  Dan Fullerton

Moment of Inertia

Slide Duration:

Section 1: Introduction
What is Physics?

7m 12s

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

1h 51s

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

23m 47s

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

36m 47s

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

30m 34s

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

30m 24s

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

23m 57s

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

23m 57s

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

20m 41s

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

32m 10s

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

20m 31s

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

24m 58s

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

37m 34s

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

28m 4s

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

54m 56s

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

16m 44s

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

13m 9s

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

46m 30s

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

28m 26s

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

21m 36s

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

32m 52s

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

24m

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

26m 9s

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

56m 58s

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

33m 2s

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

1h 1m 12s

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

34m 59s

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

28m 11s

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

28m 11s

Intro
0:00
Question 1
0:15
Part A: I
0:16
Part A: II
0:46
Part A: III
1:13
Part B
1:40
Part C
2:49
Part D: I
4:46
Part D: II
5:15
Question 2
5:46
Part A: I
6:13
Part A: II
7:05
Part B: I
7:48
Part B: II
8:42
Part B: III
9:03
Part B: IV
9:26
Part B: V
11:32
Question 3
13:30
Part A: I
13:50
Part A: II
14:16
Part A: III
14:38
Part A: IV
14:56
Part A: V
15:36
Part B
16:11
Part C
17:00
Part D: I
19:56
Part D: II
21:08
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• ## Related Books 1 answer Last reply by: Professor Dan FullertonFri Mar 11, 2016 12:41 PMPost by David LÃ¶fqvist on March 11, 2016Doesn't the thickness of the rod play a roll? Or is that just because of air resistance? 1 answer Last reply by: Professor Dan FullertonTue Mar 1, 2016 6:17 AMPost by Ore Okusanya on February 29, 2016hey dan, in mechanics doing this topic moments, i find it hard to distinguish which moments is going either clockwise or anticlockwise. 1 answer Last reply by: Professor Dan FullertonMon Jan 4, 2016 6:33 AMPost by Shehryar Khursheed on January 2, 2016Hello Mr. Fullerton,I have a question about the moment of inertia of the cylinder. I noticed that you used an integral from 0 to R. However, I was wondering whether or not you can go the other way, and keep R constant while changing L; that is, going from the top to the bottom instead of inside out as you did. I will explain the process I went through:1) (rho)=M/(pi*R^2*L)2) Now, i'm going to divide the cylinder into tiny cylinders (can be considered disks), with a thickness dl that will go from the top (L=0) to the bottom of the cylinder (L=L)3) (rho)dv= dm4) dv= pi*R^2*dl 5) integral(r^2*dm)= inegral((R^2)(rho*pi*R^2*dl))6) now taking the constants out:  rho*pi*R^4*integral(dl) from 0 to L7) after solving the integral, I got I=MR^2, which is the moment of inertia of a disk, not a cylinderMy question is did I do something wrong while solving for the moment of inertia this way? Or are you not allowed to do it the way I did it, keeping R constant but changing L? If so, why? 1 answer Last reply by: Professor Dan FullertonSun Dec 20, 2015 8:50 AMPost by Jim Tang on December 19, 2015hey dan,I'm confused about how you set up dm.for the rod problem, how are you incorporating the area of the rod if you don't use pi r^2? i can't see how dx is the volume here?also, for the cylinder, you lost when you said area but you said 2 pi r, which is circumference.i think I'm missing the intuitive understand of dm, but i can't see how the area is incorporated here in both these scenarios. 1 answer Last reply by: Professor Dan FullertonTue May 5, 2015 7:26 PMPost by Xinyuan Xing on May 5, 2015In example you add an excessive /3 under Î»ï¼Œwhich may lead to confusion. Btw it's a nice video for students to understand the mathematic procedure for the derivation of physics formula, helps me a lot, too. 1 answer Last reply by: Professor Dan FullertonMon Apr 6, 2015 6:18 AMPost by Jason Kim on April 5, 2015On the last example in this video how is it that you do not have to account for moment of inertia of two spheres that are attached to the rod? 2 answersLast reply by: Thadeus McNamaraSat Jan 3, 2015 5:21 PMPost by Thadeus McNamara on December 31, 2014I dont understand the step that starts with dm= at 14:19 1 answerLast reply by: Thadeus McNamaraWed Dec 31, 2014 4:55 PMPost by Thadeus McNamara on December 31, 2014at around 11:25 you are finding the moment of inertia for a rod spinning about its center. i understand that your bounds of integration are from L/2 to -L/2. The first time I did it, i used bound of integration from L/2 to 0. I then multiplied that answer by 2. How come my way does not work?

### Moment of Inertia

• Inertial mass is an object’s ability to resist a linear acceleration. Moment of Inertia (or rotational inertia) is an object’s resistance to a rotational acceleration.
• Objects that have most of their mass near their axis of rotation have smaller rotational inertias than objects with more mass farther from their axis of rotation.
• The moment of inertia can be found by summing the product of the mass of each object in the system and the square of its distance from the axis of rotation for all the objects in the system.

### Moment of Inertia

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

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

### Transcription: Moment of Inertia

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

I’m Dan Fullerton and in this lesson we are going to talk about moment of inertia.0004

Our objectives include determining by inspection which set of symmetrical objects has the greatest moment of inertia.0008

Determining by what factor an object’s moment of inertia changes if its dimensions are increased by a consistent factor.0015

Calculating the moment of inertia for various objects0022

and stating applying the parallel axis theorem which we use to find a moment of inertia about a different point for an object.0026

Let us talk about types of inertia for a moment.0033

An inertial mass or translational inertia is an object's ability to resist a linear acceleration.0037

When we talk about things rotating, you also know that if you try and rotate something it has some resistance to being rotated as well.0044

It has a resistance to the angular acceleration, we will call that the moment of inertia or rotational inertia.0051

It is an object's resistance to rotational acceleration.0058

Objects that have most of their mass near their axis of rotation have a smaller rotational inertia,0063

an object that have most of their mass farther from their axis of rotation.0067

Now inertial mass, we are going to give the symbol M, we have been using that.0072

For moment of inertia, we are going to call that capital I or rotational inertia.0075

The rotational analogue is linear inertia.0080

Let us start our exploration by looking at the kinetic energy of a rotating disk.0083

If we look at this disk and if it starts to spin, it is pretty obvious that it is in motion.0089

Therefore, it must have some kinetic energy but we do not know how to deal with that yet.0094

All we have talked about is the kinetic energy of objects moving translationally at different points in space.0098

Now parts of this are rotating, parts are maintaining their position.0105

The way we are going to do this is, let us start by defining the direction for angular velocity ω.0110

Let us assume that is spinning about its center point with some angular velocity and we will draw a point for a center here.0116

Let us take and let us start by finding just the kinetic energy of a little tiny piece of our uniform disk here.0125

As we do this, let us figure out that the entire disk has a radius capital R, we have a vector RI,0135

to our little piece of mass of our disk MI, that is moving with some velocity at this point in time VI.0146

With that, I think we can start to look at the kinetic energy of just that little tiny piece of disk.0156

The kinetic energy of that little piece I is going to be ½ × its mass MI × its speed Vi².0163

We also know for objects rotating that linear velocity is its angular velocity × its radius.0174

We can write this as the kinetic energy of that little piece I is ½ × its mass × the square root of its angular velocity × RI².0182

But then if we wanted the total kinetic energy of the entire disk, we have to add up this kinetic energy for little tiny pieces of the disk to get the total.0197

Our total kinetic energy for it rotating is going to be the infinite sum for all those little pieces I of KI,0206

which is just going to be the sum over I of ½ MI ω² RI².0217

As I look at this, I have a couple constants here that I can pull out of the summation.0229

The ½ is a constant, it looks like the ω, the angular velocity is going to be the same for any points.0234

We can pull those out and say that this is going to be ω² / 2 to say our total K=ω² / 2 × the sum overall I of MI RI².0240

For my next step, what I am going to do is I'm going to define some constant capital I, constant for this problem, which we are going to call the moment of inertia.0258

The sum of all MR² we will call I.0268

I could rewrite this then as our total kinetic energy is going to be ½ I, some of them are squared, ω².0273

When I do that summation, our total kinetic energy is ½ the moment of inertia × the squared of the angular velocity.0289

If I know the moment of inertia, I do not have to worry about all these little pieces.0295

Notice how similar this is to our formula for translational kinetic energy ½ MV².0300

Instead of linear velocity, we now have angular velocity.0310

Instead of inertial mass, we are using rotational inertia or moment of inertia and we still have the ½ vector.0314

We have added one more variable that switches when we go to the rotational world instead of talking about mass and we talk about rotational inertia or moment of inertia.0321

How do we calculate this more generally?0332

Moment of inertia is the sum of all MR² for an object or if you got an infinite sum, we can take the integral of R² × the differential of mass.0335

The mass of one tiny piece.0345

As you go through this unit, it is helpful to memorize some moments of inertia of common objects.0350

Some of those include, well here is the general one, for anything you can use the sum overall MI R².0355

For a disk ½ MR², for a solid sphere 2/5 MR², for a rod about its center point 1/12 ML², L being its length.0362

For a hoop that is MR², for a hollow sphere its 2/3 mass × squared radius and for rod about its end it is 1/3 ML².0374

We will derive a couple of these but as you go through the course, it is usually helpful to memorize a couple of these to save you some time on some problems.0384

If you are asked to derive it, you already know the answers so you know you are doing things correctly.0392

These ones you will definitely want to know, the solid sphere, hollow sphere, hoops, and disk,0397

they are all good ones to have in the back your mind.0402

Let us take a look at calculating MR² or moment of inertia with a couple of these.0406

Find the moment of inertia I of two 5 kg bowling balls joined by meter long rod of negligible mass when rotated about the center of the rods, rotated from right there.0412

Almost looks like a barbell.0422

Compare that to the moment of inertia of the object when rotated about one of the masses.0424

Before we even start, let us think about which one of these we think is going to be tougher to rotate.0429

It is going to be tougher to rotate this about its center point or this about its end?0434

Just by common sense and experience things I have seen in my life, I'm going to guess that this one is a little bit hard to rotate.0438

We are going to find that out.0445

Let us start here on the left, we will call this M1, this is M2, we will call this R1 and we will call R2.0447

Our moment of inertia is the sum of our MR² which is going to be M1 R1² + M2 R2².0459

The mass one is 5 so 5 kg in that distance if the whole thing is in meter, R1 must be half of meter so 0.5 m² + M2 5 × its length 0.5².0473

Moment of inertia is going to be 10 × 0.5² or 2.5 kg m².0489

Same setup, it rotates in about a different point on the right.0500

Moment of inertia is still the sum of all our MR² so we are going to have M1 R1² + M2 R2² where this is going to be R1.0504

It looks like R2 is going to be 0, moment of inertia is going to be 5 × 1m² + 5 × 0² or just 5 kg m².0517

Same object but rotated about a different point, we have twice the moment of inertia here on the right which makes sense.0534

As we said before we started here that we thought that it would be hard to give a rotational acceleration 2.0540

How about a slightly more complicated object?0549

Find the moment of inertia of the uniform rod about its end and about its center.0552

As we do this one, what we are going to do as far as a strategy is we are going to break the rod up into little tiny pieces0558

with some mass VM at some distance R from our rotation point.0567

We will define the linear mass density as the total mass of our rod divided by its length.0575

If we do that then, the differential of mass, the amount of mass that in that little tiny bit,0582

DM is going to be the linear mass density × dx as we integrate from 0 to L.0588

If we do this about its end, our moment of inertia is going to be using our formula R² dm which is going to be the integral from 0 to L.0597

Our R is just our x coordinate that is x² and dm we said was λ dx.0612

This implies then that the moment of inertia is going to be equal to, λ is a constant in this case so we can pull λ out of the integral sign,0622

integral from 0 to L of x² dx, integral of x² is x³/3 evaluated from 0 to L which is going to be λ L³.0630

But we also said that λ was M/L so our moment of inertia I is going to be, we will replace λ with M/L, we still have an L³/3.0647

I can make a ratio of 1 that becomes L² so we end up with ML² / 3.0661

There is the moment of inertia of that rod rotated about its end.0671

Let us do it about its center.0676

Same basic calculation but set up just a little bit differently.0688

Moment of inertia is R² dm which is now going to be the integral from -L /2 to L /2 calling our center point here 0 of x² λ dx,0691

which implies then that our moment of inertia is going to be, we can pull our λ out again and0708

we are going to have x³/3 evaluated from-L /2 to L/2, which is going to be λ /3 × we will have L /2³/3 - -L /2³/3.0713

Our moment of inertia is going to be equal to λ/3 all of this is going to be equal to λ³/8 - - λ³/8, 2 λ³/8, or λ³/4.0733

Once again, we can take a look at our λ which we defined as M/L.0752

If λ is M/L that will be M/L, for λ we have still got a L³ and we got a 12 down here L that becomes L².0757

And I end up with 1/12 ML², much smaller moment of inertia to rotate about that center point.0769

You can verify those as when I said you probably have to memorize from our previous formula screen.0779

All right looking at another object, let us take a look at a solid cylinder.0786

Find the moment of inertia of a uniform solid cylinder about an axis through its center.0791

This is kind of our soda can that is spinning through its point on the center.0795

We are going to assume it has a uniform density because it is a uniform solid cylinder.0799

Its volume mass density is going to be its total mass divided by its volume, which would be its mass divided by, its volume will be the area × its length which is π R² L.0804

Our strategy is going to be as we integrate it, we are going to take little tiny pieces of the can and think of them as very thin slices.0824

As we integrate from the tiny once all the way out, we are going to get our total cylinder.0834

We need to figure out the differential of mass that is in one tiny piece of the can of that size.0839

To take that, imagine we take this hollow can, what you are going to do is cut it and spread it out to find its mass.0845

To do that then the differential of mass, the mass included there is going to be the area of the rectangle of material we make × its thickness.0853

Its area is going to be the circumference 2π R × its length L × the mass density and then the thickness0864

is going to be our little dr is we integrate from 0 all the way out to R.0877

We are going to make these infinitesimally thin.0882

Our differential of mass inside our little hollow, it is in that piece of little hollow cylinder is 2π RL × our volume mass density × dr the thickness.0885

As if we have cut that and spread it out to make a tiny thin rectangle of material.0898

Once we got that set up, the actual integration piece is pretty straightforward.0903

Our moments of inertia is the integral of R² dm which is going to be the integral from R =0 to R,0909

the radius of our entire cylinder of R² and our dm we just defined as 2π RL ρ dr.0921

Our moment of inertia then, we can pull out our constants.0937

2π is a constant, our volume mass density ρ is a constant because it is uniform, L is a constant.0941

That will leave us with the integral from 0 to R of, we have here R³ dr which is 2π ρl.0950

The integral of R³ is R⁴/4 evaluated from 0 to R which implies then that our moment of inertia is going to be 2π ρ L.0960

We will have R⁴ / 4 -0⁴/4 which is 0.0973

But we also know that our ρ is M/π R² L.0981

We will substitute that in so this is going to be equal to, we will not imply, we will say that is equal to, we have 2π, our ρ is M/ir² L.0990

We also still have our L R⁴/4 so this implies then that our moment of inertia is going to be, let us see what we can cancel out of here.1010

We have got an L, we have got an L, we got a π and π, R² and R⁴ that becomes R in the second, 2 becomes a 2.1022

I end up with MR² /2 moment of inertia for our solid cylinder.1034

You get the idea of the procedure you go to define moments of inertia of these continuous or more complicated objects.1046

Let us take a look at the parallel axis theorem, this is a really cool,1054

If you know the moment of inertia, I of any objects through an axis that intersects the center of mass of the object, we will call that axis L.1064

You can find a moment of inertia around any axis that is parallel to that current axis of rotation we will call L prime.1066

If this is our object of some sort, we know its moment of inertia through L.1083

We want to know its moment of inertia through L prime at some distance D, away in parallel to that initial one.1088

Assuming the initial one goes to the center of mass, the way to find the moment of inertia about L prime is just going to be the moment of inertia1094

about that center of mass + mass × the distance between those two axis².1101

Let us take a look at how we can use that to solve a problem.1116

Find the moment of inertia of a rod of mass M and length L about one end of the rod using the parallel axis theorem.1121

We have already done this sort of problem before but let us find about the end since we already know what it is, about its center.1128

About its center, we know that the moment of inertia about the center of mass is 1/12 ML² around about its center.1135

And this distance must be L/2 because we are going to move from here to here.1145

Our distance D is L/2 so the moment of inertia of the rod about its end is the moment of inertia about its center of mass + mass × the square of its distance,1152

which is, we have moment of inertia about the center of mass 1/12 ML² + mass and our D is L /2² which will be ML² / 12 + ML² / 4,1165

which implies that the moment of inertia about the end is going to ML² / 12 + this will be 3 ML² /12 to give ourselves a common denominator1183

which is 4 ML² / 12 or ML² /3.1196

Another way you can find a moment of inertia about an object once you know its moment of inertia about the center of mass,1203

as long as that new axis is parallel to the one where you know already, it is nice, straightforward, easy calculation.1209

Alright, to calculate the moment of inertia of a hollow sphere with a mass of 10 kg and a radius of 0.2 m.1218

Here we are going to assume that you have memorized your moment of inertia for common objects.1226

So this would be 2/3 M R² for a hollow sphere which is 2/3 × mass 10 kg and our radius 0.2 m² or 0.27 kg m².1232

How about for a long, thin rod?1255

Find the moment of inertia for a long, thin rod with the mass of 2 kg and a length of 1 m rotating about the center of its length.1258

Let us take a look and assume it is uniform so that is 1/12 ML² about its center which is 1/12 × 2 kg × 1 m² or about 0.17 kg m².1268

What is its moment of inertia when rotating about its end?1288

That is just going to be 1/3 ML² or 1/2 × at 2 kg × L² 1 which is just going to be 0.67 kg m².1291

An object with uniform mass density is rotated about an axle which may be in position A, B, C, or D.1312

Rank the objects moment of inertia from smallest to largest based on axle position.1319

From smallest to largest, we are going where it is easiest to accelerate it rotationally towards its toughest to accelerate rotationally.1326

As we know it is going to be easiest when you got it at the center point here, C we have the smallest moment of inertia.1334

As you move away from that center point, it gets tougher and tougher C, B, D and finally if you are rotating it about A.1340

I would go C, B, D, A, for the ranking of the moment of inertia from smallest to largest.1351

Alright let us do one more, a uniform rod of length L has a moment of inertia I 0 when rotated about its midpoint.1358

A sphere of mass M is added to each of the rod, what is the new moment of inertia of the rod ball system?1365

Over here, moment of inertia is I 0, here we need to figure out its new moment of inertia.1372

The moment of inertia on the right is going to be the moment of inertia of the rod + we have to add up our M².1380

The moment of inertia will be I 0 + we have M × its distance from our center point that is going to be L /2.1391

We got L /2² + same thing on the right hand side ML /2².1403

I is going to be I initial + ML² /4 + ML² /4.1411

Our moment of inertia is going to be I initial + ML² /2.1420

Alright, hopefully that gets you a good start on moment of inertia or rotational inertia.1430

We will be using it quite extensively in the next few lessons.1434

Thanks for joining us at www.educator.com and make it a great day everyone.1437

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