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Dan Fullerton

Dan Fullerton

Energy & Conservative Forces

Slide Duration:

Table of Contents

I. Introduction
What is Physics?

7m 12s

Intro
0:00
Objectives
0:11
What is Physics?
0:27
Why?
0:50
Physics Answers the 'Why' Question
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
Graphical Vector Addition
2:54
Graphical Vector Addition
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
II. 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
Example XV: Quadratic Solution
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
Radians and Degrees
0:32
Degrees
0:35
Radians
0:40
Example I: Radians and Degrees
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
III. 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
IV. 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
V. 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
VI. 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
VII. Rotational Motion
Rotational Kinematics

32m 52s

Intro
0:00
Objectives
0:07
Radians and Degrees
0:35
Once Around a Circle: In Degrees
0:36
Once Around a Circle: In Radians
0:48
Measurement of Radian
0:51
Example I: Radian and Degrees
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
VIII. 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
IX. 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
X. 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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Lecture Comments (12)

4 answers

Last reply by: Parth Shorey
Sun Oct 4, 2015 9:10 PM

Post by Parth Shorey on October 1, 2015

How come you didn't factor in g on example II?

1 answer

Last reply by: Professor Dan Fullerton
Tue Apr 7, 2015 7:42 PM

Post by Luvivia Chang on April 6, 2015

Hello Professor Dan Fullerton.
I have difficulty determining whether there is a "-"infront of the whole formula in both mechanics and E&M. Every time I use a formula, I have to think for a long time about "positive or neegative?"
What can I do?

1 answer

Last reply by: Professor Dan Fullerton
Tue Nov 18, 2014 8:32 PM

Post by Thadeus McNamara on November 18, 2014

how did you get from 16:05 to 16:10? (change in U to just Ug)
also 18:10 to 18:15 (change in U to just Us)

0 answers

Post by Thadeus McNamara on November 18, 2014

^^^ this is at around the 16  minute mark

1 answer

Last reply by: Professor Dan Fullerton
Tue Nov 18, 2014 8:31 PM

Post by Thadeus McNamara on November 18, 2014

why is the integration between r and infinite? Shouldn't the infinite be first and then the r second since infinite is bigger?

Energy & Conservative Forces

  • Energy is the ability or capacity to do work. Work is the process of moving an object. Therefore, energy is the ability or capacity to move an object.
  • Energy can be transformed from one type to another. You can transfer energy from one object to another by doing work.
  • The work done on a system by an external force changes the energy of the system.
  • Kinetic energy is energy of motion, or the ability or capacity of a moving object to move another object.
  • Potential energy (U) is energy an object possesses due to its position or state of being.
  • A single object can have only kinetic energy, as potential energy requires an interaction between two or more objects.
  • The internal energy of a system includes the kinetic energy of the objects that comprise the system and the potential energy of the configuration of the objects that comprise the system.
  • Changes in a system’s internal structure can result in changes in internal energy.
  • Conservative forces are forces in which the work done on an object is independent of the path taken. Alternately, a force in which the work done moving along a closed path is zero. The work done by a conservative force is directly related to a negative change in potential energy W=-ΔU.
  • For a conservative force, the component of force along a line l is equal to the opposite of the derivative of the potential energy with respect to that line.

Energy & Conservative Forces

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

  • Intro 0:00
  • Objectives 0:08
  • Energy Transformations 0:31
    • Energy Transformations
    • Work-Energy Theorem
  • Kinetic Energy 1:12
    • Kinetic Energy: Definition
    • Kinetic Energy: Equation
  • Example I: Frog-O-Cycle 2:07
  • Potential Energy 2:46
    • Types of Potential Energy
    • A Potential Energy Requires an Interaction between Objects
  • Internal energy 3:50
    • Internal Energy
  • Types of Energy 4:37
    • Types of Potential & Kinetic Energy
  • Gravitational Potential Energy 5:42
    • Gravitational Potential Energy
  • 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
    • Type of Conservative Forces
    • Types of Non-conservative Forces
  • Work Done by Conservative Forces 13:28
    • Work Done by Conservative Forces
  • Newton's Law of Universal Gravitation 14:18
    • Gravitational Force of Attraction between Any Two Objects with Mass
  • Gravitational Potential Energy 15:27
    • Gravitational Potential Energy
  • Elastic Potential Energy 17:36
    • Elastic Potential Energy
  • Force from Potential Energy 18:51
    • Force from Potential Energy
  • Gravitational Force from the Gravitational Potential Energy 20:46
    • Gravitational Force from the Gravitational Potential Energy
  • Hooke's Law from Potential Energy 22:04
    • Hooke's Law from Potential Energy
  • Summary 23:16
    • Summary
  • 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

Transcription: Energy & Conservative Forces

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

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

Our objectives include defining energy, describing various types of energy,0008

talking about some alternative definitions of conservative force,0014

describing some examples of conservative and non conservative forces.0019

Finally using a relationship between force and potential energy to find forces and potential energy functions.0023

Let us dive right in by talking about what is energy.0030

Energy is the ability or capacity to do work.0033

Work is the process of moving object.0037

If we put those together energy is the ability or capacity to move on the object.0040

Energy can be converted to different types.0047

It can be transformed from one type to another.0049

You see that all the time.0051

You transfer energy from one object to another by doing work.0053

We talked about the work energy theorem in our last lesson.0057

The work done on a system by an external force changes the energy of that system.0060

We are going to be talking about lots of energy transformations in the next couple lessons0066

but one of those is going to come up again and again is kinetic energy.0071

Kinetic energy is energy of motion or if energy is the ability or capacity to move an object.0075

Kinetic energy that is the ability or capacity of a moving object to move another object.0081

Something that is moving has the ability to move something else.0088

The baseball coming toward your nose with a bunch of Velocity has kinetic energy.0091

It has kinetic energy because it has the ability to move something else.0096

Basically the bones in your nose.0100

If that hits your nose it is going to squish it up.0101

It is going to cause something else to move.0103

Kind of a messy example but that is the idea.0106

Kinetic energy is the capacity or ability of that moving object to move something else.0108

The formula for kinetic energy is ½ mass × the square of your speed.0116

Units of course energy Joules.0122

Let us take a look at a quick example here.0126

A frog speeds along a frog o cycle at a constant 30 m/s.0128

If the mass of the frog and motorcycle is 5 kg find the kinetic energy of the frog cycle system.0133

Kinetic energy is ½ MV² to be ½ × the mass 5 kg × our speed 30 m/s² 900 × 5=4500.0141

½ of that is going to be 2250 joules.0155

Let us talk now about potential energy.0166

Potential energy which gets the symbol capital U is energy an object possesses based on its position or its speed of being.0168

There are lots of different types of potential energy.0177

Gravitational potential energy will talk about quite a bit in this course.0180

It is the energy an object possesses because of its position in the gravitational field.0183

Elastic potential energy we talked about a little bit with Hooke's law already.0189

The energy that you have from some sort of elastic displacement something like a spring or an elastic band.0193

Chemical potential energy, electric potential energy, nuclear potential energy, all different forms of potential energy.0202

It is important to point out that a single object in isolation can only have kinetic energy.0210

In order to have some from a potential energy you need to have an interaction between objects.0218

You need to have at least two objects in your system or something they have potential energy.0222

As we talk about these we are also going to run across the term internal energy every now and then.0230

The internal energy of the system includes the kinetic energy of the objects that make up that system0235

and potential energy of the configuration of the objects that make up the system.0241

For example if you think of the temperature, the heat and temperature of an object that0246

is based on the speed with which the molecules inside are moving around.0253

That is an internal energy because it includes the kinetic energy of the objects comprising that system.0258

We can make changes internally that system.0264

The changed the internal energy of the object.0267

Change in a systems internal structure can result in changes in the internal energy.0271

Let us see if we can put some of us together as we talk about different types of energy.0277

I like to break energy up in the two main types potential and kinetic.0281

Some of these cross the boundary between each one.0285

If we talk about electrical energy depending on how you are looking at moving charges create current.0288

Moving there is kinetic energy.0296

Light moving photons as they do not have mass but they do have energy.0299

Light is another goofy one.0304

We could talk about is kinetic energy.0305

Wind is moving air molecules.0308

Thermal energy the motion of molecules and atoms making up of an object.0310

Sound vibrating air or vibrating molecules and waves.0316

Potential side we have chemical potential energy, gravitational potential energy, we can talk about electrical potential energy,0322

in terms of voltage charges held at different levels.0329

Nuclear potential energy, elastic potential energy, and tons of others.0335

Let us start with gravitational potential energy Ug.0343

If we have some object we will start here at some arbitrary point we will call y = 00347

and we take it across a meandering path until eventually gets to another position.0354

If this is our +y direction well we have changed its gravitational potential energy.0360

How much if we change it?0366

Well let us take a look.0368

The work done in moving it there which should be the amount of potential energy that has its final point.0370

As we are dealing with the conservative force we will talk about that in a minute.0377

As we go from y=0 to Y equals some final value let us call that the h level.0381

So that is h.0389

F • dr define their work which would be the integral from 0 to h are force.0391

To lift that up we have to overcome the force of gravity so that is going to be MG and DR.0398

We are worried about the Dy position.0405

MG should be constant as long as we are in the same relative level, the same gravitational field if we are going miles and miles up.0409

A very big distance into the atmosphere G might change.0417

As long as we are relatively close to use pretty constant.0420

We will pull those out MG we should say work equals MG integral from 0 to H.0423

Dy which is just going to be MGH.0430

As long as are the constant gravitational field gravitational potential energy can be written as MGH you have probably seen that before.0434

Let us take a look and example here.0447

In the diagram we have 155 N box on a ramp.0449

Applied force F causes the box to slide from point A to point B.0453

What is the total amount of gravitational potential energy gained by the box?0458

As I look at this change in gravitational potential energy is going to be MG Δ h or change in height.0464

Which is going to be 155 N that describes its weight or the force of gravity MG on it.0475

Δ H and goes up 1.8 m.0481

That would be 279 joules pretty straightforward.0486

Let us take a look at graph in question.0496

The hippopotamus is throwing vertically upward.0498

I do not know why and I really do not know how.0500

Which pair of graphs best represent the hippos kinetic energy in gravitational potential energy as functions of its displacement while it rises?0503

As it goes higher and higher as it rises, it is going to slow down so we have to expect that we are going to see kinetic energy decreasing.0513

On the other hand as it gets higher and higher its gravitational potential energy should be increasing.0521

The kinetic energy is being transformed into gravitational potential energy.0527

Which graph we have that displays that the best?0532

That looks like one as we get further and further displacements on its way up kinetic energy goes to 0 at its highest point where it stops.0535

That is where we have our maximum of gravitational potential energy.0541

The answer must be 1.0545

Looking at a slightly more involved question.0549

A pendulum of mass M swings on the light string of length L.0551

If the mass hanging directly down is set to 0. Of gravitational potential energy0556

find a gravitational potential energy the pendulum as a function of θ and L.0561

It looks like if I break this up a little bit or really doing is we are changing the height of our pendulum as it swings from there to there.0568

There is our change the Δ Y.0586

We have to figure out what that is.0591

To do that I am going to take a look at my triangle here and realize that our hypotenuse here is going to be equaled L.0594

We got the adjacent side of our angle right here and this would be the opposite side.0601

To help figure out that height let us say that the cos θ SOHCAHTOA.0607

Cos is adjacent over hypotenuse.0614

That is a adjacent over hypotenuse which implies that this adjacent side, this piece right here from there to there is going to be the hypotenuse × cos θ.0618

Since our hypotenuse is the length of our string L that is going to be L cos θ.0638

Our Δ y is going to be the entire length L - adjacent side which is L cos θ.0646

I'm going to find that Δ y is going to be L × (1 - cos θ).0660

If I wanted a change in potential energy I can write Δ Ug = Mg Δ Y.0667

Which would be Mg × L 1 - cos θ.0677

We have our gravitational potential energy as a function of θ and L for our swinging pendulum.0689

Let us talk for a minute about conservative forces.0698

A force in which the working on an object is independent of the path is known as a conservative force.0701

Gravity for example.0706

If I left a pen straight up or go to the side and run it around all over the place, the only thing that matters0707

for the total change in the energy or the total work done is the initial and final points.0714

You can also write that define a conservative forces of force in which the work done moving along the close path is 0.0721

As long as you come back to where you start whoever you get there and net work done must be 0.0727

Or it is a force in which the work that is directly related to a negative change in the potential energy.0733

The work that is equal to the opposite of the change in the potential energy.0738

Some examples of these.0743

Let us talk about some conservative forces and some non conservative forces.0746

Conservative forces would be things like gravity.0762

We are only worry about the initial and final points or elastic forces.0767

When we talk about electricity in the following course on ENM columbic or electric force.0775

Non conservative forces are one you typically think of as the law C forces for example.0783

Friction the energy is ever destroy your loss that can be converted less useful forms.0789

Friction drag forces, retarding forces, air resistance, fluid resistance, non conservative forces.0795

There are some properties of conservative forces.0807

For the work done by conservative forces, let us say it is a work done by a conservative force0811

is equal to the opposite of the change in its potential energy.0817

Which implies then that Δ U equals the opposite of the work done by the conservative force.0822

Which implies that the Δ U equals the opposite of the integral from some initial position to some final position of F•Dr.0833

And we can use this in a bunch of different ways.0849

We are going to apply it with a couple different types of conservative forces in the next few slides.0851

Let us take a look at in the context of newton's law the universal gravitation.0857

You have probably seen this before and other courses.0861

I do not know if it is formally introduced here yet but the gravitational force between two objects is - G universal gravitational constant.0863

It is a fudge factor to make the unit is work out equal to 6.67 × 10⁻¹¹ Nm² / kg².0872

A constant × first mass × second mass divided by the square of the distance between their centers.0882

The r hat just tells you the direction of r hat in the direction of the unit vector from the first object0888

to the second in the negative tells you that it is going to be attractive there.0896

We can look at it as here we have mass 1.0901

Somewhere over here we have a mass 2.0905

The distance between their centers we can draw that right in nice and quick.0908

There it is that would be our r vector.0916

R hat would just be a unit vector in that direction.0919

If we start with a Newton’s law of universal gravitation we can use what we just found out about conservative forces to find a potential energy.0926

The universal gravitational potential energy.0935

Change in potential energy is minus the work done by conservative forces is going to be - the work done by gravity0941

or - the integral as we go from infinity to r some point of - GM 1 M2 /r² with respect to r.0950

As I do this integration I can pull my constant out.0964

Potential energy due to gravity is going to be GM 1 M 2 should not change for the purposes of this problem.0969

This will be GM1 M2 our negative and negative cancel out.0977

Integral from infinity to some r of DR/ r² or r⁻² Dr.0982

Let us write it that way just to make it a little easier to see.0991

GM 1 M2 integral from infinity to r of r⁻² DR is going to be GM 1M2.0993

The integral of r⁻² is going to be 1/r.1003

Evaluated from infinity to r so potential energy due to gravity is just going to be GM 1M2 with 1.1008

We will plug that in - GM 1M21 1/ infinity will be 0.1024

I'm going to come up with - GM 1 M2 /r.1029

And that is our formula for gravitational potential energy between two objects separated by some distance r.1041

We are able to derive it knowing the force because it is a conservative force.1050

We can do the same thing with the elastic potential energy.1055

Remember the force in the spring by Hooke’s law – KX.1060

The work done is negative change in potential energy and that is a conservative force or - the integral from 0 to X of F• DR1065

as we are finding the work done which is - the integral from 0 to X of – KXDX.1081

The potential energy stored in the spring is the integral from 0 to X of KXDX which is K plot a constant integral from 0 to X.1092

0 to X of XDX or K × X²/2 evaluate from 0 to X which is just going to end up being ½ KX 2.1105

There is a potential energy stored in the spring.1120

If we can go from force to potential energy we should be able to get force from potential energy.1125

Let us take a look at that.1132

As soon as we have an object on some path dr or we could also call this DL that might be just as common dl.1134

As it travels along this path some forces acting on a net force can be changing.1141

If we want to find V potential energy to this force along the path what we can do is break it up1147

and these little tiny pieces DL and find the potential energy for each one of those by finding the work done through each one of those.1155

The differential potential energy that tiny amount of potential energy is the opposite of that little tiny bit of work done by the conservative force.1166

Or - F dotted with will use dl there which if we want to force in the direction DL this is going to be - F cos θ DL.1168

If you remember our dot product definition.1192

What we are going to do is we are going to call this F cos θ we are going to call that the force in the direction of L.1196

We will call that fl.1202

Once we define that the little tiny bit of energy, potential energy is - F cos θ DL which we could also write as – FLDL.1205

That means that FL the force in the direction of that displacement is just going to be - du DL.1224

The opposite of the derivative of the potential energy function with respect to L gives you that force.1234

We can go from potential energy to force.1241

Let us do that with gravity.1246

FL equals - du DL and their potential energy function for gravity is - GM 1M2/r.1250

Let us say we are looking for force in the direction r.1263

Force r equals - the derivative with respect to R of our potential energy function - GM 1M2/r.1267

We can pull out our constants GM 1M2 × the derivative with respect to r of 1/r1277

which means that the force in the direction of R is going to be GM 1M2.1288

The derivative of 1/ R is -1/ r² or - GM 1M2 /r² which is what we had initially force = - GM 1M2/r² in that direction of r hat.1294

We have gone from force to potential energy.1315

We can go back the other way using this formula.1318

Let us take a look at Hooke’s law.1323

Here we are going to take a look at a spring and we are going to give its potential energy function as ½ KX².1326

As you extend it to the right you have more potential energy.1335

The area under the graph from Ug and no kinetic.1340

If you are let it go with the object, it gets to its center pointed it no longer has elastic kinetic energy instead it so elastic potential energy.1346

It is all kinetic.1354

We get to the other side it slows down and stops with all potential again it stretch out or compressed spring.1355

And then you come back.1364

You have this constant back and forth potential kinetic energy.1366

We can look at this in terms of the force realizing that - du DL =Fl.1372

The force in the direction of the L is – d/dx for spring of ½ KX² which is just going to be – KX.1380

We have gone the other way again.1392

To put a quick summary together.1396

If we start with potential energy something like gravitational potential energy is MGH.1398

We can look at the force - Du DL.1407

A force the derivative of this with respect to h is going to be – MG.1415

There is our formula for weight.1425

Assuming you are in a constant gravitational field.1428

If you want to get more general with Newton’s universal gravitation, potential energy due to gravity is - GM 1M2/r.1431

The force using - dudl is going to be - GM 1M2/r².1441

Newton’s law of universal gravitation.1449

Or if we start with the spring where the stored potential energy this spring is ½ KX² in a linear spring1452

that follows Hooke’s law we can find the force by taking the opposite of the derivative that with respect to x which is – KX.1460

We can go that direction.1470

We previously went that direction.1474

Let us do a couple examples.1479

5 kg sphere’s position is given by the function x of T = 3T³ -2T.1482

The term in the kinetic energy of the sphere at time T = 3s.1488

Kinetic energy is ½ MV².1496

It would be helpful to know the velocity.1500

If we know x (t) is t3³- 2t the velocity is the first derivative of x which will be 9t² -2.1503

The velocity of t = 3s is just going to be 9 × 9=81 – 2 = 79 m/s.1519

We can go back to our kinetic energy function.1528

Kinetic energy is ½ × 5kg × speed 79 m/s or about 15600 joules.1531

An example where we find force from potential energy.1547

Another 5kg sphere’s potential energy U is described by this function U(x) = 4x² + 3x – 2.1550

Determine the force on the particle at x = 2m.1559

Force = -du dl in this case it is going to be – d /dx of 4x² + 3x – 2= -derivative of 4x² 8x.1564

The derivative of +3x is 3, -2 is 0.1581

Which is -8x – 3.1587

If we want the force at x = 2m that is going to be -8 × 2m -3.1591

-16 – 3 = -19 N.1604

One last problem here.1614

Work on a spinning disc.1616

A 2kg disc moves in uniform circular motion on a frictionless horizontal table.1618

Attach to the point of rotation by a 10cm spring but the spring constant of 50 N/m.1624

When stationary the spring has a length of 8cm.1631

While it is turning it is extended 2cm.1634

How much work is performed on the disc by the spring as the disc moves through 1 full revolution?1638

The force in order to have it moving in a circle must be that way.1647

The velocity at any given point in time is that way.1651

They are perpendicular.1654

If we do not have any force in the direction of the displacement you cannot do any work.1657

F cos θ cos 90 is 0°.1663

Therefore the work done is 0.1666

A tricky question there.1673

Hopefully that gets you a good start on energy and conservative forces.1675

Thank you so much for joining us here at www.educator.com.1679

I hope to see you again real soon.1682

Make it a great day everybody.1683

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