  Dan Fullerton

Torque

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 FullertonWed Dec 20, 2017 6:05 AMPost by James Glass on December 20, 2017Hello, for the see-saw problem, how would angular acceleration be determined if the see-saw was not in equilibrium ... for example if the fulcrum was in the center of the board would I = I(cm) + M(tortoise)R^2 + M(hare)R^2 be used? 1 answer Last reply by: Professor Dan FullertonMon Jan 4, 2016 2:28 PMPost by Shehryar Khursheed on January 4, 2016Will angular acceleration always be in the same direction as the net torque, similar to how linear acceleration is always in the same direction as the net force? 3 answers Last reply by: Professor Dan FullertonFri Sep 18, 2015 4:25 PMPost by Parth Shorey on September 18, 2015Is it possible that net force is zero but the net torque is not ? 1 answer Last reply by: Professor Dan FullertonWed Sep 16, 2015 4:51 AMPost by Parth Shorey on September 15, 2015I still don't understand the see saw question? How do you know what formula to apply? 1 answer Last reply by: Professor Dan FullertonThu Apr 9, 2015 6:02 AMPost by Micheal Bingham on April 8, 2015Wonderful Lecture! I have a question concerning example II. I get a little bit confused to where I place my angles when problem-solving, how do we know what angle in our vector component triangle equals to 45? I drew transversal lines and concluded the angle adjacent to the right angle equates to 45. 1 answer Last reply by: Professor Dan FullertonFri Mar 20, 2015 6:55 PMPost by Mohsin Alibrahim on March 20, 2015Professor Fullerton In example 7, how did you determine that the force 1kg is in the middle ? 1 answer Last reply by: Professor Dan FullertonThu Nov 6, 2014 10:37 AMPost by Scott Beck on November 6, 2014Do you mean torque for thumb in the third bullet point on the slide about "Finding Direction Using the Right-Hand Rule"

### Torque

• Torque is a force that causes an object to turn. It is the rotational analogue of force.
• Torque must be perpendicular to the displacement to cause a rotation.
• The further away the force is applied from the point of rotation, the more leverage you obtain. This distance is known as the lever arm.
• The direction of the torque vector is perpendicular to both the position vector and the force vector, and can be found using the right hand rule.
• Positive torques cause counter-clockwise rotational acceleration, and negative torques cause clockwise rotational acceleration.
• Similar to Newton’s 2nd Law (Fnet=ma), the net torque is equal to the product of an object’s moment of inertia and its angular acceleration.
• Static equilibrium is a condition in which the net force and net torque acting on an object are zero, and the system is at rest. Dynamic equilibrium is a condition in which the net force and net torque are zero, but the system is moving at constant translational and rotational velocity.

### Torque

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 0:06
• Torque 0:18
• Definition of Torque
• Torque & Rotation
• Lever Arm ( r )
• Example: Wrench
• Direction of the Torque Vector 1:45
• Direction of the Torque Vector
• Finding Direction Using the Right-hand Rule
• Newton's 2nd Law: Translational vs. Rotational 2:20
• Newton's 2nd Law: Translational vs. Rotational
• Equilibrium 3:17
• Static Equilibrium
• Dynamic Equilibrium
• 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
• Example VIII: Part B
• Example VIII: Part C
• Example VIII: Part D

### Transcription: Torque

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

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

Our objectives include calculating the torque on a rigid object and applying conditions of equilibrium to analyze a rigid object under the influence of a variety of forces.0007

Let us start off by defining torque.0018

Torque which gets the symbol of a big letter co is a force that causes an object to turn and it is a vector.0020

Torque must be perpendicular to the displacement in order to cause a rotation.0027

And the further away the force is applied from the point of rotation, the more leverage it obtains.0031

This distance is known as the lever arm r.0036

If you look here, for an example on a wrench using that to turn this piece over here, replying the force at some angle θ with B we call the line of action.0040

The distance from the center of our rotation to where we are applying the force is our lever arm r and0051

the only force that is really going to matter here is this piece of the force, the one that is perpendicular to the line of action.0056

That is going to be F sin θ if we draw that over here maybe it will be a little easier to see F sin θ at some distance r when we are trying to find the torque.0065

You probably know by experience if you try and apply a lot of force here, that is not going to do a whole lot.0075

Apply that same force further away, you get more rotation that is because you have more torque.0081

Torque is we the r vector, the distance from that point where you are applying the force, that vector, crossed with your force vector.0088

Let us do a number of cross products if we want the magnitude of the torque that is going to be r F sin θ.0097

Now the direction of the torque vector again is a little bit counterintuitive.0104

It is perpendicular to both the position vector r and the force vector f.0109

You find the direction using the right hand rule, point the fingers of your right hand in the direction of the line of action you are0114

and then bend your fingers in the direction of the force.0122

Your thumb points in the direction of the positive torque, that is the direction of your torque vector.0124

Positive torques causes counterclockwise rotation and negative torques cause clockwise rotation according to the standard sign convention.0130

Where this starts to become really interesting, we have been doing all these parallels between translational rotational motion,0141

from velocity to angular velocity, from mass inertial mass to rotational inertia.0148

We have another one of those now for torque, net force in the translational world corresponds to net torque in the rotational world.0154

Newton’s second law of the translational version, net force = mass × acceleration.0164

In the rotational world, net torque= moment of inertia × angular acceleration.0170

Just point out the parallels, linear acceleration to angular acceleration, inertial mass to rotational inertia or moment of inertia, and force to torque.0175

That is going to allow us to solve and analyze a whole new set of problems and situations.0189

Let us go over equilibrium again because we are going to start with some equilibrium problems.0198

Static equilibrium implies that the net force and the net torque of an object is 0 and the system is at rest.0203

Dynamic equilibrium implies that the net force and net torque is 0, the system is moving at constant translational and rotational velocity.0211

It is moving but no net force or net torque.0220

Let us start off with a see saw problem, the 10 kg tortoise, that is a big tortoise, sits on a see-saw 1m from the fulcrum,0225

where must the 2 kg hare sit in order to maintain static equilibrium?0234

What is the force on the fulcrum?0238

Let us draw a diagram here of our seesaw first and we will put some fulcrum there.0240

We know that our tortoise, its 1m from the fulcrum, so that distance there will be 1m.0248

Over here, at this end, we are going to have our tortoise.0253

Let us see, that means that the force from the tortoises is going to be its force due to gravity MG 10 kg × G we could just write this as 10 G for the force.0260

We have got a 2kg hare where does it have to sit to maintain equilibrium?0274

We will say that is going to be somewhere over here, we do not know exactly where that is going to be.0279

Its force is going to be 2 G and we will call this distance x.0284

For the purposes of this problem, we will ignore the mass of the fulcrum itself, its mass is the perfect fulcrum, the magic fulcrum.0290

In order to solve this, one of the things I'm going to look at first is, understanding that it is in equilibrium, it is not rotating if they are balanced.0299

Therefore, we can write that the net torque which is equal to moment of inertia × α must equal 0.0310

We can replace our torques with the net torque with the some of our torques.0320

We have over here a 10 G force at a distance 1 m that is in the counterclockwise direction so that would be a positive torque0324

that is going to be from our tortoise, the force 10 G × its distance at which it acts 1 m and its perpendicular.0334

We do not have to worry about that angle component.0342

We have, due to our hare, we have a clockwise torque and that will be negative, so minus the force 2 G × the distance from our center of rotation x all of that has to equal 0.0346

I have 10 G-2 Gx = 0 or 10 G = 2 Gx, x must equal 5m.0363

We have a follow-up question, what is the force on the fulcrum?0377

For that we can look at Newton’s second law in the translational world, net force = mass × acceleration equal 0.0380

We look at our forces, we have over here, if we call up positive, we have -10 G from our tortoise.0388

We have -2 G from our hare and we have some force up from our fulcrum so + the force of our fulcrum and all that must equal 0.0396

Therefore, the force of our fulcrum must equal 12 G which is going to be 12 × G 10 m / s² is going to be 120 N.0410

A fairly straightforward example but we will do some more here.0428

Let us take a look at a beam, we have a beam of total mass M and length L, with the moment of inertia about its center of ML² / 12.0431

The beam is attached to a frictionless hinge and angle of 45° and allowed to swing freely.0441

Find the beams angular acceleration.0447

The first thing I notice is it is giving us the moment of inertia about the center point not about the hinge.0451

If you remember the moment of inertia of a uniform rod about the end, you could use that but let us just get some practice with a parallel axis theorem.0457

Let us say that the moment of inertia about the N is the moment of inertia about the center of mass + mass × the shift² where this will be our distance D.0465

That is going to be, we have ML² / 12 + M D is L /2².0479

Put that together and I end up with 1/3 ML² so that is the moment of inertia in the current configuration we have for our beam here.0490

Let us define a couple of things as we look at the problem.0500

We said this is distance D, we have here the force of gravity on the center of our beam where it acts which is MG.0503

But if we are looking at torques, only the piece that is perpendicular to our line of action counts.0514

We are really after that component, that is going to be MG cos θ because that is our angle θ at 45° which matches our angle θ over here.0522

We have got that figure out, MG cos θ, we know our distance D, we can go write our Newton’s second law equation for rotational motion.0537

Net torque = moment of inertia × angular acceleration and I look at our net torques we have, let us see at the clockwise direction,0546

some negative we have -MG cos θ, that force × the distance of which it acts over 2 must be equal to Iα.0557

Which implies then that α must be equal to -MG cos θ L /2 × the moment of inertia is going to be -MG cos θ L/2.0572

We found our moment of inertia over here was 1/3 ML² so that is going to be 1/2 ML².0590

With just a little bit of simplification here, we have got an L, we got an L², we have M and M we can cancel out, that gives us -3 G cos θ/2 L.0601

There is our angular acceleration using the parallel axis theorem to find the moment of inertia and Newton’s second law for rotation.0625

We have done some Atwood problems with the ideal pulleys, now let us talk about real pulleys.0635

We have a light string attached to a mass M wrapped around a pulley that has some mass Mt and radius R, find the acceleration of the mass.0640

Alright, to do this what I'm going to start by drawing my pulley.0651

There it is, it has some radius R and the forces acting on it, in the places where they are acting,0655

if that is our tangent T, we have T acting that direction, we have the weight of the pulley.0664

Mass of the pulley × the acceleration due to gravity and we must have some force of the pivot here, a normal force that is acting up.0672

There is our pulley diagram.0681

Starting there, let us take a look at Newton’s second law, net torque = Iα.0686

As I look at our torque, our torque is going to be, we have T at a distance R and that is perpendicular.0697

Our torque is going to be RT, I'm going to worry about magnitudes for now.0704

Our moment of inertia for a disk is ½ MR² so our moment of inertia is going to be ½ MT R².0709

Our torque, our T = ½ MT R² × α of course, which implies then that our tension T divide R from both sides is going to be equal to ½ MT Rα.0721

But we also are looking for linear acceleration, we get angular acceleration, remember that α is equal to A /R or A= Rα.0741

Which we can replace Rα with A to find that our tension is ½ mass of our pulley × A.0752

Alright, now let us draw a free by the diagram for our mass here.0761

We have our object, we have our tension up and we have force of gravity down.0765

In writing Newton’s second law, we called down the positive y direction MG - T must equal MA or MG - we know our tension now is ½ MPA must equal MA or MG = ½ MPA + MA.0775

I can pull out an A there to find that acceleration is going to be equal to MG /M + MP /2.0804

We found the acceleration of the mass now that we have a real pulley that has some mass and rotational inertia.0817

Looking a little bit more detail at torque, we have a system of 3 wheels fixed to each other that is free to rotate about an axis through its center here.0828

Forces are exerted on the wheels as shown, what is the magnitude of the net torque on the wheels?0836

Our net torque is just the sum of all our individual torques.0843

Let us add those up, starting with this one up here.0847

We have a force of 2 F acting at a distance of 2R and it is perpendicular so we have cos IR my sin of 90° which is 1.0849

Since it is causing a clockwise torque, let us make sure we call that negative.0862

We also have a force over here of 2 F and a distance 1.5 R still 90° but this one is in the counterclockwise direction so that is positive.0867

We have a 2 F force that looks like it is at 1R.0880

Let me draw that a little bit more carefully × 1R and we have our 3 F force which is operating at a radius of 1.5 R,0886

also causing a counterclockwise rotation so that is positive.0896

Our net torque M is -4 FR + 2 FR + 3 × ½ is 4.5 FR or net torque= 6 ½ - 4 2.5 FR.0900

Let us do a ranking test, a constant force F is applied for 5s at various points of the uniform density object below.0928

Rank the magnitude of the torque exerted by the force on the object about an axis located at the center of mass from smallest to largest.0937

We are going to have the greatest torque when we are the furthest away and most perpendicular.0946

As I look at these different spots, it looks like we are going to have our maximum torque when we start with the minimum,0952

when we are applying net force right where the center B, then we will go to C, then we will go to A, because that is an angle.0958

Finally D, because we got that one that is perpendicular or most close to perpendicular compared to the axis of rotation here.0966

So B, C, A, D, would give us the ranking of the torque from smallest to largest, assuming we are rotating about that point in the center.0976

We can also look at ranking angular acceleration.0985

A variety of masses are attached to different points to a uniform beam attached to a pivot, write the angular acceleration of the beam from largest to smallest.0988

If we want the largest angular acceleration, we want the most force, the furthest away from the axis of rotation.0997

That is going to be at D, where we have 2 M at the very end and then M right beside it.1006

D will be the most then it looks like C is the next most, we have got M at the very end and 2 M just inside that.1010

Then, it looks like we are probably looking at A where we have 2 M at the very end and finally we have B 3 M half of the distance.1020

D, C, A, B would give us the greatest angular acceleration from largest to smallest because we are looking at the ranking of the torques.1028

Let us do a cafe sign example, a 3 kg cafe sign is hone from a 1 kg horizontal pole as shown and a wire is attached to prevent the sign from rotating.1039

We are trying to find the tension in the wire.1049

Let me just redraw that a little more simply over here and use a ruler just to make things nice and neat.1053

If there is our pull that goes with their sign, it looks like it is a 4 m long pull so 1, 2, 3, 4 m.1062

As I look at the different forces acting on it, it looks like we have a force that is 1 kg.1071

The force from the center of mass, its gravitational force is going to be 1 kg × G or 1 G at the center.1079

We have a 3 kg mass that is right over here, so that will be 3 G and we have a tension from the wire over here.1087

We will draw that at the very end where it is acting.1101

There is our tension and that angle right there is 30°.1104

Since, it is an equilibrium we know the net torque must be 0 so we will start there.1110

Net torque equal 0 which implies then, as we end add up the torques that is going to be we will have T sin 30 × the distance over which it acts,1116

over that 4 m, that is counterclockwise and we will call that positive.1130

We also have -3 G acting at 3m, negative because it is causing a clockwise torque.1135

We have a -1 G at 2m negative because it is also causing a clockwise torque.1144

Putting this together, sin 30 is 2, so that will be 2 T.1152

We have 2 T – 9 G - 11 G so T is going to be equal to 11 G /4 sin 30 or 2 which is going to be 54 N.1157

Let us finish up by looking at an old AP problem.1178

Here we have the 2008 free response number 2 problem, you can find it here at the link on top.1185

This looks mighty familiar, we got a horizontal rod with some length and mass.1198

The left of the rod is attached to the hinge and we got a spring scale attached to our wire in order to determine the tension in the wire.1203

First thing we are asked to do is to diagram, draw and label vectors to show all the forces acting on the rod.1212

Let us start by drawing a rod here, something like that.1220

And as I look, we are going to have the weight of the rod itself MG.1225

We are going to have the weight of our block on the end, we will call that mg.1230

We have a tension here at some angle 30° and we also have a force from the hinge which in order to balance all this out must be going somewhere up into the right.1238

That is the force of our hinge.1253

M we will call 2kg, m is 0.5 kg and the whole thing has a length of 0.6 m.1259

There is A, looking at part B, calculate the reading on the spring scale.1270

The net torque has to be equal to 0 which implies that let us add up our torque, we have got a TL sin 30.1278

We are going to assume that pivot is around here so that length is L – we will have mgl negative1288

because it is causing a clockwise torque - MGL /2 with a mass of our bar causing its torque, its force.1299

All that has to equal 0.1308

Therefore, our tension must equal, we have G L / M × M + M /2 divided by L sin 30° which implies that our tension must be L sin 30 is ½1310

that will be 2 G, L/L cancel out × M + M /2 which is going to be 2 × 10 m/s² × our little mass 0.5 + M/2 2kg/ 2 is 1kg so 20 × ½ is going to be 30 N.1329

There is part B, moving on to part C, the rotational inertia of a rod about its center is 1/12 ML² where M is the mass of the rod and L is its length.1355

Find rotational inertia of the rod block system about the hinge.1366

For C, the moment of inertia of our system is going to be the moment of inertia of the rod + the moment of inertia of the block.1372

We can find the moment of inertia of the rod about its center of mass is 1/12 ML² if we want that about the hinge, we can use the parallel axis theorem.1381

Just in case you did not remember what the moment of inertia of a rod is about its end.1394

The moment of inertia of the rod about the hinge is going to be the moment of inertia of the rod about its center of mass + Md²1399

because we got a parallel axis and our initial was about the center of mass.1408

That is going to be 1/12 ML² + M × L/2² which is ML² /12 + ML² /4 or the moment of inertia of the rod about the hinge is just ML² /3.1412

We know the moment of inertia of the block is ml².1434

When I put that all together, the moment of inertia of the system is 1/3 ML² + ml² which is going to be L² × M /3 + m1441

which is going to be 0.6² × 2kg /3 + ½ kg = 0.42 kg m².1459

One more part to the problem, part D, if the cord that supports the rod is cut near the end of the rod, calculate the initial angular acceleration of the rod block system.1477

Net torque = moment of inertia × angular acceleration.1491

Therefore, our angular acceleration is the net torque/ the moment of inertia which is MGL + MG L /2 all divided by the moment of inertia.1497

We no longer are worried about that tension, the wire, because we cut it.1513

Which is going to be equal to, we can factor out a GL /I m + M /2 which is going to be 10 m/s² × length 0.6/moment of inertia 0.42 kg m² × 0.5 kg +1518

2/2 is going to be 1, for a total angular acceleration of 21.4 radiance/s².1540

Hopefully, that gets you a good feel for torque and Newton’s second law for things that are rotating.1555

We will get more into that in our next lesson on rotational dynamics.1562

Thank you for watching www.educator.com.1565

We will see you again soon and make it a great day everyone.1567

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