  Dan Fullerton

Rotational Kinematics

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 FullertonTue Jan 9, 2018 7:45 AMPost by Zeeshan Shaikh on January 8, 2018When you are deriving the centripetal acceleration formula.  You state that r is equal to the sum of both the x and y component.  How do you do this?  You can't do pythagorean theorem.  I just don't understand how you can say r equals the sum of the x and y component.  Does it have to do with the fact you are using unit vectors? 1 answer Last reply by: Professor Dan FullertonThu Jan 5, 2017 2:14 PMPost by James Glass on January 5, 2017Back to "For the free response problem VII APC 2003 FR3, why is the initial Potential Energy UInitial = Ubucket + Uprojectile, it seems like the potential energies are in opposition to each other and I thought it would be UInitial = Ubucket - Uprojectile." Understanding energy does not have direction helps make the Uinitial = Ubucket + Uprojectile seem true.... but logically it seems like the bucket would fall slower due to the mass x gravity of the projectile side... thus decreasing the kinetic energy available later.  There must be a fundamental oversight by me because the AP guide has the same answer. 2 answersLast reply by: James GlassThu Jan 5, 2017 1:18 PMPost by James Glass on January 5, 2017For the free response problem VII APC 2003 FR3, why is the initial Potential Energy UInitial = Ubucket + Uprojectile, it seems like the potential energies are in opposition to each other and I thought it would be UInitial = Ubucket - Uprojectile. 1 answer Last reply by: Professor Dan FullertonSun Dec 20, 2015 8:43 AMPost by Jim Tang on December 19, 2015These lectures click so much more with the AP frq at the end. Your physics 1 and 2 would be so good if you had them, maybe some B frq? Anyways, great lectures! 1 answer Last reply by: Professor Dan FullertonMon Mar 9, 2015 6:07 AMPost by Philip Schultz on March 9, 2015Where are the practice problems like advertised? 0 answersPost by Professor Dan Fullerton on December 31, 2014Since r is a constant, it gets pulled out of the integration, for the rest of the derivation, I am using the chain rule.  (Derivative of rcos (wt) = -rsin(wt)*d/dt(wt) = -wrsin(wt) 1 answerLast reply by: Thadeus McNamaraWed Dec 31, 2014 3:26 PMPost by Thadeus McNamara on December 31, 2014at around 9:20, can you rexplain how you found the derivative of r? why arent you using chain rule

### Rotational Kinematics

• Once around a circle is 360 degrees, or two pi radians. A radian measures a distance around an arc equal to the length of the arc’s radius.
• Linear position is given by the r vector. The position around a curved path is represented by s. Angular positions / displacements are given by theta (θ).
• The linear displacement can be found by multiplying the angular displacement by the radius.
• Linear speed / velocity is given by the v vector. Angular speed and velocity are represented by omega (ω).
• Linear acceleration is given by the a vector. Angular acceleration is given by alpha (α).
• Translational (linear) kinematics parallels rotational kinematics. Equations for one mirror the equations for the other.

### Rotational Kinematics

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
• Once Around a Circle: In Degrees
• Once Around a Circle: In Radians
• Example I: Radian and Degrees 1:08
• Example I: Convert 90° to Radians
• Example I: Convert 6 Radians to Degree
• Linear vs. Angular Displacement 1:43
• Linear Displacement
• Angular Displacement
• Linear vs. Angular Velocity 2:04
• Linear Velocity
• Angular Velocity
• Direction of Angular Velocity 2:28
• Direction of Angular Velocity
• Converting Linear to Angular Velocity 2:58
• Converting Linear to Angular Velocity
• Example II: Angular Velocity of Earth 3:51
• Linear vs. Angular Acceleration 4:35
• Linear Acceleration
• Angular Acceleration
• Example III: Angular Acceleration 5:09
• Kinematic Variable Parallels 6:30
• Kinematic Variable Parallels: Translational & Angular
• Variable Translations 7:00
• Variable Translations: Translational & Angular
• Kinematic Equation Parallels 7:38
• Kinematic Equation Parallels: Translational & Rotational
• Example IV: Deriving Centripetal Acceleration 8:29
• Example V: Angular Velocity 13:24
• Example V: Part A
• Example V: Part B
• Example VI: Wheel in Motion 14:39
• Example VII: AP-C 2003 FR3 16:23
• Example VII: Part A
• Example VII: Part B
• Example VII: Part C
• Example VIII: AP-C 2014 FR2 25:35
• Example VIII: Part A
• Example VIII: Part B
• Example VIII: Part C
• Example VIII: Part D
• Example VIII: Part E

### Transcription: Rotational Kinematics

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

I am Dan Fullerton and in this lesson we are going to talk about rotational kinematics.0003

To begin with our objective, understand and apply relationships between translational and rotational kinematics.0008

Write and apply relations among the angular acceleration, angular velocity,0016

and angular displacement of an object rotating about a fixed axis with constant angular acceleration.0021

Use the right hand rule to determine the direction of the angular velocity vector.0027

I know a couple of these are going to be a bit of review but as we get back in the rotation in more depth,0036

it will probably take a minute or 2 to make sure we have got these fundamentals down.0041

In degrees, once around the circle is 360° and once around the circle is 2π in radians.0045

The radians measures the distance around an arc equivalent to the length of the arc’s radius.0052

That distance around δ S is circumference or 2π r or if you are measuring diameter it would just be π × d.0058

Let us do a couple conversions again very quickly, 90° radians, 90° if we want that in radians, 2π radians is equal to 360°.0068

If we start off with 6 radians, 2π radians is 360° and those cancel out 360 ° × 6/2π is 344°.0086

We have got our conversions, angular vs. linear displacement.0103

Linear position displacement we have talked about as δ r and δ s.0107

Angular position or displacement we give by δ θ.0111

As you go around the circle, you have increasing amounts of θ, where S the linear distances r × θ or δ S is our δ θ.0114

If we talked about velocity in the same way, linear speed or velocity is given by the V vector.0125

Angular speed or velocity is given by the squiggly W, the ω vector where velocity is the derivative of position with respect to time.0130

Angular velocity is a derivative of angular position or displacement with respect to time.0141

We talk about these angular vectors, the direction is given by the right hand rule, something that is very non intuitive.0148

If we think about an object going around a path like this, the radius to the side, take the right hand0154

or wrap the fingers of your right hand in the direction the object is moving around that circular path0160

and your thumb will give you the direction of the positive angular velocity vector.0165

The angular velocity vector does not point in the direction the object is actually moving.0170

Converting linear to angular velocity, we have velocity as the rate of change of position or displacement with respect to time.0177

But we know that S is our r × θ, the radius × θ.0188

Therefore, we can write that this is equal to D / DT of the derivative of r θ.0195

But r is a constant, our radius is not changing so we can write this then as V = r D θ dt which is r ω.0203

D θ dt is our ω, so V = r ω or if we want ω, ω = V/ r.0216

Doing an example where we look at the angular velocity of the Earth.0230

Find the magnitude of Earth’s angular velocity in radians per second.0234

Ω is δ θ /δ T which is 2 π radians / 24 hours which is going to be π radians/ 12 hours but we know that 1 hour his 3600s to get this in more standard units.0239

Ω would be equal to 7.27 × 10⁻⁵ radians /s.0261

All the things that we have done before but useful to get just as firm foundation before we get a little bit more in depth here.0269

If we want to talk about linear vs. angular acceleration, if linear acceleration is given by A vector, angular acceleration is given by the Α vector,0277

where if A is the derivative of velocity, the angular acceleration is a derivative of angular velocity.0284

In this case, it is how quickly you are changing your angular velocity is what we call angular acceleration.0293

Just like we did with the angular velocity, as far as finding the direction, the direction of the angular acceleration vector is also given by a similar right hand rule.0300

Let us do an angular acceleration problem.0310

Our friend rides a unicycle, if the unicycle wheel begins at rest and accelerates uniformly0313

in a counterclockwise direction to an angular velocity of 15 rpm to the time of 6s,0317

Find the angular acceleration of the unicycle wheel.0323

First, let us convert rpm to radians/ s, 15 rpm is 15 revolutions / 60s but there are 2 π radians in each revolution.0328

That is going to be 1.57 radians /s.0345

Our angular acceleration is our change in angular velocity with respect to time which will be our final - our initial angular velocity with respect to time,0353

Or 1.57 radians/ s ÷ 6s which will be 0.26 radians /s² and because it is accelerating counterclockwise that is what we are going to call a positive angular acceleration.0365

As we go through and look at rotational kinematics, it is helpful to talk about some of these variables.0390

We are talking about angular, we have δ θ, a linear velocity translational velocity V, angular velocity ω.0403

Translational or linear acceleration A and angular acceleration Α and time is the same across both of these paradigms.0409

Where it starts to get useful is when we look at the variable translations, you will start to see a pattern.0419

If S = r θ, V = r ω, A = r α.0425

All we are doing is just multiplying the angular version by the radius to get the linear or the translational version.0431

Similarly, θ = s /r, ω = V /r, α = A /r.0439

You take the linear version divided by the radius to get the angular version.0444

Of course, time is time regardless of which paradigm you are doing.0448

When we get to kinematic problems, this makes our formulas much simpler.0453

Our kinetic equations that we have derived earlier V = V initial + AT, if we want to look at the rotational equivalent,0459

all we do is we replace any velocities with angular velocity.0466

We replace any displacements with angular displacements.0470

We replace any acceleration with angular accelerations.0473

This becomes ω = initial ω + Α × T or δ x = V knot t + ½ at² becomes δ θ = ω knot t + ½ α t² .0477

Or finally, V² = V initial² + 2a δ x, ω² = ω initial² + 2α δ θ.0492

You are just replacing the variables but the form of those kinetic equations and they are used that is exactly the same.0501

We have also talked about how we derive that centripetal acceleration.0510

Probably, we are taking a minute and doing it again just to make sure we have it down.0514

So if we look at some specific point that you get after some angular displacement θ, that sometime from T0 to T,0518

we could call its x position will be r cos θ and its y position would be r sin θ.0526

Our r vector is going to be r cos where θ = ω t, that would be ω t I hat + r sin ω t j hat.0535

Velocity is just going to be the derivative of that, it is the derivative of r with respect to t,0552

that is going to be the derivative with respect to t of r cos ω t i hat + r sin ω t j hat.0558

Which implies then that V is equal to, if we take the derivative of this we will get the derivative of the first + the derivative of the second.0574

That is going to be V equal to, we will have r I hat, derivative of cos is going to be opposite of the sin.0582

This will be in r I hat × ω – sin ω t.0592

We have our term over here, we will have + r j hat ω cos ωt.0602

If we rearrange them a little bit and make it look a little bit more formal, V = -ω r sin ω t I hat + ω r cos ω t j hat.0616

There is our velocity but we can take that a step further and let us do that.0640

If V (t), let us give a little room for that, if V (t)= - ω r sin ω t I hat + ω t cos ω t j hat.0652

And our acceleration is going to be the derivative velocity with respect to time which is going to be0671

the derivative of all of this is going to be - ω² r cos ω t I hat - ω² r sin ωt j hat.0678

Which implies then that acceleration = - ω² and then we are left with r cos ω t I hat + r sin ω t j hat.0698

If we recalled this is our initial r vector.0719

A is - ω² r or we also know ω = V /r so that means A = - V /r² × r or V² /r - V² /r.0726

Now the negative sign, why are we worried about that?0751

We are talking about the centripetal acceleration, we are defining toward the center of the circle as positive so that would V² /r.0754

When we are talking about the vectors, we define it this way where r is from the center out to that position point while the centripetal acceleration is opposite of that.0762

That is where the negative comes.0771

R goes from the center to the circle where as the acceleration is from the object to the center.0775

It is easier as we do this and stop worrying about our vector signs and directions, A is V² /r.0792

Let us do an example here.0803

An object of mass M moves in a circular path of radius r according to θ = 2 t2 + t + 4, where θ is measured in radians and t is in seconds.0805

Find the angular velocity of the object that equals to seconds.0817

As I look at this, ω = d θ dt which is going to be the derivative with respect to time of 2t³ + t + 4, which is going to be 6t² + 1,0822

which implies then since t = 2s, that ω T = 2s is going to be 6 × 2² + 1 or 25 radians/s.0839

Find the object’s speed at this time.0857

The velocity, the speed at T = 2s is going to be r ω at t = 2s which is just going to be 25r m/s.0860

Let us take a look at an example with the wheel.0878

The wheel of radius r and mass capital M undergoes a constant angular acceleration of magnitude α,0881

what is the speed of the wheel after it is completed one complete turn assuming it started from rest?0890

This is a kinematics problem so let us figure out what we know.0896

Our initial angular velocity is 0 because it starts from rest, we are trying to find its final angular velocity0901

or trying to find its final linear velocity but angular velocity we will get to it later.0909

We know our displacement is 2π once around the circle and we have some angular acceleration α.0914

With what I know, I would go to my kinematic equation by rotational version ω final² = ω initial² + 2 α δ θ or ω final² = ω initial² is 0 so this becomes 2 × α δ θ is 2π.0924

This is 4π α, ω final is just going to be √4π α.0952

If we want that, our speed, that is going to r ω that would be r√4π α.0964

Let us finish up by doing a couple of AP problems from old past AP exams.0978

We will take a look first at the 2003 exam Mechanics question 3.0984

Take a minute, pull that out, you can find it here at the link above or google it, download it and give it a shot and then come back here and see what we have got.0989

It looks like we first are plotting some data points.1000

I plotted the data points first and we are supposed to draw the best fit curve.1003

It is kind of a goofy, easy, initial question there.1009

I would draw at something like this and to draw a curve, the shape, I have this kind of like that.1014

It says using that best fit curve determine the distance traveled by the projectile if a 250 kg is placed in the counterweight bucket.1023

To do that, all I do is go up here to wherever it happens to be 250 kg and come here and read off on the graph and I get an x of about 33 m.1034

All right, going to part B.1051

For part B, says students are assuming that the mass of the arm, the cup, and the counterweight bucket can be neglected and then1060

they develop a model for x as a function of mass using x = Vxt, where Vx is the horizontal velocity of the projectile as it flies off the top of the cup and T is the time.1069

First off, how many seconds after leaving the cup the projectile strike the ground?1081

That sounds like a kinematics question to me, so B1 we have an initial, if we look vertically, initial vertical velocity is 0.1086

We do not know our final vertical velocity, δ y is 15m, our acceleration is 10 m /s².1098

We are calling down the positive Y direction and T that is what we are trying to find.1106

I would use δ y = V initial t + ½ Ay T², where V initial is 0 so T is going to be 2 δ y / √A which is 2 × 15m / 10m/s², 30/10² that is about 1.73s.1113

Let us go onto part B2, derive the equation that describes the gravitational potential energy of the system1145

relative to the ground assuming the mass in the counterweight bucket is M.1153

For B2, as I look at that, our initial potential energy is going to be equal to the potential energy in the bucket + the potential energy of our projectile1158

which is going to be mass B, G × the height of B + mass of the projectile G × the height of the projectile, which is, let us see what we have, 10 m/s² for G × 3m × M + 10 kg.1169

And all of that is going to be equal to 300 K + 30 M.1192

Now let us take a look at part 3, derive the equation for the velocity as it leaves the cup.1204

We are getting a little bit more involved here.1210

Part 3, our final potential energy + our final kinetic energy must equal V initial and we are looking at when it leaves the cup there.1214

We can say that our final potential is going to have to equal, we have got 1 × 10 × 110 M, 1 × 10 × M + 15 × 10 m/s² × 10 =1500 + 10 M.1226

Our kinetic final, we know is ½, which implies there we are just coming up with it again, we are just coming up with the different pieces.1251

Our kinetic finals is going to be ½ × mass × square root of our velocity V x² + ½ mass of our bucket × the velocity of our bucket².1262

Our initial potential, we said was 300 + 30 M from part B2 up above.1278

Putting all of this together, we have 300 + 30 M must be equal to that 10 M + 1500 from up here + we have got 5 Vx² + ½ M VB².1286

The key to solving this problem at this point is realizing that both ends of that catapult, they are swinging with the same angular velocity.1313

If that is the case, ω B must equal ω A and since V = ω r and ω = V /r, we can write that VB / 2 must equal Vx / 121322

and then we get our relationship between the velocity of bucket and our Vx.1337

VB must equal Vx/6 so now we can go and we can put that back in our blue equation there to solve for the velocity as it leaves that Vx.1343

300 + 30 M = 10 M + 1500 + 5 Vx² + ½ M and VB² now is just going to be Vx² / 36.1356

It is an algebra exercise, 20 M = we take that 300 out, 20 M = 1200 + 5 Vx² + M Vx² / 72 which implies that 20 M - 1200 is going to be equal to Vx² × 5 + M / 72.1383

Or getting vxx all by itself, Vx is going to be equal to 20 M - 1200 ÷ 5 + M / √72.1420

And there is probably a way to simplify that further but that looks like plenty to me.1434

Onto part C, complete the theoretical model by writing a relationship for x as a function of the counterweight mass using the results from B1 and B3.1444

That is just x = velocity × time from our horizontal kinematics which is just going to be, time was 1.73.1457

This is 1.73 × √20 M - 1200/5 +√M / 72.1467

Part C2, compare the experimental and theoretical values of x for a counterweight bucket mass of 300 kg.1482

Theoretically, when we plug in for M = 300kg in our formula, I come up with about 39.6m and we write it off the graph,1492

if we read it off the graph actual M at 300kg was about 37m.1507

Where did that difference come from?1514

The difference could be from a lot of things, you could have had friction at the axis, you could have air resistance.1517

Remember, how we neglected the masses of the arm, the bucket, and the cup, any of those can all contribute to this difference there.1522

That covers that free response, let us take a look at one more here.1530

Let us go to the 2014 Mechanics exam free response number 2.1536

Again there is the link there, we can google it, take a minute and print it out, check it out, and try it, and come back here and see how you do.1540

It is kind of an interesting set up on this one.1552

The first thing I have to do is find an expression for the height of the ramp in terms of the V knot M and fundamental constants.1554

I would use conservation of energy where the final kinetic energy must be equal to the initial gravitational potential energy1561

or ½ M × what they call V initial² = MGH.1570

Therefore, just solving for H that is going to be V knot² / 2G, pretty straightforward for part A.1579

Move on to B, short time after passing point T, the block is in contact with the wall and moves with the speed of V.1591

Is the vertical component of the net force on the block upward, downward, or 0?1599

There is no vertical acceleration so Ay = 0, which means you can say that the net force in the y direction must equal 0 so 0.1604

In part 2, it says on that figure, draw an arrow starting on the block to indicate1620

the direction of the horizontal component of the net force on the moving block when it is that position shown.1625

We have got it moving in a circular path and it is right here at that position.1631

When it is at that position, we have the normal force acting directly to the left so that is the direction of the normal force1641

but we also have some amount of frictional force that is opposing the motion.1649

When I put those 2 together, we will add them up tip to tail normal friction, I get something that must be down into the left for my net force.1655

I would draw something like that for my net force.1664

That leads us to part C, determine an expression for the magnitude of the normal force exerted on the block by the wall as a function of velocity.1673

The net centripetal force is MV² /r and the only centripetal component is provided by the normal force that is M.1685

Therefore, I would say that N = MV² /r.1695

The frictional force is perpendicular to that is not going to come into play when we are talking about the centripetal force.1699

We have here part D, derive an expression for the magnitude of the tangential acceleration of the block at the instantaneous speed V.1708

Tangential acceleration has to do with our frictional force.1721

Our frictional force is μ × the normal force and we just found the normal force so that means that our frictional force is going to be me MV² /r and1725

that must be equal to mass × the tangential acceleration or the magnitude of the tangential acceleration is just going to be μ V² /R.1739

It looks like we got one more part of the question.1757

Let us give ourselves a little more room here.1760

For part E, derive an expression for the velocity as a function of time after passing point T.1763

To start with, we know that the magnitude of our tangential acceleration is – μ V² /r, that negative because the speed is in the opposite direction of the acceleration.1771

We also know that acceleration is the derivative of velocity with respect to time, dv dt must equal – μ V² / R or dv / V² is going to be equal –μ/R dt.1785

If we want to get our velocity then it looks like we are going to have to do a little bit of integration.1810

We will integrate this so I’m going rewrite this first as the integral of V⁻² dv evaluated from some velocity V initial to final V = - μ /R1815

integral from T = 0 to T of dt which implies then the integral of V⁻² is -1 /V.1832

– V⁻¹ evaluated from V knot to V = - μ / R × T or I could write that as -1 / V -1 /V initial = -μ /RT1844

which implies then that, let us rearrange the order there, put that negative through.1864

1/ V knot -1 / V = - μ/ R T or getting a common denominator, multiplying everything by V knot, V knot /V knot - V knot / V = -V knot μ T / R.1869

Let us see if we can do little bit more rearrangement.1893

That is just 1 so we could say that – V knot / V = - V knot μ T / R -1 which implies then, let us multiply that to -1.1896

V knot / V is equal to, I’m going to write 1 as R/R.1913

R/R + V knot μ T /R or V knot / V is equal to R + V knot μ T / R which implies then, if V knot / V is that / that and V /V knot is the denominator/numerator.1918

V / V knot = R /r + V knot μ T.1941

V all by itself is V knot × r /r + V knot μ T.1949

All the work for that, alright hopefully that gets you a pretty good start on rotational kinematics.1962

Thank you so much for spending time with us here at www.educator.com and make it a great day everyone.1968

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