge
10:46
Calculate the Electric Field Vector Due to a Point Charge
10:52
Calculate the Potential Difference Due to a Point Charge
11:05
To Find the Potential Difference Due to Multiple Point Charges
11:16
Example 4: Potential Due to a Point Charge
11:52
Example 5: Potential Due to Point Charges
13:04
Parallel Plates
16:34
Configurations in Which Parallel Plates of Opposite Charge are Situated a Fixed Distance From Each Other
16:37
These Can Create a Capacitor
16:45
E Field Due to Parallel Plates
17:14
Electric Field Away From the Edges of Two Oppositely Charged Parallel Plates is Constant
17:15
Magnitude of the Electric Field Strength is Give By the Potential Difference Between the Plates Divided by the Plate Separation
17:47
Capacitors
18:09
Electric Device Used to Store Charge
18:11
Once the Plates Are Charged, They Are Disconnected
18:30
Device's Capacitance
18:46
Capacitors Store Energy
19:28
Charges Located on the Opposite Plates of a Capacitor Exert Forces on Each Other
19:31
Example 6: Capacitance
20:28
Example 7: Charge on a Capacitor
22:03
Designing Capacitors
24:00
Area of the Plates
24:05
Separation of the Plates
24:09
Insulating Material
24:13
Example 8: Designing a Capacitor
25:35
Example 9: Calculating Capacitance
27:39
Example 10: Electron in Space
29:47
Example 11: Proton Energy Transfer
30:35
Example 12: Two Conducting Spheres
32:50
Example 13: Equipotential Lines for a Capacitor
34:48
Current & Resistance

21m 14s

Intro
0:00
Objectives
0:06
Electric Current
0:19
Path Through Current Flows
0:21
Current is the Amount of Charge Passing a Point Per Unit Time
0:25
Conventional Current is the Direction of Positive Charge Flow
0:43
Example 1: Current Through a Resistor
1:19
Example 2: Current Due to Elementary Charges
1:47
Example 3: Charge in a Light Bulb
2:35
Example 4: Flashlights
3:03
Conductivity and Resistivity
4:41
Conductivity is a Material's Ability to Conduct Electric Charge
4:53
Resistivity is a Material's Ability to Resist the Movement of Electric Charge
5:11
Resistance vs. Resistivity vs. Resistors
5:35
Resistivity Is a Material Property
5:40
Resistance Is a Functional Property of an Element in an Electric Circuit
5:57
A Resistor is a Circuit Element
7:23
Resistors
7:45
Example 5: Calculating Resistance
8:17
Example 6: Resistance Dependencies
10:09
Configuration of Resistors
10:50
When Placed in a Circuit, Resistors Can be Organized in Both Serial and Parallel Arrangements
10:53
May Be Useful to Determine an Equivalent Resistance Which Could Be Used to Replace a System or Resistors with a Single Equivalent Resistor
10:58
Resistors in Series
11:15
Resistors in Parallel
12:35
Example 7: Finding Equivalent Resistance
15:01
Example 8: Length and Resistance
17:43
Example 9: Comparing Resistors
18:21
Example 10: Comparing Wires
19:12
Ohm's Law & Power

10m 35s

Intro
0:00
Objectives
0:06
Ohm's Law
0:21
Relates Resistance, Potential Difference, and Current Flow
0:23
Example 1: Resistance of a Wire
1:22
Example 2: Circuit Current
1:58
Example 3: Variable Resistor
2:30
Ohm's 'Law'?
3:22
Very Useful Empirical Relationship
3:31
Test if a Material is 'Ohmic'
3:40
Example 4: Ohmic Material
3:58
Electrical Power
4:24
Current Flowing Through a Circuit Causes a Transfer of Energy Into Different Types
4:26
Example: Light Bulb
4:36
Example: Television
4:58
Calculating Power
5:09
Electrical Energy
5:14
Charge Per Unit Time Is Current
5:29
Expand Using Ohm's Law
5:48
Example 5: Toaster
7:43
Example 6: Electric Iron
8:19
Example 7: Power of a Resistor
9:19
Example 8: Information Required to Determine Power in a Resistor
9:55
Circuits & Electrical Meters

8m 44s

Intro
0:00
Objectives
0:08
Electrical Circuits
0:21
A Closed-Loop Path Through Which Current Can Flow
0:22
Can Be Made Up of Most Any Materials, But Typically Comprised of Electrical Devices
0:27
Circuit Schematics
1:09
Symbols Represent Circuit Elements
1:30
Lines Represent Wires
1:33
Sources for Potential Difference: Voltaic Cells, Batteries, Power Supplies
1:36
Complete Conducting Paths
2:43
Voltmeters
3:20
Measure the Potential Difference Between Two Points in a Circuit
3:21
Connected in Parallel with the Element to be Measured
3:25
Have Very High Resistance
3:59
Ammeters
4:19
Measure the Current Flowing Through an Element of a Circuit
4:20
Connected in Series with the Circuit
4:25
Have Very Low Resistance
4:45
Example 1: Ammeter and Voltmeter Placement
4:56
Example 2: Analyzing R
6:27
Example 3: Voltmeter Placement
7:12
Example 4: Behavior or Electrical Meters
7:31
Circuit Analysis

48m 58s

Intro
0:00
Objectives
0:07
Series Circuits
0:27
Series Circuits Have Only a Single Current Path
0:29
Removal of any Circuit Element Causes an Open Circuit
0:31
Kirchhoff's Laws
1:36
Tools Utilized in Analyzing Circuits
1:42
Kirchhoff's Current Law States
1:47
Junction Rule
2:00
Kirchhoff's Voltage Law States
2:05
Loop Rule
2:18
Example 1: Voltage Across a Resistor
2:23
Example 2: Current at a Node
3:45
Basic Series Circuit Analysis
4:53
Example 3: Current in a Series Circuit
9:21
Example 4: Energy Expenditure in a Series Circuit
10:14
Example 5: Analysis of a Series Circuit
12:07
Example 6: Voltmeter In a Series Circuit
14:57
Parallel Circuits
17:11
Parallel Circuits Have Multiple Current Paths
17:13
Removal of a Circuit Element May Allow Other Branches of the Circuit to Continue Operating
17:15
Basic Parallel Circuit Analysis
18:19
Example 7: Parallel Circuit Analysis
21:05
Example 8: Equivalent Resistance
22:39
Example 9: Four Parallel Resistors
23:16
Example 10: Ammeter in a Parallel Circuit
26:27
Combination Series-Parallel Circuits
28:50
Look For Portions of the Circuit With Parallel Elements
28:56
Work Back to Original Circuit
29:09
Analysis of a Combination Circuit
29:20
Internal Resistance
34:11
In Reality, Voltage Sources Have Some Amount of 'Internal Resistance'
34:16
Terminal Voltage of the Voltage Source is Reduced Slightly
34:25
Example 11: Two Voltage Sources
35:16
Example 12: Internal Resistance
42:46
Example 13: Complex Circuit with Meters
45:22
Example 14: Parallel Equivalent Resistance
48:24
RC Circuits

24m 47s

Intro
0:00
Objectives
0:08
Capacitors in Parallel
0:34
Capacitors Store Charge on Their Plates
0:37
Capacitors In Parallel Can Be Replaced with an Equivalent Capacitor
0:46
Capacitors in Series
2:42
Charge on Capacitors Must Be the Same
2:44
Capacitor In Series Can Be Replaced With an Equivalent Capacitor
2:47
RC Circuits
5:40
Comprised of a Source of Potential Difference, a Resistor Network, and One or More Capacitors
5:42
Uncharged Capacitors Act Like Wires
6:04
Charged Capacitors Act Like Opens
6:12
Charging an RC Circuit
6:23
Discharging an RC Circuit
11:36
Example 1: RC Analysis
14:50
Example 2: More RC Analysis
18:26
Example 3: Equivalent Capacitance
21:19
Example 4: More Equivalent Capacitance
22:48
Magnetic Fields & Properties

19m 48s

Intro
0:00
Objectives
0:07
Magnetism
0:32
A Force Caused by Moving Charges
0:34
Magnetic Domains Are Clusters of Atoms with Electrons Spinning in the Same Direction
0:51
Example 1: Types of Fields
1:23
Magnetic Field Lines
2:25
Make Closed Loops and Run From North to South Outside the Magnet
2:26
Magnetic Flux
2:42
Show the Direction the North Pole of a Magnet Would Tend to Point If Placed in the Field
2:54
Example 2: Lines of Magnetic Force
3:49
Example 3: Forces Between Bar Magnets
4:39
The Compass
5:28
The Earth is a Giant Magnet
5:31
The Earth's Magnetic North pole is Located Near the Geographic South Pole, and Vice Versa
5:33
A Compass Lines Up with the Net Magnetic Field
6:07
Example 3: Compass in Magnetic Field
6:41
Example 4: Compass Near a Bar Magnet
7:14
Magnetic Permeability
7:59
The Ratio of the Magnetic Field Strength Induced in a Material to the Magnetic Field Strength of the Inducing Field
8:02
Free Space
8:13
Highly Magnetic Materials Have Higher Values of Magnetic Permeability
8:34
Magnetic Dipole Moment
8:41
The Force That a Magnet Can Exert on Moving Charges
8:46
Relative Strength of a Magnet
8:54
Forces on Moving Charges
9:10
Moving Charges Create Magnetic Fields
9:11
Magnetic Fields Exert Forces on Moving Charges
9:17
Direction of the Magnetic Force
9:57
Direction is Given by the Right-Hand Rule
10:05
Right-Hand Rule
10:09
Mass Spectrometer
10:52
Magnetic Fields Accelerate Moving Charges So That They Travel in a Circle
10:58
Used to Determine the Mass of an Unknown Particle
11:04
Velocity Selector
12:44
Mass Spectrometer with an Electric Field Added
12:47
Example 5: Force on an Electron
14:13
Example 6: Velocity of a Charged Particle
15:25
Example 7: Direction of the Magnetic Force
16:52
Example 8: Direction of Magnetic Force on Moving Charges
17:43
Example 9: Electron Released From Rest in Magnetic Field
18:53
Current-Carrying Wires

21m 29s

Intro
0:00
Objectives
0:09
Force on a Current-Carrying Wire
0:30
A Current-Carrying Wire in a Magnetic Field May Experience a Magnetic Force
0:33
Direction Given by the Right-Hand Rule
1:11
Example 1: Force on a Current-Carrying Wire
1:38
Example 2: Equilibrium on a Submerged Wire
2:33
Example 3: Torque on a Loop of Wire
5:55
Magnetic Field Due to a Current-Carrying Wire
8:49
Moving Charges Create Magnetic Fields
8:53
Wires Carry Moving Charges
8:56
Direction Given by the Right-Hand Rule
9:21
Example 4: Magnetic Field Due to a Wire
10:56
Magnetic Field Due to a Solenoid
12:12
Solenoid is a Coil of Wire
12:19
Direction Given by the Right-Hand Rule
12:47
Forces on 2 Parallel Wires
13:34
Current Flowing in the Same Direction
14:52
Current Flowing in Opposite Directions
14:57
Example 5: Magnetic Field Due to Wires
15:19
Example 6: Strength of an Electromagnet
18:35
Example 7: Force on a Wire
19:30
Example 8: Force Between Parallel Wires
20:47
Intro to Electromagnetic Induction

17m 26s

Intro
0:00
Objectives
0:09
Induced EMF
0:42
Charges Flowing Through a Wire Create Magnetic Fields
0:45
Changing Magnetic Fields Cause Charges to Flow or 'Induce' a Current in a Process Known As Electromagnetic Induction
0:49
Electro-Motive Force is the Potential Difference Created by a Changing Magnetic Field
0:57
Magnetic Flux is the Amount of Magnetic Fields Passing Through an Area
1:17
Finding the Magnetic Flux
1:36
Magnetic Field Strength
1:39
Angle Between the Magnetic Field Strength and the Normal to the Area
1:51
Calculating Induced EMF
3:01
The Magnitude of the Induced EMF is Equal to the Rate of Change of the Magnetic Flux
3:04
Induced EMF in a Rectangular Loop of Wire
4:03
Lenz's Law
5:17
Electric Generators and Motors
9:28
Generate an Induced EMF By Turning a Coil of Wire in a magnetic Field
9:31
Generators Use Mechanical Energy to Turn the Coil of Wire
9:39
Electric Motor Operates Using Same Principle
10:30
Example 1: Finding Magnetic Flux
10:43
Example 2: Finding Induced EMF
11:54
Example 3: Changing Magnetic Field
13:52
Example 4: Current Induced in a Rectangular Loop of Wire
15:23
Section 6: Waves & Optics
Wave Characteristics

26m 41s

Intro
0:00
Objectives
0:09
Waves
0:32
Pulse
1:00
A Pulse is a Single Disturbance Which Carries Energy Through a Medium or Space
1:05
A Wave is a Series of Pulses
1:18
When a Pulse Reaches a Hard Boundary
1:37
When a Pulse Reaches a Soft or Flexible Boundary
2:04
Types of Waves
2:44
Mechanical Waves
2:56
Electromagnetic Waves
3:14
Types of Wave Motion
3:38
Longitudinal Waves
3:39
Transverse Waves
4:18
Anatomy of a Transverse Wave
5:18
Example 1: Waves Requiring a Medium
6:59
Example 2: Direction of Displacement
7:36
Example 3: Bell in a Vacuum Jar
8:47
Anatomy of a Longitudinal Wave
9:22
Example 4: Tuning Fork
9:57
Example 5: Amplitude of a Sound Wave
10:24
Frequency and Period
10:47
Example 6: Period of an EM Wave
11:23
Example 7: Frequency and Period
12:01
The Wave Equation
12:32
Velocity of a Wave is a Function of the Type of Wave and the Medium It Travels Through
12:36
Speed of a Wave is Related to Its Frequency and Wavelength
12:41
Example 8: Wavelength Using the Wave Equation
13:54
Example 9: Period of an EM Wave
14:35
Example 10: Blue Whale Waves
16:03
Sound Waves
17:29
Sound is a Mechanical Wave Observed by Detecting Vibrations in the Inner Ear
17:33
Particles of Sound Wave Vibrate Parallel With the Direction of the Wave's Velocity
17:56
Example 11: Distance from Speakers
18:24
Resonance
19:45
An Object with the Same 'Natural Frequency' May Begin to Vibrate at This Frequency
19:55
Classic Example
20:01
Example 12: Vibrating Car
20:32
Example 13: Sonar Signal
21:28
Example 14: Waves Across Media
24:06
Example 15: Wavelength of Middle C
25:24
Wave Interference

20m 45s

Intro
0:00
Objectives
0:09
Superposition
0:30
When More Than One Wave Travels Through the Same Location in the Same Medium
0:32
The Total Displacement is the Sum of All the Individual Displacements of the Waves
0:46
Example 1: Superposition of Pulses
1:01
Types of Interference
2:02
Constructive Interference
2:05
Destructive Interference
2:18
Example 2: Interference
2:47
Example 3: Shallow Water Waves
3:27
Standing Waves
4:23
When Waves of the Same Frequency and Amplitude Traveling in Opposite Directions Meet in the Same Medium
4:26
A Wave in Which Nodes Appear to be Standing Still and Antinodes Vibrate with Maximum Amplitude Above and Below the Axis
4:35
Standing Waves in String Instruments
5:36
Standing Waves in Open Tubes
8:49
Standing Waves in Closed Tubes
9:57
Interference From Multiple Sources
11:43
Constructive
11:55
Destructive
12:14
Beats
12:49
Two Sound Waves with Almost the Same Frequency Interfere to Create a Beat Pattern
12:52
A Frequency Difference of 1 to 4 Hz is Best for Human Detection of Beat Phenomena
13:05
Example 4
14:13
Example 5
18:03
Example 6
19:14
Example 7: Superposition
20:08
Wave Phenomena

19m 2s

Intro
0:00
Objective
0:08
Doppler Effect
0:36
The Shift In A Wave's Observed Frequency Due to Relative Motion Between the Source of the Wave and Observer
0:39
When Source and/or Observer Move Toward Each Other
0:45
When Source and/or Observer Move Away From Each Other
0:52
Practical Doppler Effect
1:01
Vehicle Traveling Past You
1:05
Applications Are Numerous and Widespread
1:56
Doppler Effect - Astronomy
2:43
Observed Frequencies Are Slightly Lower Than Scientists Would Predict
2:50
More Distant Celestial Objects Are Moving Away from the Earth Faster Than Nearer Objects
3:22
Example 1: Car Horn
3:36
Example 2: Moving Speaker
4:13
Diffraction
5:35
The Bending of Waves Around Obstacles
5:37
Most Apparent When Wavelength Is Same Order of Magnitude as the Obstacle/ Opening
6:10
Single-Slit Diffraction
6:16
Double-Slit Diffraction
8:13
Diffraction Grating
11:07
Sharper and Brighter Maxima
11:46
Useful for Determining Wavelengths Accurately
12:07
Example 3: Double Slit Pattern
12:30
Example 4: Determining Wavelength
16:05
Example 5: Radar Gun
18:04
Example 6: Red Shift
18:29
Light As a Wave

11m 35s

Intro
0:00
Objectives
0:14
Electromagnetic (EM) Waves
0:31
Light is an EM Wave
0:43
EM Waves Are Transverse Due to the Modulation of the Electric and Magnetic Fields Perpendicular to the Wave Velocity
1:00
Electromagnetic Wave Characteristics
1:37
The Product of an EM Wave's Frequency and Wavelength Must be Constant in a Vacuum
1:43
Polarization
3:36
Unpoloarized EM Waves Exhibit Modulation in All Directions
3:47
Polarized Light Consists of Light Vibrating in a Single Direction
4:07
Polarizers
4:29
Materials Which Act Like Filters to Only Allow Specific Polarizations of Light to Pass
4:33
Polarizers Typically Are Sheets of Material in Which Long Molecules Are Lined Up Like a Picket Fence
5:10
Polarizing Sunglasses
5:22
Reduce Reflections
5:26
Polarizing Sunglasses Have Vertical Polarizing Filters
5:48
Liquid Crystal Displays
6:08
LCDs Use Liquid Crystals in a Suspension That Align Themselves in a Specific Orientation When a Voltage is Applied
6:13
Cross-Orienting a Polarizer and a Matrix of Liquid Crystals so Light Can Be Modulated Pixel-by-Pixel
6:26
Example 1: Color of Light
7:30
Example 2: Analyzing an EM Wave
8:49
Example 3: Remote Control
9:45
Example 4: Comparing EM Waves
10:32
Reflection & Mirrors

24m 32s

Intro
0:00
Objectives
0:10
Waves at Boundaries
0:37
Reflected
0:43
Transmitted
0:45
Absorbed
0:48
Law of Reflection
0:58
The Angle of Incidence is Equal to the Angle of Reflection
1:00
They Are Both Measured From a Line Perpendicular, or Normal, to the Reflecting Surface
1:22
Types of Reflection
1:54
Diffuse Reflection
1:57
Specular Reflection
2:08
Example 1: Specular Reflection
2:24
Mirrors
3:20
Light Rays From the Object Reach the Plane Mirror and Are Reflected to the Observer
3:27
Virtual Image
3:33
Magnitude of Image Distance
4:05
Plane Mirror Ray Tracing
4:15
Object Distance
4:26
Image Distance
4:43
Magnification of Image
7:03
Example 2: Plane Mirror Images
7:28
Example 3: Image in a Plane Mirror
7:51
Spherical Mirrors
8:10
Inner Surface of a Spherical Mirror
8:19
Outer Surface of a Spherical Mirror
8:30
Focal Point of a Spherical Mirror
8:40
Converging
8:51
Diverging
9:00
Concave (Converging) Spherical Mirrors
9:09
Light Rays Coming Into a Mirror Parallel to the Principal Axis
9:14
Light Rays Passing Through the Center of Curvature
10:17
Light Rays From the Object Passing Directly Through the Focal Point
10:52
Mirror Equation (Lens Equation)
12:06
Object and Image Distances Are Positive on the Reflecting Side of the Mirror
12:13
Formula
12:19
Concave Mirror with Object Inside f
12:39
Example 4: Concave Spherical Mirror
14:21
Example 5: Image From a Concave Mirror
14:51
Convex (Diverging) Spherical Mirrors
16:29
Light Rays Coming Into a Mirror Parallel to the Principal Axis
16:37
Light Rays Striking the Center of the Mirror
16:50
Light Rays Never Converge on the Reflective Side of a Convex Mirror
16:54
Convex Mirror Ray Tracing
17:07
Example 6: Diverging Rays
19:12
Example 7: Focal Length
19:28
Example 8: Reflected Sonar Wave
19:53
Example 9: Plane Mirror Image Distance
20:20
Example 10: Image From a Concave Mirror
21:23
Example 11: Converging Mirror Image Distance
23:09
Refraction & Lenses

39m 42s

Intro
0:00
Objectives
0:09
Refraction
0:42
When a Wave Reaches a Boundary Between Media, Part of the Wave is Reflected and Part of the Wave Enters the New Medium
0:43
Wavelength Must Change If the Wave's Speed Changes
0:57
Refraction is When This Causes The Wave to Bend as It Enters the New Medium
1:12
Marching Band Analogy
1:22
Index of Refraction
2:37
Measure of How Much Light Slows Down in a Material
2:40
Ratio of the Speed of an EM Wave in a Vacuum to the Speed of an EM Wave in Another Material is Known as Index of Refraction
3:03
Indices of Refraction
3:21
Dispersion
4:01
White Light is Refracted Twice in Prism
4:23
Index of Refraction of the Prism Material Varies Slightly with Respect to Frequency
4:41
Example 1: Determining n
5:14
Example 2: Light in Diamond and Crown Glass
5:55
Snell's Law
6:24
The Amount of a Light Wave Bends As It Enters a New Medium is Given by the Law of Refraction
6:32
Light Bends Toward the Normal as it Enters a Material With a Higher n
7:08
Light Bends Toward the Normal as it Enters a Material With a Lower n
7:14
Example 3: Angle of Refraction
7:42
Example 4: Changes with Refraction
9:31
Total Internal Reflection
10:10
When the Angle of Refraction Reaches 90 Degrees
10:23
Critical Angle
10:34
Total Internal Reflection
10:51
Applications of TIR
12:13
Example 5: Critical Angle of Water
13:17
Thin Lenses
14:15
Convex Lenses
14:22
Concave Lenses
14:31
Convex Lenses
15:24
Rays Parallel to the Principal Axis are Refracted Through the Far Focal Point of the Lens
15:28
A Ray Drawn From the Object Through the Center of the Lens Passes Through the Center of the Lens Unbent
15:53
Example 6: Converging Lens Image
16:46
Example 7: Image Distance of Convex Lens
17:18
Concave Lenses
18:21
Rays From the Object Parallel to the Principal Axis Are Refracted Away from the Principal Axis on a Line from the Near Focal Point Through the Point Where the Ray Intercepts the Center of the Lens
18:25
Concave Lenses Produce Upright, Virtual, Reduced Images
20:30
Example 8: Light Ray Thought a Lens
20:36
Systems of Optical Elements
21:05
Find the Image of the First Optical Elements and Utilize It as the Object of the Second Optical Element
21:16
Example 9: Lens and Mirrors
21:35
Thin Film Interference
27:22
When Light is Incident Upon a Thin Film, Some Light is Reflected and Some is Transmitted Into the Film
27:25
If the Transmitted Light is Again Reflected, It Travels Back Out of the Film and Can Interfere
27:31
Phase Change for Every Reflection from Low-Index to High-Index
28:09
Example 10: Thin Film Interference
28:41
Example 11: Wavelength in Diamond
32:07
Example 12: Light Incident on Crown Glass
33:57
Example 13: Real Image from Convex Lens
34:44
Example 14: Diverging Lens
35:45
Example 15: Creating Enlarged, Real Images
36:22
Example 16: Image from a Converging Lens
36:48
Example 17: Converging Lens System
37:50
Wave-Particle Duality

23m 47s

Intro
0:00
Objectives
0:11
Duality of Light
0:37
Photons
0:47
Dual Nature
0:53
Wave Evidence
1:00
Particle Evidence
1:10
Blackbody Radiation & the UV Catastrophe
1:20
Very Hot Objects Emitted Radiation in a Specific Spectrum of Frequencies and Intensities
1:25
Color Objects Emitted More Intensity at Higher Wavelengths
1:45
Quantization of Emitted Radiation
1:56
Photoelectric Effect
2:38
EM Radiation Striking a Piece of Metal May Emit Electrons
2:41
Not All EM Radiation Created Photoelectrons
2:49
Photons of Light
3:23
Photon Has Zero Mass, Zero Charge
3:32
Energy of a Photon is Quantized
3:36
Energy of a Photon is Related to its Frequency
3:41
Creation of Photoelectrons
4:17
Electrons in Metals Were Held in 'Energy Walls'
4:20
Work Function
4:32
Cutoff Frequency
4:54
Kinetic Energy of Photoelectrons
5:14
Electron in a Metal Absorbs a Photon with Energy Greater Than the Metal's Work Function
5:16
Electron is Emitted as a Photoelectron
5:24
Any Absorbed Energy Beyond That Required to Free the Electron is the KE of the Photoelectron
5:28
Photoelectric Effect in a Circuit
6:37
Compton Effect
8:28
Less of Energy and Momentum
8:49
Lost by X-Ray Equals Energy and Gained by Photoelectron
8:52
Compton Wavelength
9:09
Major Conclusions
9:36
De Broglie Wavelength
10:44
Smaller the Particle, the More Apparent the Wave Properties
11:03
Wavelength of a Moving Particle is Known as Its de Broglie Wavelength
11:07
Davisson-Germer Experiment
11:29
Verifies Wave Nature of Moving Particles
11:30
Shoot Electrons at Double Slit
11:34
Example 1
11:46
Example 2
13:07
Example 3
13:48
Example 4A
15:33
Example 4B
18:47
Example 5: Wave Nature of Light
19:54
Example 6: Moving Electrons
20:43
Example 7: Wavelength of an Electron
21:11
Example 8: Wrecking Ball
22:50
Section 7: Modern Physics
Atomic Energy Levels

14m 21s

Intro
0:00
Objectives
0:09
Rutherford's Gold Foil Experiment
0:35
Most of the Particles Go Through Undeflected
1:12
Some Alpha Particles Are Deflected Large Amounts
1:15
Atoms Have a Small, Massive, Positive Nucleus
1:20
Electrons Orbit the Nucleus
1:23
Most of the Atom is Empty Space
1:26
Problems with Rutherford's Model
1:31
Charges Moving in a Circle Accelerate, Therefore Classical Physics Predicts They Should Release Photons
1:39
Lose Energy When They Release Photons
1:46
Orbits Should Decay and They Should Be Unstable
1:50
Bohr Model of the Atom
2:09
Electrons Don't Lose Energy as They Accelerate
2:20
Each Atom Allows Only a Limited Number of Specific Orbits at Each Energy Level
2:35
Electrons Must Absorb or Emit a Photon of Energy to Change Energy Levels
2:40
Energy Level Diagrams
3:29
n=1 is the Lowest Energy State
3:34
Negative Energy Levels Indicate Electron is Bound to Nucleus of the Atom
4:03
When Electron Reaches 0 eV It Is No Longer Bound
4:20
Electron Cloud Model (Probability Model)
4:46
Electron Only Has A Probability of Being Located in Certain Regions Surrounding the Nucleus
4:53
Electron Orbitals Are Probability Regions
4:58
Atomic Spectra
5:16
Atoms Can Only Emit Certain Frequencies of Photons
5:19
Electrons Can Only Absorb Photons With Energy Equal to the Difference in Energy Levels
5:34
This Leads to Unique Atomic Spectra of Emitted and Absorbed Radiation for Each Element
5:37
Incandescence Emits a Continuous Energy
5:43
If All Colors of Light Are Incident Upon a Cold Gas, The Gas Only Absorbs Frequencies Corresponding to Photon Energies Equal to the Difference Between the Gas's Atomic Energy Levels
6:16
Continuous Spectrum
6:42
Absorption Spectrum
6:50
Emission Spectrum
7:08
X-Rays
7:36
The Photoelectric Effect in Reverse
7:38
Electrons Are Accelerated Through a Large Potential Difference and Collide with a Molybdenum or Platinum Plate
7:53
Example 1: Electron in Hydrogen Atom
8:24
Example 2: EM Emission in Hydrogen
10:05
Example 3: Photon Frequencies
11:30
Example 4: Bright-Line Spectrum
12:24
Example 5: Gas Analysis
13:08
Nuclear Physics

15m 47s

Intro
0:00
Objectives
0:08
The Nucleus
0:33
Protons Have a Charge or +1 e
0:39
Neutrons Are Neutral (0 Charge)
0:42
Held Together by the Strong Nuclear Force
0:43
Example 1: Deconstructing an Atom
1:20
Mass-Energy Equivalence
2:06
Mass is a Measure of How Much Energy an Object Contains
2:16
Universal Conservation of Laws
2:31
Nuclear Binding Energy
2:53
A Strong Nuclear Force Holds Nucleons Together
3:04
Mass of the Individual Constituents is Greater Than the Mass of the Combined Nucleus
3:19
Binding Energy of the Nucleus
3:32
Mass Defect
3:37
Nuclear Decay
4:30
Alpha Decay
4:42
Beta Decay
5:09
Gamma Decay
5:46
Fission
6:40
The Splitting of a Nucleus Into Two or More Nuclei
6:42
For Larger Nuclei, the Mass of Original Nucleus is Greater Than the Sum of the Mass of the Products When Split
6:47
Fusion
8:14
The Process of Combining Two Or More Smaller Nuclei Into a Larger Nucleus
8:15
This Fuels Our Sun and Stars
8:28
Basis of Hydrogen Bomb
8:31
Forces in the Universe
9:00
Strong Nuclear Force
9:06
Electromagnetic Force
9:13
Weak Nuclear Force
9:22
Gravitational Force
9:27
Example 2: Deuterium Nucleus
9:39
Example 3: Particle Accelerator
10:24
Example 4: Tritium Formation
12:03
Example 5: Beta Decay
13:02
Example 6: Gamma Decay
14:15
Example 7: Annihilation
14:39
Section 8: Sample AP Exams
AP Practice Exam: Multiple Choice, Part 1

38m 1s

Intro
0:00
Problem 1
1:33
Problem 2
1:57
Problem 3
2:50
Problem 4
3:46
Problem 5
4:13
Problem 6
4:41
Problem 7
6:12
Problem 8
6:49
Problem 9
7:49
Problem 10
9:31
Problem 11
10:08
Problem 12
11:03
Problem 13
11:30
Problem 14
12:28
Problem 15
14:04
Problem 16
15:05
Problem 17
15:55
Problem 18
17:06
Problem 19
18:43
Problem 20
19:58
Problem 21
22:03
Problem 22
22:49
Problem 23
23:28
Problem 24
24:04
Problem 25
25:07
Problem 26
26:46
Problem 27
28:03
Problem 28
28:49
Problem 29
30:20
Problem 30
31:10
Problem 31
33:03
Problem 32
33:46
Problem 33
34:47
Problem 34
36:07
Problem 35
36:44
AP Practice Exam: Multiple Choice, Part 2

37m 49s

Intro
0:00
Problem 36
0:18
Problem 37
0:42
Problem 38
2:13
Problem 39
4:10
Problem 40
4:47
Problem 41
5:52
Problem 42
7:22
Problem 43
8:16
Problem 44
9:11
Problem 45
9:42
Problem 46
10:56
Problem 47
12:03
Problem 48
13:58
Problem 49
14:49
Problem 50
15:36
Problem 51
15:51
Problem 52
17:18
Problem 53
17:59
Problem 54
19:10
Problem 55
21:27
Problem 56
22:40
Problem 57
23:19
Problem 58
23:50
Problem 59
25:35
Problem 60
26:45
Problem 61
27:57
Problem 62
28:32
Problem 63
29:52
Problem 64
30:27
Problem 65
31:27
Problem 66
32:22
Problem 67
33:18
Problem 68
35:21
Problem 69
36:27
Problem 70
36:46
AP Practice Exam: Free Response, Part 1

16m 53s

Intro
0:00
Question 1
0:23
Question 2
8:55
AP Practice Exam: Free Response, Part 2

9m 20s

Intro
0:00
Question 3
0:14
Question 4
4:34
AP Practice Exam: Free Response, Part 3

18m 12s

Intro
0:00
Question 5
0:15
Question 6
3:29
Question 7
6:18
Question 8
12:53
Section 9: Additional Examples
Metric Estimation

3m 53s

Intro
0:00
Question 1
0:38
Question 2
0:51
Question 3
1:09
Question 4
1:24
Question 5
1:49
Question 6
2:11
Question 7
2:27
Question 8
2:49
Question 9
3:03
Question 10
3:23
Defining Motion

7m 6s

Intro
0:00
Question 1
0:13
Question 2
0:50
Question 3
1:56
Question 4
2:24
Question 5
3:32
Question 6
4:01
Question 7
5:36
Question 8
6:36
Motion Graphs

6m 48s

Intro
0:00
Question 1
0:13
Question 2
2:01
Question 3
3:06
Question 4
3:41
Question 5
4:30
Question 6
5:52
Horizontal Kinematics

8m 16s

Intro
0:00
Question 1
0:19
Question 2
2:19
Question 3
3:16
Question 4
4:36
Question 5
6:43
Free Fall

7m 56s

Intro
0:00
Question 1-4
0:12
Question 5
2:36
Question 6
3:11
Question 7
4:44
Question 8
6:16
Projectile Motion

4m 17s

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0:13
Question 2
0:45
Question 3
1:25
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2:00
Question 5
2:32
Question 6
3:38
Newton's 1st Law

4m 34s

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0:15
Question 2
1:02
Question 3
1:50
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2:26
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2:54
Question 7
3:11
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3:29
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3:47
Question 10
4:02
Newton's 2nd Law

5m 40s

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0:55
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Newton's 3rd Law

3m 44s

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0:17
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2:11
Question 6
2:29
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Friction

6m 37s

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0:47
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Question 7
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5:50
Ramps and Inclines

6m 13s

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0:18
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1:01
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3:11
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5:08
Circular Motion

5m 17s

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0:21
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1:01
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2:33
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3:10
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Question 7
3:56
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4:33
Gravity

6m 33s

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1:05
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4:41
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Momentum & Impulse

9m 29s

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Question 9
7:47
Conservation of Momentum

9m 33s

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Work & Power

6m 2s

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0:29
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1:36
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4:49
Springs

7m 59s

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2:26
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3:37
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4:39
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5:28
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Energy & Energy Conservation

8m 47s

Intro
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1:27
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5:19
Question 9
5:37
Question 10
7:12
Question 11
7:40
Electric Charge

7m 6s

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0:10
Question 2
1:03
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1:32
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2:12
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3:01
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3:49
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4:24
Question 8
4:50
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5:32
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5:55
Question 11
6:26
Coulomb's Law

4m 13s

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0:14
Question 2
0:47
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1:25
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2:25
Question 5
3:01
Electric Fields & Forces

4m 11s

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0:51
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1:30
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3:12
Electric Potential

5m 12s

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0:42
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2:22
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2:49
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4:02
Electrical Current

6m 54s

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0:42
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4:37
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4:59
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5:50
Resistance

5m 15s

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0:53
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2:31
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3:21
Question 6
4:06
Ohm's Law

4m 27s

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0:33
Question 3
0:59
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1:32
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2:50
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3:19
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3:50
Circuit Analysis

6m 36s

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2:16
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3:18
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5:51
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6:00
Magnetism

3m 43s

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0:31
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1:19
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1:35
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2:36
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3:03
Wave Basics

4m 21s

Intro
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0:13
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0:36
Question 3
0:47
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1:13
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1:27
Question 6
1:39
Question 7
1:54
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2:22
Question 9
2:51
Question 10
3:32
Wave Characteristics

5m 33s

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0:23
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1:04
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2:01
Question 4
2:50
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3:12
Question 6
3:57
Question 7
4:16
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Wave Behaviors

3m 52s

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0:40
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1:04
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1:17
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1:39
Question 6
2:07
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2:41
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3:09
Reflection

3m 48s

Intro
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0:50
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1:29
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1:46
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3:08
Refraction

2m 49s

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0:29
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1:03
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1:24
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2:01
Diffraction

2m 34s

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1:37
Question 6
2:04
Electromagnetic Spectrum

7m 6s

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Question 1
0:24
Question 2
0:39
Question 3
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Question 4
1:51
Question 5
2:03
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Question 7
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4:30
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5:04
Question 11
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Question 12
6:16
Wave-Particle Duality

5m 30s

Intro
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Question 2
0:34
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0:53
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1:54
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2:16
Question 6
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3:45
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4:13
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4:33
Energy Levels

8m 13s

Intro
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Question 1
0:25
Question 2
1:18
Question 3
1:43
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2:08
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3:17
Question 6
3:54
Question 7
4:40
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5:54
Question 10
6:41
Question 11
7:14
Mass-Energy Equivalence

8m 15s

Intro
0:00
Question 1
0:19
Question 2
1:02
Question 3
1:37
Question 4
2:17
Question 5
2:55
Question 6
3:32
Question 7
4:13
Question 8
5:04
Question 9
5:29
Question 10
5:58
Question 11
6:48
Question 12
7:39
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Lecture Comments (31)

1 answer

Last reply by: Professor Dan Fullerton
Tue Apr 26, 2016 8:05 AM

Post by Adam Peng on April 26, 2016

How to determine a sphere is a shell or a solid? For example, is basketball a shell?

1 answer

Last reply by: Professor Dan Fullerton
Wed Apr 13, 2016 6:18 PM

Post by Zhe Tian on April 13, 2016

For example 1, why are we considering the two bowling balls to be point masses? I thought the moment of inertia would be 2/5mR^2 because they are spheres...

1 answer

Last reply by: Professor Dan Fullerton
Thu Mar 17, 2016 10:05 AM

Post by john lee on March 17, 2016

Excuse me professor Dan Fullerton,
what does the vector of the angular momentum mean?

1 answer

Last reply by: Professor Dan Fullerton
Tue Feb 23, 2016 3:10 PM

Post by James Glass on February 23, 2016

In example 1, the one with the 2 bowling balls, could you add the radius of the bowling balls to the radius of the meter stick if you knew the radius of each bowling ball? JGlass

1 answer

Last reply by: Professor Dan Fullerton
Sat Oct 18, 2014 7:05 AM

Post by Foaad Zaid on October 18, 2014

In example 9 around the 32:50 mark you say that rotational kinetic energy doubles, while angular velocity is cut in half. Do you mean moment of inertia. I thought that the I was cut in half?

1 answer

Last reply by: Professor Dan Fullerton
Sat Oct 18, 2014 7:02 AM

Post by Foaad Zaid on October 17, 2014

Also for the first example part (b) why is the distance between the two masses r1?

1 answer

Last reply by: Professor Dan Fullerton
Sat Oct 18, 2014 7:01 AM

Post by Foaad Zaid on October 17, 2014

Hello, for the first example, I'm not understanding how the 2 objects are rotating? Do you mind just clarifying this for me please? Thank you. :)

2 answers

Last reply by: Jungle Jones
Sat Aug 23, 2014 10:23 AM

Post by Jungle Jones on August 22, 2014

In example 9, the ice skater, for the second part, about the rotational kinetic energy.
If the moment of inertia is cut in half, that means that I becomes .5I, and w becomes 2w.
So when you evaluate for Krot, shouldn't it be (.5)(.5I)(2w)^2 = only Iw?
I don't see why you didn't put the 1/2 for I.

1 answer

Last reply by: Professor Dan Fullerton
Fri Aug 22, 2014 7:23 PM

Post by Jungle Jones on August 22, 2014

In example 2, when finding the angular acceleration, how did you get the units as rad/s2?
The torque had units of N.m and the moment of inertia had units of kg.m2, so wouldn't the acceleration units be N/(kg.m)?

1 answer

Last reply by: Professor Dan Fullerton
Fri Aug 22, 2014 7:22 PM

Post by Jungle Jones on August 22, 2014

Why is the lowercase r used for the general moment of inertia equation, but the uppercase R used for the equations for the other common objects?

1 answer

Last reply by: Professor Dan Fullerton
Mon Jul 7, 2014 6:01 PM

Post by robert moreno on July 7, 2014

on example 6, why is final inertia 2?

1 answer

Last reply by: Professor Dan Fullerton
Fri Dec 6, 2013 6:18 AM

Post by Shrinivas Sadachar on December 5, 2013

Hello,
 I had a question regarding the Round-About example.
Why can't I use the kinematic equation (below) after finding alpha(angular acceleration)?
omega(final) squared = omega initial squared + 2 times alpha times change in theta?

Thanks

1 answer

Last reply by: Professor Dan Fullerton
Tue Jun 18, 2013 11:52 AM

Post by Ikze Cho on June 18, 2013

is inertia not measured in Kg?
i thought that Mass is a measure of inertia

1 answer

Last reply by: Professor Dan Fullerton
Tue Apr 2, 2013 6:52 AM

Post by Jawad Hassan on April 1, 2013

Hi,

Just wanted to thank you for this vids, verry good.
right on point with lots of examples.

1 answer

Last reply by: Professor Dan Fullerton
Sat Mar 30, 2013 8:23 AM

Post by Kamiko Darrow on March 30, 2013

Why is net torque zero?

Rotational Dynamics

  • Moment of inertia, or rotational inertia, is a measure of how hard it is to change the angular velocity of an object.
  • Newton's 2nd Law states that the acceleration of an object is equal to the net force applied divided by the object's inertial mass. Newton's 2nd Law for Rotation states that the angular acceleration of an object is equal to the net torque applied divided by the object's moment of inertia.
  • Linear momentum is the product of an object's inertial mass and linear velocity, and is conserved in a closed system. Angular momentum (L) is the product of an object's moment of inertia and its angular velocity about the center of mass, and is also conserved in a closed system.
  • For an object in an elliptical orbit, the product of its radius and its linear velocity at any point in the orbit will remain constant assuming no external forces or torques are applied.
  • Rotational kinetic energy is equal to half product of the object's moment of inertia and the square of its angular velocity.

Rotational Dynamics

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
  • Types of Inertia 0:39
    • Inertial Mass (Translational Inertia)
    • Moment of Inertia (Rotational Inertia)
  • Moment of Inertia for Common Objects 1:48
  • Example 1: Calculating Moment of Inertia 2:53
  • Newton's 2nd Law - Revisited 5:09
    • Acceleration of an Object
    • Angular Acceleration of an Object
  • Example 2: Rotating Top 5:47
  • Example 3: Spinning Disc 7:54
  • Angular Momentum 9:41
    • Linear Momentum
    • Angular Momentum
  • Calculating Angular Momentum 10:51
    • Direction of the Angular Momentum Vector
    • Total Angular Momentum
  • Example 4: Angular Momentum of Particles 14:15
  • Example 5: Rotating Pedestal 16:51
  • Example 6: Rotating Discs 18:39
  • Angular Momentum and Heavenly Bodies 20:13
  • Types of Kinetic Energy 23:41
    • Objects Traveling with a Translational Velocity
    • Objects Traveling with Angular Velocity
  • Translational vs. Rotational Variables 24:33
  • Example 7: Kinetic Energy of a Basketball 25:45
  • Example 8: Playground Round-A-Bout 28:17
  • Example 9: The Ice Skater 30:54
  • Example 10: The Bowler 33:15

Transcription: Rotational Dynamics

Hi, folks. I am Dan Fullerton and I would like to welcome you back to Educator.com.0000

Our next topic -- Rotational Dynamics.0004

Our objectives are going to be to understand the moment of inertia or rotational inertia of an object or system -- depends upon the distribution of mass within the object or system, to determine the angular acceleration of an object when an external torque or force is applied.0006

We will calculate the angular momentum for a point particle, utilize the Law of Conservation of angular momentum and analyzing the behavior of rotating rigid bodies, and finally calculate the kinetic energy of a rotating body.0025

With that, let us talk about types of inertia.0039

So far, we have talked about inertial mass or translational inertia, which is an object's ability to resist the linear acceleration.0041

Well, in the rotational world, we have an analog of that as well. It's called moment of inertia or rotational inertia.0050

That is an object resistance to a rotational acceleration or an angular acceleration.0057

Now, objects of that most of their mass near their center of rotation tend to have smaller rotational inertias than objects with more mass farther from their axis of rotation.0063

Think of a figure skater spinning on the ice. While their arms are out, they tend to go slower.0071

To go faster they pull their arms in; they are shrinking their moment of inertia as they do that.0077

Smaller moment of inertia means easier to accelerate.0082

The formula for moment of inertia is the sum of mass times the square of the radius.0086

Now, if you have an object that is more complex than a simple particle, you have to add up all of the little bitty pieces of mass times the square of their distance from that axis of rotation.0093

Add them all up and you get the moment of inertia.0104

Let us talk about the moment of inertia for a couple of common objects.0108

For any object, if you take the sum of the all masses times the square of the distance from the axis of rotation that formula will work for any object.0111

But that is not always easy to apply, so for some common objects -- things like a disc, the moment of inertia is 1/2 times the mass of the disc times the square of its radius, assuming it is a uniform mass density distribution.0120

A hoop on the other hand is mr2. A solid sphere is 2/5 mr2.0134

A hollow sphere on the other hand, where all of the mass is on the outside almost like a spherical shell, is 2/3 mr2.0143

A rod rotated about its center is going to be about 1/12 mL2 where L is the length of the rod, but if you rotate it about its end, then it becomes 1/3 mL2.0149

The moment of inertia goes up because more of the mass is situated away from that axis further away from that axis of rotation.0165

Let us take a look of how could we calculate moment of inertia.0174

We have two 5 kg bowling balls joined by a meter long rod and we are going to say that rod is of negligible mass.0177

If we rotate it about the center of the rod we can find its moment of inertia this way.0184

The moment of inertia is going to be the sum for all the different particles of mr2, which in this case -- let us call this m1 and we will call this m2.0189

We will call this distance r1 and this distance r2.0200

That is going to be m1(r1)2 + m2(r2)2.0208

In this case, m1 here is going to be 5 and if this whole distance -- we call 1 m then r1 must be 1/2, so that is .52 + m2(5) × r(.5)2.0217

This gives me a moment of inertia equal to 2.5 kg × m2.0237

Now, let us take the same object and rotate it now by the end under one of the bowling balls -- so putting more of the mass further away.0247

That to me, just theoretically, I would think you know that is going to be harder to spin.0256

I am thinking we are going to have a larger moment of inertia. Let us find out.0261

Once again, moment of inertia, capital I, is the sum of mr2, which will be m1r12 + m2r22.0266

Once again, m1, m2, but now r1 is this entire distance, 1 m and r2 is going to be 0.0277

So I end up with mass1(5) × r1(1)2 + mass2(5) × distance from the axis of rotation, 020287

That is just going to be 5 kg-m2.0297

So the moment of inertia here doubled compared to when we spun it about its center of mass.0302

We can take a look at this in terms of Newton's Second Law as well.0310

Newton's Second Law said that the net force on an object was equal to its mass -- its linear inertia times the acceleration.0314

The angular acceleration of an object, on the other hand, was the net torque applied divided by the object's moment of inertia.0324

Again we have the same parallels -- force, torque. Linear inertia -- rotational inertia. Linear acceleration -- rotational acceleration0331

It all works the same way. Let us take another example here. Let us talk about a rotating top.0345

A top with moment of inertia .001 kg-m2 is spun on a table by applying a torque of .01N-m for 2 seconds.0352

If the top starts from rest find the final angular velocity of the top.0362

Well, let us figure out what information we know to begin with.0367

The initial angular velocity is 0. It starts at rest.0371

We are trying to find the final angular velocity.0374

We do not know the angular displacement; we do not know α, and we don't know time.0379

Pardon me. We do know time, it is 2 s.0388

Well, it is sure be helpful to know that angular acceleration.0390

Let us take a look and say that the net torque is equal to I(α) -- Newton's Second Law for Rotation.0394

That means then that alpha is going to be the net torque divided by the moment of inertia and our net torque was .01N-m and our moment of inertia .001 kg-m2.0403

That tells me then that my angular acceleration must be 10 rad/s2.0419

I can plug that in over here for my alpha as 10 rad/s2.0427

Now, I can use my kinematics to find what final angular velocity is.0435

Final angular velocity is initial angular velocity plus alpha times time0443

That is going to be -- well this is 0, so 10 rad/s2 × 2s is going to give us a final angular velocity of 20 rad/s.0449

That is Newton's Second Law in kinematics, all put together -- This time though for rotation.0468

Let us take another example.0475

What is the angular acceleration experience by a uniform solid disc of mass 2 kg and radius .1 m when the net torque of 10N-m is applied?0476

Assume the disc spins about its center, which we can see from the diagram there as well.0486

Well, net torque is moment of inertia or rotational inertia times angular acceleration.0492

Now, because this is a disk we can look up its moment of inertia which is going to be 1/2mr2, where there is our (r) and it has some total mass (m).0499

The net torque equals -- well I, we have 1/2mr2 × α.0516

Therefore, alpha must be equal to 2 times our net torque divided by mr2.0529

Now, we can substitute in our values to find that alpha is equal to 2 times our net torque (10N-m) divided by the mass (2 kg) times the square of the radius .1 m2.0540

20/2 × .12 = .01, which should give us 1000 rad/s2.0556

The same basic sort of problem -- now, we are just solving for angular acceleration and we had to go look up the formula for the moment of inertia, which you saw a couple of slides ago for some common objects.0569

Linear momentum -- the product of an object's inertial mass and its velocity -- is conserved in a closed system.0583

That is the conservation of linear momentum. We have talked about that already.0588

Linear momentum describes how difficult it is to stop a moving object.0594

There is an analogy in the rotational world, too.0597

Angular momentum -- a vector (capital L), which is the product of an object's moment of inertia or rotational inertia and its angular velocity about the center of mass -- is also conserved in a closed system when there are not any external torques.0600

That describes how difficult it is to stop a rotating object.0606

We have angular momentum equals moment of inertia times angular velocity.0620

That fits right along with our analogy, linear momentum equals linear inertia (mass) times linear velocity.0627

Here are the analogs -- angular momentum, linear momentum; rotational inertia, linear inertia; angular velocity, linear velocity -- same sort of parallels again.0636

How do we calculate angular momentum?0652

Well, what we are going to do is, we are going to talk about a mass moving along with some velocity (v) at some position(r) about point (Q).0655

Angular momentum depends on their point of reference.0663

We are going to start by setting a reference point (Q).0669

In that case, the object has some angular momentum (L) about (Q) and we could find that by multiplying the vectors (r) and (p) with the vector cross product -- the vector product, which will give us another vector, which is a lot like we did the talking about torque.0671

The angular momentum vector (r) cross (p) -- we determine its direction by the right-hand rule.0687

Point the fingers of your right hand in the direction of (r) where (r) is the vector from your reference point to the object.0693

Now, bend your fingers in the direction of the velocity. Your thumb then will point in the direction of the positive angular momentum.0701

It is another right-hand rule, so that would be into or out of the plane of the page.0709

In this case, if I point the fingers of my right hand in the direction of (r), bend them in the direction of (v), my thumb is going to point into the plane of the page or the screen here.0710

The direction of the angular momentum vector would be into the plane of the page.0723

Its magnitude is given by (mvr) sin(θ) -- mass times velocity times its distance (r) times the sine of the angle between this continued line and velocity -- very, very similar to torque.0730

We have two ways to find out our angular momentum.0745

Now, total angular momentum -- if we have a bunch of particles -- is just the sum of all the individual angular momenta.0749

Let us take a look quickly at a special case here -- what about for an object traveling in a circle?0755

Now, if we have some mass traveling in a circle with some velocity at a given point (V) and it is located some radius (r) from the center of the circle -- and let us call that point (C), our reference point...0762

...Then the angular momentum about point (C) is going to be (mvr) sin(θ).0775

But notice because (v) is always going to be tangent to the circle and (r) is always 90 degrees from that -- sin(θ) is always going to be 90 degrees -- sin 90 degrees is 1.0787

So that is just going to be (mvr), but, also remember when we do our translation between linear and angular variables that (v) is equal to = omega(r).0796

I can replace (v) with omega(r) so that is (m) omega(r) times another (r) or (r)2.0811

If I rewrite that, I could rewrite that as omega (mr)2, but if you recall for a point particle mr2 is the moment of inertia.0819

(L) about point (C) is equal to omega times (I), or as we wrote it earlier that is I(ω).0832

That is where that comes from.0844

Angular momentum is equal to rotational inertia or moment of inertia times angular velocity.0847

Let us take a look at how we could calculate angular momentum for a couple of particles.0855

We are trying to find the angular momentum for a 5 kg point particle located at 2-2 with a velocity of 2 m/s to the East.0860

We want to find it about three different points though, so first, let us find it about this point (O).0868

The angular momentum about point (O) -- and let us just stick with this magnitude now to make life nice and simple.0874

The magnitude of the angular momentum about point (O) is going to be equal to (mvr) sin(θ), where our mass is 5 and our velocity is 2 m/s.0881

Our distance from our point -- well if this is 2 and this is 2, the Pythagorean Theorem says right here that our hypotenuse must be 2 square roots of 2.0895

The sin of θ -- well that is going to be an angle here of 45 degrees and that is equal to square root of 2 over 2.0910

When I do all of this -- 5 × 2 = 10 × 2 = 20 and square root of 2 × and the square root of 2/2 = 1, so I end up with 20 kg-m2/s.0918

Now, let us find it about point (P).0934

Angular momentum about point (P) -- same formula (mvr) sin(θ).0937

Our mass is still same 5 and our velocity is still 2.0944

Now, about point (P) though -- our (r) distance is just 2 units (2) and the sine here is going to be sin 90 degrees which is 1, so 5 × 2 = 10 × 2 = 20 × 1 = 20 -- 20 kg-m2/s.0949

So for the moment of inertia about (O) and about (P), you get the same thing.0971

Now, let us do it about point (Q) -- Moment of inertia about point (Q) is going to be (mvr) sin(θ), but in this case, about point (Q), notice our (r) vector and (v) vector are in the same direction -- the angle between them then is 0.0976

Since the sine of 0 degrees equals 0, the angular momentum about point (Q) is going to be 0.0995

Angular momentum depends on your point of reference.1005

Let us take a look at an example with a rotating pedestal.1012

Angelina spins on a rotating pedestal with an angular velocity of 8 rad/s.1016

Bob throws her an exercise ball which increases her moment of inertia from 2kg-m2 to 2 1/2 kg-m2.1021

What is Angelina's angular velocity after she catches exercise ball?1029

We are going to neglect any external torque from the ball just to keep the problem simple.1033

Well, what I do here is realize by conservation of angular momentum -- since she is spinning about her center, her axis of rotation -- we can say that the total angular momentum before she catches the ball must be equal to the total angular momentum after she catches the ball.1038

So, (L)initial equals (L)final, but angular momentum is moment of inertia times angular velocity initial, so that must equal moment of inertia final times angular velocity final.1055

Well, initial moment of inertia, we know is 2 and final is going to be 2 1/2, so omega must change.1075

In this case, (I)initial is 2, (ω)initial is 8, so that must equal (I)final (2.5) times whatever her final angular velocity is.1082

16 divided by 2 1/2 -- I am going to come up with an angular velocity of 6.4 rad/s.1097

By increasing her rotational inertia, her angular velocity decreases.1100

Let us take some example of some rotating discs.1120

We have a disc with moment of inertia 1 kg-m2 spinning about an axle through its center.1122

It has an angular velocity of 10 rad/s.1130

An identical disc which is not rotating is slid along the axle until it makes contact with the first disc.1132

If the 2 discs then stick together, what is their combined angular velocity?1139

Well, I will go back to conservation of angular momentum, which will work because they are rotating about their centers of mass.1144

Initial angular momentum equals final angular momentum or initial moment of inertia and initial angular velocity must equal final moment of inertia, final angular velocity.1150

I want to know what the final angular velocity is.1166

That is going to be equal to I(0), omega(0) over I-final.1168

I-initial was 1, omega-initial was 10, and I-final -- well if we double that, it is going to go from 1 kg-m2 to 2 kgm2, so 10/2 = 5 rad/s.1177

This should make some amount of intuitive sense -- one objects spinning at 10 rad/s, the other is still, but identical object, and you put them together -- What happens?1195

Again you get the twice the mass, twice the rotational inertia, and half the angular velocity.1204

Let us talk about angular momentum with respect to heavenly bodies.1214

Really what we are talking about here is orbits.1216

We want to develop a relationship for the velocity and radius of a planet in an elliptical orbit about any point in that orbit.1219

Now, right away when we look at this, we know angular momentum must be conserved because there is no external torque in the system.1227

We will go put something like a planet over here, and call up the mass.1240

It has some velocity right at that point.1245

At this point, it has some (r) vector r(1), we will call that v(1) and there is our mass.1251

Another point in time -- say it is down over here -- now it has velocity (2) and it has a different position vector r(2).1259

Since the net torque is 0 though, the total angular momentum must be the same.1273

The angular momentum about point (S) is going to be -- well when it is at point (1) that it is going to be (m1v1r1) sin(θ)(1) where that angle there is θ(1).1278

But that also must be equal to the angular momentum over here at point (2) -- (m2v2r2) sin(θ)(2).1295

But the mass is the same. That has not changed.1307

We can divide out the mass and then state that (v1r1) sin(θ)(1) must equal (v2r2) sin(θ)(2).1311

Our relationship between the velocity is the distance from the Sun and the angle at any point in that orbit.1328

Now for the special case, when the planet is at this point, which is known as the apogee point -- so let us call that point (A) or when it is over here at perigee, you can call that point (P).1335

Well, at those points we have a special situation, because if you look here -- the velocity and the (r) vectors -- we are going to have an angle of 90 degrees and the same thing over here.1352

We have (r) versus velocity and our angle here again is (θ) 90 degrees, so at apogee and perigee, we can simplify this even further.1365

The velocity at apogee times the radius of the apogee times the sine of theta at apogee must equal the velocity at perigee times the radius with a position vector at perigee times the sine of θ(P).1380

But since these are both 90 degrees and the sine of 90 degrees is 1, we can simplify this to say that the velocity at (A) times the length of the position vector at (A) must equal the velocity at perigee times the position vector at perigee.1394

That works when we are at these special points where we have got that 90 degree angle.1409

It is a nice relationship between velocity and the position vector and the angle.1414

All right, let us talk now about types of kinetic energy.1422

We briefly talked about the kinetic energy of an object as the energy an object has due to its state of motion.1425

The translational kinetic energy we talked about was 1/2 mv2 -- mass times the square of speed.1432

Objects traveling with a translational energy must have a translational kinetic energy.1439

Similarly again another parallel to rotational motion, objects that are spinning must have a rotational kinetic energy.1443

Again as we look here, rotational kinetic energy is 1/2 instead of mass or linear inertia -- we have rotational inertia or moment of inertia.1452

Instead of linear velocity squared, we have angular velocity squared.1462

Same parallels again just swapping the linear variables for the rotational variables.1467

If we wanted to put this all together into a nice table -- displacement in the translational world, we called δ(s) or δ(x) depending on what we were talking about.1475

In the angular world, δ(θ) - angular displacement; velocity (v) - angular velocity ω; linear acceleration A - angular acceleration α; and time, the same in both worlds...1483

...Force (F) linear, the angular equivalent torque and mass or moment of inertia -- (m) in the translational world is (I) -- rotational inertia in the angular world.1498

In our equations, we can expand too -- (S) = r(θ), θ equals (s) over (r).1509

We have done this translations between linear and angular quantities before.1515

Time is the same, but now Newton's Second Law -- F = ma and torque = I(α).1521

For momentum -- linear momentum (P) equals mass times velocity and angular momentum equals moment of inertia times angular velocity.1527

And kinetic energy -- kinetic energy is 1/2 mv2 and rotational kinetic energy is 1/2I(ω)2.1535

Let us put this together to talk about the kinetic energy of a basketball.1545

A .62 kg basketball flies through the air with a velocity of 8 m/s.1549

Find its translational kinetic energy.1554

Well, kinetic energy to the translation linear kinetic energy is 1/2 mv2, which is going to be 1/2 times our mass (.62 kg) times our velocity (8 m/s2) or 19.84 and the units of energy are joules (J).1557

The same basketball -- knowing that its radius is .38 m -- also spins about its axis as it is traveling with an angular velocity of 5 rad/s.1582

Let us determine its moment of inertia and its rotational kinetic energy.1591

Well, we can model it as a hollow sphere and going back to our table of formulas for moments of inertia, the moment of inertia of a hollow sphere is 2/3 mr2.1596

That is going to be 2/3 times its mass (.62) times its radius (.382) or .0597 kg-m2.1609

Determine its rotational kinetic energy.1629

Well, kinetic energy for rotational motion is 1/2 I(ω2).1630

For our moment of inertia, we just determined as .0597 and its angular velocity is 5 rad/s, so 52 -- multiply that out and I come up with 0.75 J.1638

What is the total kinetic energy of the basketball?1657

Well, to get its total kinetic energy, all we are going to do is we are going to combine its translational and its rotational.1661

Total kinetic energy is the translational kinetic energy plus the rotational kinetic energy, so that is going to be 19.84 J + 0.75 J or about 20.6 J in total.1668

That is kinetic energy of a rotating object that is also moving translationally.1691

Let us take a look at a playground roundabout again.1698

A roundabout on the playground with a moment of inertia of 100 kg-m2 -- (I) = 100 kg-m2 -- starts at rest and is accelerated by a force of 150N at a radius of 1 m from its center.1700

If the force is applied at an angle of 90 degrees from the line of action for a time of.5 s that equals 90 degrees times half of a second, what is the final rotational velocity of a roundabout?1723

Well, as I look here, any time I start seeing final angular velocity in initial, I am starting to think about 'You know probably looking at a kinematics equation.'1740

But it would sure be nice to have the angular acceleration.1749

Well to do that I probably need to go to Newton's Second Law for Rotation.1752

Net torque equals I(alpha), therefore, alpha is going to be equal to our net torque over our moment of inertia.1757

I do not have net torque, but I do have force, radius and the angle.1768

Our net torque is going to be F(r) sin(θ) over our moment of inertia.1772

Now, I can substitute in to find angular acceleration equal to our force -- 150 times our radius (1) times the sine of 90 degrees and that is going to be 1 all over our moment of inertia -- 100.1779

So I get 150/100 or 1.5 rad/s2.1796

We want the final rotational velocity of the roundabout though.1808

I am going to go back now to my kinematics for rotation and say that final angular velocity is initial angular velocity + alpha angular acceleration times time.1811

That is going to be 0 + α -- we just determined is 1.5 rad/s2 times our time of 0.5 s.1822

Therefore, our final angular velocity is going to be 1.5 × 1/2 or 0.75 rad/s.1834

Let us take a look at another one.1852

The ice skater is a famous problem in physics around rotational dynamics and moment of inertia.1853

Here without getting into the numbers, we have an ice skater that spins with a specific angular velocity.1860

She brings her arms and legs closer to her body reducing her moment of inertia to half of its original value.1865

What happens to her angular velocity?1871

Well, as the skater pulls her arms and legs in, moment of inertia is going to decrease to the point that it is half of its original value, but angular momentum remains constant.1873

As she spins around, her center of mass remains constant.1887

Why? There is no external torque in this problem.1892

Therefore, angular momentum about the center of mass -- the axis of rotation to the center of mass -- is conserved.1899

If (L) equals I(ω) and we are going to cut (I) in half, (L) must remain the same -- it is conserved.1905

In that case, omega must double, so if we cut that in half omega doubles.1917

All right, that explains what happens to our angular velocity, but what about a rotational kinetic energy?1925

Well, for that, let us go to our formula for rotational kinetic energy. It is 1/2 I (ω)2.1931

In this case again, we have 1/2 -- (I) became a lot smaller. It got cut in half, but omega doubled.1941

Do not forget omega is squared, so if that cut in half and that got doubled -- 1/2 × 2 × 2 -- we are going to double the rotational kinetic energy of the entire system.1951

Kinetic energy rotational doubles while angular velocity gets cut in half.1966

Wait. Where did that energy come from?1975

This rotational energy doubled as she pulled her arms in.1978

Well, the skater must have done work to pull her arms in.1981

That must have required a force applied for some distance in order to do that.1984

That is where we got this extra rotational kinetic energy.1988

Let us take a look at the example of a bowler.1995

Gina rolls a bowling ball of mass 7 kg -- m = 7 kg and radius (10.9 cm), which is .109 m, down a lane with a velocity of 6 m/s.1998

Find the rotational kinetic energy of the bowling ball assuming it does not slip.2012

What is its total kinetic energy?2016

Well, the first thing I am going to do -- it is a solid bowling bowl, and we will assume the mass is uniformly distributed.2020

I am going to find the moment of inertia of the bowling ball by modeling it as a solid sphere.2025

Moment of inertia first for a solid sphere is 2/5 mr2.2031

That will be 2/5 times its mass (7) times the square of its radius (.1092) or about 0.033 kg-m2.2037

Now, it would be helpful to find its angular velocity and we can do that by recognizing angular velocity as its linear velocity divided by the radius, assuming it is not slipping and we can make that assumption. It does not slip.2054

That is going to be our 6 m/s divided by its radius (.109 m) or about 55 rad/s.2066

Well, from here let us find its rotational kinetic energy.2080

Rotational kinetic energy is 1/2 I(ω2) or 1/2 × our (I) .033 kg-m2× our angular velocity (55 rad/ms2) or about 50 J.2086

What about its total kinetic energy?2111

Well, kinetic energy -- total is going to be 1/2 mv2 + the rotational, 1/2 I (ω2), which is going to be 1/2 × our mass (7) × the velocity (62) +...2114

Well, 1/2 I(ω2) -- we all ready said was 50 J, so 1/2 × 7 × 62, 36 + 50, I come out with about 176 J for its total kinetic energy -- rotational + translational.2135

Thanks for watching Educator.com.2154

Hopefully this gets you started with rotational motion and conservation of angular momentum and putting that all together with rotational dynamics as well.2156

Make it a great day. We will see you again.2164

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