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Dan Fullerton

Dan Fullerton

Simple Harmonic Motion

Slide Duration:

Table of Contents

I. Introduction
What is Physics?

7m 38s

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

24m 12s

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

25m 5s

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

30m 11s

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

36m 13s

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

20m 32s

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

10m 52s

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

10m 16s

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

34m 55s

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

5m 58s

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

17m 49s

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

35m 27s

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

26m 6s

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

21m 59s

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

7m 18s

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

26m 37s

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

32m 56s

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

15m 33s

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

11m 21s

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

36m 6s

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

31m 20s

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

20m 15s

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

23m 20s

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

58m 30s

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

19m 48s

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

18m 7s

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

7m

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

20m

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

24m 17s

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

24m 15s

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

22m 29s

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

38m 24s

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

35m 58s

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

21m 14s

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

10m 35s

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

8m 44s

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

48m 58s

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

24m 47s

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

19m 48s

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

21m 29s

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

17m 26s

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

26m 41s

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

20m 45s

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

19m 2s

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

11m 35s

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

24m 32s

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

39m 42s

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

23m 47s

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

14m 21s

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

15m 47s

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

38m 1s

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

37m 49s

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

16m 53s

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

9m 20s

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

18m 12s

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

3m 53s

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

7m 6s

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

6m 48s

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

8m 16s

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

7m 56s

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

4m 17s

Intro
0:00
Question 1
0:13
Question 2
0:45
Question 3
1:25
Question 4
2:00
Question 5
2:32
Question 6
3:38
Newton's 1st Law

4m 34s

Intro
0:00
Question 1
0:15
Question 2
1:02
Question 3
1:50
Question 4
2:04
Question 5
2:26
Question 6
2:54
Question 7
3:11
Question 8
3:29
Question 9
3:47
Question 10
4:02
Newton's 2nd Law

5m 40s

Intro
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Question 1
0:16
Question 2
0:55
Question 3
1:50
Question 4
2:40
Question 5
3:33
Question 6
3:56
Question 7
4:29
Newton's 3rd Law

3m 44s

Intro
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Question 1
0:17
Question 2
0:44
Question 3
1:14
Question 4
1:51
Question 5
2:11
Question 6
2:29
Question 7
2:53
Friction

6m 37s

Intro
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Question 1
0:13
Question 2
0:47
Question 3
1:25
Question 4
2:26
Question 5
3:43
Question 6
4:41
Question 7
5:13
Question 8
5:50
Ramps and Inclines

6m 13s

Intro
0:00
Question 1
0:18
Question 2
1:01
Question 3
2:50
Question 4
3:11
Question 5
5:08
Circular Motion

5m 17s

Intro
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0:21
Question 2
1:01
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1:50
Question 4
2:33
Question 5
3:10
Question 6
3:31
Question 7
3:56
Question 8
4:33
Gravity

6m 33s

Intro
0:00
Question 1
0:19
Question 2
1:05
Question 3
2:09
Question 4
2:53
Question 5
3:17
Question 6
4:00
Question 7
4:41
Question 8
5:20
Momentum & Impulse

9m 29s

Intro
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0:19
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2:17
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3:25
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3:56
Question 5
4:28
Question 6
5:04
Question 7
6:18
Question 8
6:57
Question 9
7:47
Conservation of Momentum

9m 33s

Intro
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0:15
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2:08
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4:03
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4:10
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6:08
Question 6
6:55
Question 7
8:26
Work & Power

6m 2s

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0:13
Question 2
0:29
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0:55
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1:36
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2:18
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3:22
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4:01
Question 8
4:18
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4:49
Springs

7m 59s

Intro
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Question 1
0:13
Question 4
2:26
Question 5
3:37
Question 6
4:39
Question 7
5:28
Question 8
5:51
Energy & Energy Conservation

8m 47s

Intro
0:00
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0:18
Question 2
1:27
Question 3
1:44
Question 4
2:33
Question 5
2:44
Question 6
3:33
Question 7
4:41
Question 8
5:19
Question 9
5:37
Question 10
7:12
Question 11
7:40
Electric Charge

7m 6s

Intro
0:00
Question 1
0:10
Question 2
1:03
Question 3
1:32
Question 4
2:12
Question 5
3:01
Question 6
3:49
Question 7
4:24
Question 8
4:50
Question 9
5:32
Question 10
5:55
Question 11
6:26
Coulomb's Law

4m 13s

Intro
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Question 1
0:14
Question 2
0:47
Question 3
1:25
Question 4
2:25
Question 5
3:01
Electric Fields & Forces

4m 11s

Intro
0:00
Question 1
0:19
Question 2
0:51
Question 3
1:30
Question 4
2:19
Question 5
3:12
Electric Potential

5m 12s

Intro
0:00
Question 1
0:14
Question 2
0:42
Question 3
1:08
Question 4
1:43
Question 5
2:22
Question 6
2:49
Question 7
3:14
Question 8
4:02
Electrical Current

6m 54s

Intro
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Question 1
0:13
Question 2
0:42
Question 3
2:01
Question 4
3:02
Question 5
3:52
Question 6
4:15
Question 7
4:37
Question 8
4:59
Question 9
5:50
Resistance

5m 15s

Intro
0:00
Question 1
0:12
Question 2
0:53
Question 3
1:44
Question 4
2:31
Question 5
3:21
Question 6
4:06
Ohm's Law

4m 27s

Intro
0:00
Question 1
0:12
Question 2
0:33
Question 3
0:59
Question 4
1:32
Question 5
1:56
Question 6
2:50
Question 7
3:19
Question 8
3:50
Circuit Analysis

6m 36s

Intro
0:00
Question 1
0:12
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2:16
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2:33
Question 4
2:42
Question 5
3:18
Question 6
5:51
Question 7
6:00
Magnetism

3m 43s

Intro
0:00
Question 1
0:16
Question 2
0:31
Question 3
0:56
Question 4
1:19
Question 5
1:35
Question 6
2:36
Question 7
3:03
Wave Basics

4m 21s

Intro
0:00
Question 1
0:13
Question 2
0:36
Question 3
0:47
Question 4
1:13
Question 5
1:27
Question 6
1:39
Question 7
1:54
Question 8
2:22
Question 9
2:51
Question 10
3:32
Wave Characteristics

5m 33s

Intro
0:00
Question 1
0:23
Question 2
1:04
Question 3
2:01
Question 4
2:50
Question 5
3:12
Question 6
3:57
Question 7
4:16
Question 8
4:42
Question 9
4:56
Wave Behaviors

3m 52s

Intro
0:00
Question 1
0:13
Question 2
0:40
Question 3
1:04
Question 4
1:17
Question 5
1:39
Question 6
2:07
Question 7
2:41
Question 8
3:09
Reflection

3m 48s

Intro
0:00
Question 1
0:12
Question 2
0:50
Question 3
1:29
Question 4
1:46
Question 5
3:08
Refraction

2m 49s

Intro
0:00
Question 1
0:29
Question 5
1:03
Question 6
1:24
Question 7
2:01
Diffraction

2m 34s

Intro
0:00
Question 1
0:16
Question 2
0:31
Question 3
0:50
Question 4
1:05
Question 5
1:37
Question 6
2:04
Electromagnetic Spectrum

7m 6s

Intro
0:00
Question 1
0:24
Question 2
0:39
Question 3
1:05
Question 4
1:51
Question 5
2:03
Question 6
2:58
Question 7
3:14
Question 8
3:52
Question 9
4:30
Question 10
5:04
Question 11
6:01
Question 12
6:16
Wave-Particle Duality

5m 30s

Intro
0:00
Question 1
0:15
Question 2
0:34
Question 3
0:53
Question 4
1:54
Question 5
2:16
Question 6
2:27
Question 7
2:42
Question 8
2:59
Question 9
3:45
Question 10
4:13
Question 11
4:33
Energy Levels

8m 13s

Intro
0:00
Question 1
0:25
Question 2
1:18
Question 3
1:43
Question 4
2:08
Question 5
3:17
Question 6
3:54
Question 7
4:40
Question 8
5:15
Question 9
5:54
Question 10
6:41
Question 11
7:14
Mass-Energy Equivalence

8m 15s

Intro
0:00
Question 1
0:19
Question 2
1:02
Question 3
1:37
Question 4
2:17
Question 5
2:55
Question 6
3:32
Question 7
4:13
Question 8
5:04
Question 9
5:29
Question 10
5:58
Question 11
6:48
Question 12
7:39
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Lecture Comments (41)

1 answer

Last reply by: Professor Dan Fullerton
Sun Jan 8, 2017 11:32 AM

Post by Khalid Khan on January 8, 2017

Why do we not use kinetic energy when we calculate total energy in example 2?

1 answer

Last reply by: Professor Dan Fullerton
Sun Jan 8, 2017 11:20 AM

Post by Khalid Khan on January 8, 2017

Can you please explain where you got pi and the square root of m/k for the period and frequency formulas? We're learning this in class, and we haven't gotten to these formulas yet, so I am just curious.

1 answer

Last reply by: Professor Dan Fullerton
Thu Jun 30, 2016 9:22 AM

Post by Peter Ke on June 29, 2016

For example 6, when you say equilibrium position do you mean by when the pendulum faces directly down. If so doesn't it mean that the KE = maximum while PE = 0, so v=sqrt(2gh) does not apply? I think I am confuse please explain.

Thank you!

3 answers

Last reply by: Professor Dan Fullerton
Sat Apr 30, 2016 12:49 PM

Post by Shikha Bansal on April 29, 2016

For example 2b)the total energy was equal to the spring potential energy only, but for 2d) it was equal to the spring potential and the kinetic energy. Is it because .5 meters is the max it can go (since velocity is zero there)? If so, how did you know ,5 m is the maximum displacement?
Also, for 2c) wasn't the 5 Joules of total energy for the situation where the displacement was .5 meters? How come you used it when the displacement was 0 meters (aka equilibrium position)?
Furthermore, for 2c) I get that the kinetic energy is max at the equilibrium position, but shouldn't the spring potential energy equal 0? So why did u write that kinetic energy was equal to the spring potential energy?
Thanks for all the help!

1 answer

Last reply by: Professor Dan Fullerton
Thu Mar 31, 2016 3:38 PM

Post by Zhe Tian on March 28, 2016

Just a technical question but are angular velocity and angular frequency the same?

4 answers

Last reply by: Professor Dan Fullerton
Tue Aug 11, 2015 8:54 PM

Post by Anh Dang on August 10, 2015

In Circular Motion vs SHM, I don't get why you use sin for the equilibrium  position and cosine for the max displacement.  Also, why do you use the y coordinates for sin and not x?

2 answers

Last reply by: Jason Wilson
Tue Dec 9, 2014 5:32 PM

Post by Jason Wilson on December 6, 2014

Example 3:  I may be simply typing incorrectly on the calculator or possibly missing an entire step.  The inverse cos -1(.1/.2) all over 2pi: is coming back on my calc as 60/6.28 = 9.55 what am I missing...I am tired I am cramming right now. :)

3 answers

Last reply by: Professor Dan Fullerton
Sat Aug 30, 2014 8:39 PM

Post by Jungle Jones on August 30, 2014

In example 2, Hooke's Law says that F=-kx, but you just used F=kx to find the spring constant. I understood that the negative sign symbolizes that it is a restoring force, but can you explain why you didn't use the negative sign here?
And do never use the negative sign when doing calculations involving the law?

1 answer

Last reply by: Professor Dan Fullerton
Sat Aug 30, 2014 8:38 PM

Post by Jungle Jones on August 30, 2014

In example 1, I did omega = sqrt(k/m) and got 20, but I got the units of 1/(s^2).
k = 2000 N/m
m = 5 kg
N = kg m/(s^2)
so N/m -> kg/(s^2)
and that divided by 5 kg becomes 1/(s^2)
But it should be just 1/s, so what did I do wrong?

4 answers

Last reply by: Professor Dan Fullerton
Tue Jul 15, 2014 2:40 PM

Post by Jamal Tischler on July 15, 2014

I tried to calculate the spring oscilation period. I used the conservation of energy and kinematics equations for the average acceleration. 1/2kA^2=1/2mv^2 =>v^2=kA^2/m.
v^2=2aA+v0^2=2aA =>kA^2/m=2aA => a= kA/2m.
v=a*t. sqrt(kA^2/m)=kA/2m*t => t= 2 sqrt(m/k). This only the motion from the amplitude to the x=0 position. The farward 3 parts are the same because the kinetic energy is constantly transformed into elastic potential.
T=4*t=8 sqrt(m/k)
The formula said T=2*pi sqrt(m/k). pi doesnt equal 4.
Can you please explain this better ?

1 answer

Last reply by: Professor Dan Fullerton
Thu Mar 13, 2014 9:59 AM

Post by Lin Jiang on March 13, 2014

For example 5, why we needn't to find the equilibrium position. 1/2KX^2, I think x should be how far it pull from equilibrium.

0 answers

Post by Professor Dan Fullerton on February 17, 2014

I use an energy analysis, realizing that at the extremes, the kinetic energy is zero (for an instant the block is still), and all its energy is potential.  So the potential energy in a spring is 0.5kx^2.  When it's at equilibrium, however, there is no stretch in the spring, so PEs=0, and all the energy must be kinetic due to conservation of energy.  So 0.5kx^2=0.5mv^2.  Solve for the velocity!

0 answers

Post by sadia mussa on February 17, 2014

how do you get the formula for speed at equilibrium position as well as finding the last question for speed in the example for harmonic oscillator analysis because I don't seem to quite understand it, thanks

1 answer

Last reply by: Professor Dan Fullerton
Mon Dec 9, 2013 9:12 AM

Post by Constance Kang on December 9, 2013

in example 1,  since w= rot(m/k) then  if i do rot (5/2000) instead of 2pi f, how come i got a different answer ?

3 answers

Last reply by: Professor Dan Fullerton
Tue Apr 30, 2013 7:24 AM

Post by Saki Amagai on April 29, 2013

For example 6, it says "find the tension in the string at the equilibrium position". I think that the net force must equal zero at the equilibrium position; since in the previous slides, it mentioned that force and acceleration all equal zero at the point... Shouldn't "T = mg" and not "T - mg = mv^2/r"?

Simple Harmonic Motion

  • Simple harmonic motion is nature's typical response to a disturbance.
  • Simple harmonic motion is motion in which a restoring force is directly proportional to the displacement of the object.
  • Linear restoring forces exerted on objects displaced from equilibrium result in simple harmonic motion.
  • Restoring forces can result in oscillatory motion.
  • Many systems in simple harmonic motion exhibit an ongoing transformation of kinetic and potential energy.

Simple Harmonic Motion

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

Transcription: Simple Harmonic Motion

Hi everyone. I am Dan Fullerton and I am thrilled to welcome you back to Educator.com.0000

Today's topic is simple harmonic motion.0004

Our objectives are going to be to sketch and analyze a graph of displacement as a function of time for an object undergoing simple harmonic motion, to write down an appropriate expression for displacement of the form A-cos(ωt) or A-sin(ωt), where ω is going to be that angular frequency.0008

We will state the relations between displacement, velocity, and acceleration and determine the points in the motion where these quantities are minimum, 0, and maximum for an object undergoing simple harmonic motion, determine the total energy of an object in simple harmonic motion and sketch graphs of kinetic and potential energies as functions of time or displacement, and determine the period of oscillation for an ideal pendulum as well as a mass on a spring with a horizontal mass and a vertical mass.0026

That is what we are going to try and accomplish here.0052

All right. What is simple harmonic motion?0056

Simple harmonic motion is nature's typical reaction to a disturbance.0059

It is all over in this world in many, many places.0064

When you disturb something it typically responds with simple harmonic motion.0068

That could be something as simple as walking past a tree -- if you brush a branch, the branch gets a force done and starts oscillating back and forth.0073

Simple harmonic motion -- it is all over .0081

A displacement which results in a linear restoring force results in simple harmonic motion.0084

And the simple items we are going to focus on for today are going to be an ideal pendulum, a pendulum where the string has no mass and a mass on a spring, going back and forth due to that spring.0088

Let us start with a review of springs.0101

When a force is applied to a spring, the spring applies a restoring force and a spring can be compressed or it can be stretched.0106

When the spring is in equilibrium, it is unstrained; it is in its happy position; it is just thrilled to be there.0113

The factors affecting the force of a spring -- well the spring constant (k) is how tough it is to compress or stretch the spring.0120

The bigger the spring constant, the tougher it is to compress or stretch and that is measured in Newton's per meter (N/m) and the displacement is always measured from equilibrium.0126

We could make a graph of the force of a spring vs. the displacement and when we do that, for an object obeying Hooke's Law, we should get a straight line where the slope of that line which is rise over run, gives us our spring constant.0136

Or we could make the same type of graph -- force of a spring versus displacement get the same basic shape and if we want to get the work done in compressing and stretching that spring, all we have to do is come back and take the area under that curve and the area will be the work.0155

Now as we analyze that work, the work that we do on it must be the potential energy stored in the spring, so that is 1/2 base times height or 1/2 our base is going to be (x) and our height is going to be the force.0178

But if (F) is (kx), then that is going to be 1/2 x × our force, because it is a spring, (kx), or you could say that the potential energy stored in the spring is 1/2 kx2.0195

That is where that formula comes from -- one way we can derive the energy stored in a compressed or stretched spring.0211

All right. Let us talk about oscillations.0221

Repeated motions back and forth are called oscillations -- something going back and forth, back and forth is oscillating.0225

Now one revolution or one round trip, one complete cycle all mean the same thing and the period of the oscillation is the time it takes for one complete cycle or one complete revolution.0232

That is going to be an important vocabulary word, period.0243

Frequency, on the other hand, (F), is the number of cycles per second and it is measured in 1/seconds or a unit known as Hertz (Hz).0247

Period (T) is the number of seconds divided by the number of cycles to give you the time for each cycle.0261

Frequency is the number of cycles divided by number of seconds to give you the number of cycles per second and they are very closely related because Period is 1/frequency and therefore frequency is 1/Period.0269

Let us take a look at the spring block oscillator.0283

Imagine that we have a mass and we are going to connect it by a spring to some immovable object like a wall.0286

If we pull it one way and let it go, it is going to go back and forth and back and forth and oscillate.0295

That is a spring-block oscillator.0297

Factors affecting the period of its oscillation are the mass of its block (m) and the spring constant (k).0299

As we analyze this in a little bit more detail, we will call this the x = 0 position, the happy position, the equilibrium position -- maximum displacement of A or -A.0306

Now the period of a spring-block oscillator, period of a spring STs is going to be 2π times the square root of the mass divided by the spring constant.0318

Now the frequency of that spring-block oscillator which is always 1/period is just going to be 1/2π square root then of k/m.0332

Now we could also rearrange this a little bit to say that 2π times the frequency equals the square root of k/m.0344

Where there is 2π times the frequency is often times called the angular frequency (ω).0355

Angular frequency = 2πF, therefore we could write the angular frequency (ω) is the square root of k/m.0366

A couple of definitions to go along with our spring-block oscillator system.0384

All right. Let us take a look at an example with this system.0390

We have a block of mass 5 kg and it is attached to a spring, whose constant is 2,000 N/m.0393

Find the period of oscillation, the frequency, and the angular frequency.0399

Well, let us start with the period.0406

Period for a spring-block oscillator -- we know is 2π square root m/k or 2π square root -- mass is 5 kg, (k) is 2,000 N/m or when I plug that into my calculator, I get about 0.314 s for my period.0408

Let us find the frequency.0435

Frequency is 1/Period, so that is going to be 1/0.314 = 3.18, 1/seconds or 3.18 Hz.0436

And the angular frequency (ω) is 2π times the frequency, 2πF or 2π × 3.18 Hz, or 20 and the units are rad/s, although radians are not an official unit -- 20 rad/s will be our angular frequency.0457

Let us take a look at how we might analyze this in even more detail.0487

Here we are going to show our spring-block oscillator, mass (m), and we are going to look at it at three different positions -- at its equilibrium, position (A), at maximum displacement to the right (B), back to (A) to its minimum displacement or maximum displacement on the left (C), and back to (A).0491

So it is going to go back and forth, (A) to (B) to (A) to (C) to (A) and back and forth and it is going to displace a distance (x) to the right or (-x) to the left.0508

Well, if we want to look and see if we have different displacements, at point (A), we have 0 displacement, that is its equilibrium.0518

When it is at (B), its displacement is (x) and when it gets all the way to (C), its displacement is (-x).0526

Let us take a look at velocity now .0536

When it is at (A), it is going to have maximum velocity because all of its energy is going to be kinetic, so this will be maximum velocity here at (A).0538

Over here at (B), it is not going to have any velocity.0548

That is where all of its kinetic energy is converted into spring potential energy, so that will be 0 and (C) is the same way just on the other side.0551

There is no velocity at the end points.0558

It goes back and forth and for a split second it stops, turns around, speeds up, slows down, slows down, slows down, stops, speeds up, speeds up, speeds up, speeds up, slows down, stops, and back and forth as it goes on its oscillating path.0560

Let us take a look then at the force.0573

Well, while it is at point (A), the force is going to be 0 because there is no force to the spring on it because there is no displacement.0576

When it is at (B), it is going to have a maximum force, but it is a restoring force bringing it back the other direction, bringing it back towards its equilibrium position, so we will call that the negative max force.0585

And at (C) it is going to have the maximum force back to the right in the positive direction, so that will be maximum.0597

Because force = mass × acceleration, or acceleration = force/mass, the acceleration charge should look extremely similar.0604

At (A) there is no acceleration, at (B) it has its negative maximum acceleration, and at (C) it has its positive max acceleration.0612

Let us look at energy now. Spring potential energy over here is (U).0623

At (A), there is no displacement, so the spring potential energy must be 0.0631

It will have its maximum spring potential energy here at (B) and (C), so there is a maximum at (B) and (C).0638

If we want to look at kinetic energy, well that is going to be basically the inverse, right?0647

At (A), it is going to have its maximum velocity so it will have its maximum kinetic energy over there at (A), so maximum kinetic energy at (A), and at (B) and (C), it stopped at those points, it has no velocity for a split second there, so 0 and 0.0652

Let us take a look at what would happen if we made a graph of some of these.0671

I am going to start by taking a look at the displacement (x) and we are going to look at it as the object goes from (A) to (B) to (A) to (C) to (A) to (B) and back again.0675

Let us make this point (A) down all of these graphs, then we will draw another line all the way down here from (A) to (B)...0687

...another one here back to (A), and it goes to (C), and back to (A).0697

I think you quickly get the idea of what is going on here.0714

From (A) of course and back to (B) and so on, on its oscillating journey.0722

Well, if we look at displacement -- at (A) its displacement is 0, so anywhere we have (A) we can fill in the dots for 0.0730

At (B), its displacement is going to be positive (x), so it must come up like that and come back to (A).0738

At (C), its displacement must be negative (x) -- back to (A) and so on.0747

A graph of the displacement versus time.0754

If we want to do that for velocity, however, let us take a look and see what we know here.0759

For velocity, we know =at (B) and (C) we have 0, so anywhere we have (B) or (C) we must have 0 velocity.0764

At (A) we have maximum velocity and initially it is going to the right, so we will make that a positive velocity as it comes through.0773

When it comes back to (A), it has velocity going in the opposite direction at the same magnitude and back up and you can quickly see the pattern here for velocity of our spring-block oscillator.0781

Let us go to force and acceleration, let us put those over here -- a nice purple color, maybe for our force.0798

If we want to look at force, we know anywhere we have an (A), the force is 0.0810

So 0 here, 0 here, 0 here, and we have a maximum force when we are at (C), so let us fill that in -- maximum force over here at (C) and the negative maximum force when we are at (B).0815

So at (B) over here, we have negative maximum force, negative maximum force, and we can quickly plot our force versus time graph looking something like that.0832

And acceleration is just going to follow that -- force = mass × acceleration, so almost a mirror image of the graph, just different values but same shape.0848

Now, let us take a look and finish off our graph by looking at potential energy and kinetic energy versus time.0865

Let us look at potential energy in the spring first over here...0877

...there is (T), we have (A) here, goes to (B), back to (A), to (C)...0878

...from (C) we go back to (A), and from (A) we go back to (B).0895

All right. Potential energy in the spring is 0 at any of the (A)'s.0906

So there you go, a 0, 0 , 0; when it is at (B) it is at a maximum, so we will fill in our points there, and when it is at (C), it is also at a maximum.0913

Potential energy is scalar; it does not have a direction, so our graphs can look kind of like this.0925

When we look at kinetic energy, let us go there right underneath it.0935

Kinetic energy is going to have values of 0 at (B) and (C) when it is not moving.0942

So for (B), (C) -- (B) is at 0 and it is going to have maximum value when it is at (A), so it is going to look almost like the inverse of the potential energy graph.0947

Something kind of like that, and if we were to add the kinetic and potentials, they would add to a constant value because we are neglecting friction in this problem, using all conservative forces so we have conservation of mechanical energy.0967

Let us take a look at solving some problems with this and we will start off with a detailed harmonic oscillator analysis.0982

A 2 kg block is attached to a spring. A force of 20 N stretches the spring to a displacement of half a meter (0.5 m). 0996 Find the spring constant.0989

Well, we know that F = kx, therefore (k) must equal F/x or 20 N/0.5 m, which is 40 N/m.0998

The total energy is going to be the spring potential energy when it is at its maximum displacement or 1/2 kx2...1015

...which is 1/2 × 40 N/m (k) and its maximum displacement (0.5m2) for a total energy of 5 J.1026

About the speed at the equilibrium position, well at that point we have converted all of that spring potential energy into kinetic.1041

So we could start solving this one by saying that the spring potential energy is equal to the kinetic energy at the equilibrium position or 1/2 mv2 and that must equal that 5 J.1048

Therefore, the velocity = 2 × 5 divided by the mass, or 2 × 5/2 square root of 5 for about 2.24 m/s.1064

And how about the speed when it is at (x) = 0.3 m?1083

Well to do that, we are going to have to look at an energy analysis again.1088

The total energy is the spring potential energy plus the kinetic, therefore the total energy = 1/2 Kx2 + 1/2 mv2.1092

And if I am going to try and find the speed, let us get (V) isolated; I will multiply both sides by 2...1107

...2 there and I will subtract the kx2, therefore mv2 = 2 total energy minus kx2.1115

Divide by (m) and take the square root to find that the velocity will be 2 × the total energy - kx2, all divided by the mass, square root, and finally I can substitute in my values.1128

I have 2 × 5 J (total energy) - 40 (spring constant) × 0.3 (x-value), our displacement2 divided by 2 (mass) and the square root of that entire thing gives me a velocity, a speed at 0.3 m of about 1.79 m/s.1144

And I will do a quick check to see if that makes sense.1165

It should be less than the speed at the equilibrium position, the maximum speed, and it is less than 2.24 m/s.1168

So that makes sense. Excellent!1175

Let us go a little bit further with this one.1177

Let us try and find the speed now at x = -0.4 m.1181

We can use the same formula we just had, but plug in a different displacement value.1185

Velocity is going to be equal to 2 times total energy minus kx2 divided by (m), square root...1191

...that is going to be 2 × 5 J, (total energy) - 40 (spring constant) × -0.42 (new displacement/2 kg (mass); square root of all of that is about 1.34 m/s.1202

As it is a little closer to its full extension, it is slowing down even more -- less than our velocity when we were at 0.3 m. That, too, makes sense.1222

Let us find the acceleration at the equilibrium position.1231

Well when we were at x = 0, the force must equal 0 by Hooke's Law, and if the force is 0, then Newton's Second Law (F = ma) tells us that the acceleration must also be 0.1234

But if we want the acceleration at 0.5 m -- well to do that I am going to use Hooke's law (F = -kx), where (k) again is 40 N/m and a displacement at 0.5 m or -20 N.1249

We can apply Newton's Second Law again -- acceleration is force divided by mass, or -20 N/0.2 kg for an acceleration of -10 m/s2.1267

What does the negative tell you?1283

When it is at its furthest positive displacement, the acceleration is back towards its equilibrium position, going in the opposite direction, hence the negative.1284

Let us take this one even further. Let us find the net force at the equilibrium position.1298

Well at the equilibrium position we already determined the acceleration was 0, so the net force there must be 0 N.1305

How about the net force at half a meter?1312

Well, we could use Hooke's Law again (F = -kx) or -40 N/m × 0.25 m (displacement) or -10 N.1315

Where does kinetic energy equal potential energy?1330

Well if our total energy is 5 J, that is going to be the spot where the kinetic energy is 2.5 J and the potential energy is 2.5 J.1334

So if that is the case, we could figure that out -- the spring potential energy there is 2.5 J -- that is 1/2 kx2, where (x) is what we are solving for.1344

Therefore, (x)2 = 2 × 2.5/40 (K), and as I solve that then, that is equal to 0.125 and if I take the square root of both sides...1356

...(x) then equals the square root of 0.125 or about 0.354 m.1373

Pay special attention here -- note at this point where the kinetic energy and potential energy are equal is not midway between the equilibrium and the maximum displacement positions.1384

It does not work out that way, you cannot just guess halfway in between the two, you actually have to go through and solve; that is not halfway between the equilibrium and maximum displacement positions.1396

So let us take a look at circular motion versus simple harmonic motion and how they are related.1407

We have already talked about rotational motion for an object moving in a circle where we have some radius or amplitude of its motion (A) -- the angle θ measured from the horizontal, angular velocity ω and its position vector are, given by its (x) coordinate A-cos(θ), its (y) coordinate A-sinθ.1411

Well you could think of that almost as a projection down in one dimension to the spring-block oscillator system.1431

As you look at that system, imagine the object moving in a circle.1438

If you could shine a light down on it so you were just getting the projection in one dimension, you would see the exact same motion as what you see here in one coordinate.1443

Let us take a look at the (x) coordinates here -- (x) = A-cos(θ), but as we learned previously, θ = ω × (T).1452

Therefore we could write that as x = A-cos(ωT).1465

We also know that ω = 2π/period (T), so we could write that as x = A--cos(2π)/period × time.1472

Or going back to this equation, we also know that ω = 2π × the frequency, so we could write that -- replacing ω with 2π × the frequency as x = A-cos(2π), frequency, × time.1487

A bunch of different ways of looking at the same simple harmonic motion.1504

Instead of having our block start over here at a maximum displacement, what happens if we want to have it start here at x = 0?1509

Well we could use the sine function for that.1518

If we are going to have the block start at x = 0 at its equilibrium position -- start at x = 0 at time (t) = 0, then we could have the exact same equation -- we are just going to replace cosine with sine.1520

So x = A-sin(θ) or A-sin(ωt).1536

All it is, is a face shift, a sliding of the graph one way or another, depending on what you are calling your starting point.1543

As we look at graphing simple harmonic motion in that system, let us take a look at what happens to our graph of x = A-cos(ωt).1552

We will graph the (x) displacement and as we do that, we are going to look at it in terms of radians at 0, at π/2, at π, at 3π/2, 2π and so on.1567

I will make those marks on our graph right now -- π/2, π, 3π/2, 2π, and we will copy the same notches down here as well.1580

All right, for x = A-cos(ωt), we will do this as a function over here of ωt.1598

What are we going to graph?1604

Well, when our argument is 0, cosine here at times 0 is going to be 1, so we are going to get (A), so our maximum value here is (A).1606

As we get to π/2 right here, well our (x) coordinate is now 0, so we come down here to 0.1618

As we get over here to 2 full π, now we are adding negative maximum displacement, -(A).1628

Let us mark that on our graph, -(A), and back to 3π/2, 2π again and the cycle repeats.1636

We get ourselves -- just like the curve we had expected -- we get our cosine curve.1645

Looking at that, if we wanted to use the sine function instead, x = A-sin(ωt), we are going to have the same maximum amplitudes of course -- (A) and -(A).1655

But we are going to start -- if we look at the y-coordinate or for the sine function -- we are going to start at 0 for our (y), so 0 at π/2, we are at maximum value here for the (y), so positive (A)...1670

...down here to 0 at π, down here to -(A) at 3π/2 and back to 0 at 2π -- so we get our sine curve.1687

But as you look at the graphs, notice they are really the same shape; they are just off set by that π/2 amount, so you can use either one depending on which starting point you prefer to work with.1698

All right. Let us take a look at an example where we are looking at the position of an oscillator.1711

A spring-block oscillator makes 60 complete oscillations in 1 minute or 60 seconds.1715

Its maximum displacement is 0.2 m.1723

What is its position at time (t = 10 s) and at what time is it at position x = 0.1 m?1727

Well let us start up here with question A.1735

If x = A-cos(ωt) and we know that ω = 2πF, we could write this as x = A-cos(2πFt).1738

If we have 60 cycles in 60 seconds, then we know that our frequency must be 1 Hz.1759

So if frequency is 1 Hz and we know that our maximum amplitude (A) is 0.2 m, we can fill in our function a little bit to say that X = 0.2 cos(2π) × 1 (frequency) × 10 s (time) or putting that all together I get about 0.2 m.1765

It is at its maximum displacement.1794

Moving down here to B, at what time is it at position x = 0.1 m?1797

All right, s = A-cos(2πFt), but now we are solving for the time, so this implies then that 2πFt must be equal to the inverse cosine of x/A.1804

Or if we want just (t) but itself let us isolate the variable we want to find -- t = the inverse cosine of x/A divided by 2πF.1827

Now we can substitute in our variables, so that will be the inverse cosine of 0.1/0.2 or 1/2/2π × 1 Hz (frequency) or about 0.167 s.1841

Now it is important to note here for an oscillating system, that is not going to be the only time it is in that position, but it is one answer to that question.1860

Let us take a look at a vertical spring-block oscillator system.1872

Once the system settles the equilibrium where we are hanging our block from a spring instead of having it on a horizontal surface, we are going to displace the mass by pulling at some amount either +A -- pull it down, let it jump up -- or -A -- lift if up a little bit, drop it and let it oscillate up and down.1877

This is a really slick derivation and a neat analysis.1893

If we looked at a free body diagram (FBD) when it is at its equilibrium position, we have gravity pulling down and the force of the spring, (ky) pulling it up, where (y) is the equilibrium point.1898

So since it is at equilibrium at that point, we could write that the net force in the y-direction is going to be (mg), calling down positive, minus (ky) and since it is at equilibrium, let us say that that is mg - ky (equilibrium point), that must equal 0.1910

Therefore we could solve to say that the y-equilibrium point must be mg/k.1933

Now when we displace it by some amount (A), the net force in the y-direction, as we pull it down, is going to be mg - K × whatever that (y) would happen to be, which is going to be Y-equilibrium point plus that (A) amount we pulled it down.1941

We can distribute that through -- multiply that (k) through -- to find that it is mg - ky-equilibrium position - kA1964

Here is the slick part though -- Notice then as we do this that we had up here mg - ky-equilibrium = 0.1976

That means this part mg - ky-equilibrium must be equal to 0 and we could rewrite this then as FnetY = -kA.1989

There is our big result. What does that mean?2005

That is the same analysis you would do for a horizontal spring system with a spring constant (k), displaced horizontally some amount (A) from its equilibrium position.2008

We just made this so much simpler.2018

In short, to analyze a vertical spring system, all you do is find the new equilibrium position of the system, treat that -- once you have taken into account the effect of gravity -- and treat that as if that is the only force you have to deal with in the system -- just the spring force.2021

So once you find its new equilibrium position you could almost pretend that you turned it on your side and it is a horizontal spring oscillator system again.2037

You do not have to continue dealing with that force of gravity.2044

A really, really slick way to analyze a vertical spring-block oscillator.2048

Find that new equilibrium position and then ignore the effects of gravity from there; treat that as your new equilibrium position, just find it using the effect of gravity first.2052

Let us take an example to make sure we have this.2066

A 5 kg block is attached to a vertical spring, with a spring constant of 500 N/m.2069

After the block comes to rest it is pulled down 3 cm and then released.2075

What is the period of oscillation?2079

Well, period is 2π square root m/k or 2π square root 5/500, or 0.628 s -- that straight forward.2082

What is its maximum displacement of the spring from its initial unstrained position?2101

Well let us first look at it when it is at rest.2106

At that point we have (mg) down for our FBD, and we have (k) times -- let us call that displacement (d) at that point, so that the net force in the y-direction is 0, since it is at equilibrium because it is just sitting there, which implies that KD = mg or D = mg/k, which is 5 kg × 10 m/s2/spring constant (500) or 0.1 m.2110

So once you are hanging it there it hangs down 0.1 m.2144

Now you are going to go displace it; you are going to pull it down 3 cm from that equilibrium position.2148

If you then pull it down 3 cm, your maximum displacement in the y-direction is going to be that 0.1 m that you had from when it came to rest due to gravity...2155

...and that extra 3 cm that you added on as you pull it down or 0.13 m or 13 cm.2170

A nice straight forward analysis, once you take into account and figure out what its new equilibrium position is with gravity and then ignore the effects of gravity and treat it as a standard horizontal spring-block oscillator system.2183

All right. Example 5 -- another fairly involved example.2199

We have a 60 kg bungee jumper stepping off a 40 m high platform.2203

The bungee cord behaves like a spring, of spring constant 40 N/m.2209

Find the speed of the jumper at heights of 15 and 30 m above the ground.2213

And as we do this, we are going to have to assume that there is no slack in the system.2217

Find the speed of the jumper at heights 15 and 30 m above the ground.2222

All right. So first thing, let us draw a diagram here.2227

Here is our jumper and the jumper is on a 40 m foot high platform, so down here somewhere is the ground.2231

We want the speed of the jumper 15 m above the ground, so we will call that position (A) -- that is 15 m above the ground.2243

Position (B) is 30 m above the ground, which means we must have another 10 m here -- 10, 15 m, 15 m.2255

All right. The key here is we are going to try and do this through conservation of energy at this point.2267

So the gravitational potential energy at the top must be equal to the gravitational potential energy at (A) plus the spring potential energy (A) -- UAG, UAS plus the kinetic energy at (A), 15 m above the ground.2273

Or mg × H-initial, the initial gravitational potential energy must equal the gravitational potential energy at (A), mg × 15 + 1/2 k.2296

At (A) the displacement is 10 + 5, so the spring is stretched 25 m, 252 (kx2) + 1/2 mv2.2312

All right. A little bit of math here.2325

We could then say that mass (60), G (10), and height (40)...2327

...must equal 60 × 10, = 600 × 15 + 1/2 × 40 (k) × 252 = 625 + 1/2 × 60 (mass) × v2...2336

or let us see here, that is going to be 24,000 = 9,000 + 12,500 + 30 v2.2355

Or 30 v2 will equal about 2,500, so I get a velocity of about 9.13 m/s at (A), so vA = 9.13 m/s.2373

We also need the velocity at 30 m above the ground.2394

To do that then, we will follow the same basic idea but we are going to have different values for our height.2399

So we will have 24,000, our initial total energy, equal to mg × (height) 30 m + 1/2 k -- at (B) the spring is only stretched 10 m, so 102 + 1/2 mv2.2405

Therefore 24,000 = 600 (mg) × 30 + 1/2 × 20 (k) × 100 = 2,000 + 1/2 mv2.2427

Therefore 4,000 = 30v2; and in solving for V, I get about 11.55 m/s.2449

Makes sense -- it is going a little bit faster at (B), a little bit slower at (A), hopefully slows down before hitting the ground.2460

All right, let us go a little bit further with this problem. Let us see what happens next.2469

How close does the jumper get to the ground?2476

Well, to do that we are going to have to figure out where the kinetic energy becomes 0.2482

So as we do this one, we will say potential energy due to gravity total equals the potential energy due to gravity at some point (C), which is where they stop, plus potential energy due to the spring at point (C).2491

No kinetic energy -- it is 0; that is where the person stops.2507

Following along with our calculations, 24,000 (total energy) = mgH + 1/2 k, and now how far has that person been displaced?2512

That is going to be 40 minus whatever height is left, so 40 - H2.2525

So that implies then that 24,000 = 600 (mg) × H + 1/2 × 40 (k), so that is going to be 20 × 40 - H2.2532

That is starting to look like a quadratic equation, so let us get it into that form.2553

24,000 = 600H + 20 × -- well, if we square this, we get 1600- 80H + H2...2557

...or 24,000 = 600H + 32,000 - 1600H + 20H2.2576

We rearrange this to fit the quadratic formula, H2 - 50H + 400 = 0.2595

A couple of ways you can solve that, but the quadratic formula is probably my favorite.2606

I come up with a height of 10 m above the ground.2609

And let us just test that to make sure we did not make any mistakes here.2615

H2, 102, 100 - 50 × 10, 100 - 500 = -400 + 400 = 0.2618

So how close does the jumper get to the ground? 10 m.2627

Let us take a look at the pendulum again.2635

Now from the perspective of oscillations and simple harmonic motion.2639

Mass (m) is attached to a light string that swings without friction about a vertical equilibrium position.2644

For all of these, we are going to assume that this θ is relatively small.2650

Well, we have length (L) here again -- as it comes down here where there is all kinetic energy, its height has changed and we have derived a couple of times now that this height (H) is going to be L - L-cos(θ)...2655

...or this is L-cos(θ) compared to our entire length of our string (L).2673

Here we have all potential energy; here we have all kinetic and this (H) as we just said is going to be L - L-cos(θ).2685

If we take a look and start analyzing this with energy involved too though, again assuming a small θ, here we have all potential energy -- U = mgH, which is mgL × 1 - cos(θ).2696

Here it is all kinetic energy, and back here again it is all potential.2716

And if we wanted to find the velocity of our pendulum when it is at this lowest point, well we could say that the kinetic energy at the bottom must equal the potential energy at its highest point -- the top.2723

Or 1/2 mv2 = mgH where (H) is L - L-cos(θ), and we can divide now our masses.2735

So v2 = 2gL × 1 - cos(θ) or just velocity itself -- I take the square root and that will be the square root of 2gL, 1 - cos(θ).2748

Going a little further here, let us look at some other quantities that might be of interest.2765

Here at the highest point, you have the maximum force, you have the maximum acceleration, you have the maximum gravitational potential energy, but you have no kinetic energy and no velocity.2769

Here on the other hand, you have no force, (F = 0), your acceleration = 0, your gravitational potential energy = 0 -- assuming that is what we are calling 0 in this problem which should make sense.2786

Our maximum kinetic energy occurs there and we have maximum velocity there.2799

And what is causing our restoring force to put this in simple harmonic motion?2805

Well our restoring force is going to be based on the gravitational force pulling this down.2810

If we look right here, we have its weight pulling it down, but that is not what is causing the displacement; it is only a portion of that.2815

The portion that is going to be perpendicular to our string, mg-sin(θ) here is what is causing our restoring force.2826

So another way that we could graph this in terms of energy, is if we looked at energy on the y-axis versus (x) position on the (x), we know of course the total energy must remain the same.2844

We are dealing with conservative forces, conservation of energy, and we are not dealing with friction at this point.2854

Here at 0 displacement, everything is kinetic energy.2860

So as we draw this U-shape, anything above the U-shape between the E (total line) and our parabola is kinetic energy.2865

Anything below it is potential.2873

The two always sum up to the total energy.2875

So if I wanted to look at another point on the graph -- let us say we wanted to look right here.2878

In this case anything up here would be kinetic and down here would be our potential.2884

Use the graph that way -- what is above the line is kinetic, what is below is potential.2895

Another way of representing the same information; it is that important.2899

So what happens when we are talking about this pendulum and we have to start dealing with frequency and period?2905

Well, the period of an ideal pendulum is 2π square root of L/g.2910

The length of the pendulum is your variable.2915

The mass on the end does not matter, the length is what matters.2918

In your grandfather clocks at home, the length is all set, that is why those are so big.2922

Or the frequency is just going to be 1/period or 1/2π square root of g/L.2929

Now for all of these, again, we have to assume that θ is small due to a small angle approximation in the mathematics.2934

So as we are looking at this, let us take a look and see what would happen if we tried to graph period versus length.2941

If you did that you would probably get something that looks kind of like that because period is proportional to the square root of (L).2951

So if you wanted to get a nice linear graph that you could do something with, if you wanted to try and determine something like the acceleration due to gravity, for example.2960

You could take a pendulum, graph the period versus the square root of the length and you should get a nice linear graph and if you take the slope of that, the slope is going to be rise/run, which is going to be T/square root of L, which turns out to be 2π/square root of (g).2971

So if you want to go to the moon, figure out what the acceleration due to gravity is, make a bunch of different pendulums of different lengths, have them go back and forth, measure their periods, come graph the period versus square root of the length, find the slope and you can calculate the acceleration due to gravity that way.2995

Let us take a look at an example.3015

We have a 1 kg mass suspended from a 30 cm string that creates a simple pendulum.3017

The mass is displaced at an angle of 12 degrees from the vertical equilibrium position.3023

First thing we have to do, is if we are going to use any of these formulas, is to make sure that θ is small, and 12 degrees is small enough for our purposes.3029

Find the frequency and period of the pendulum.3037

Well, the period is 2π square root L/g, or 2π square root (L) 0.3 m/10 m/s2 (g)...3040

...or about 1.09 s and frequency then is just 1/period or 1/1.09 s, which is going to be about 0.92 Hz.3056

All right, find the height of the pendulum above equilibrium when at maximum displacement.3070

Well we know the height is L × 1 - cos(θ), so that is going to be 0.3 × 1 - cos(12 degrees) or about 0.0066 m.3076

And find the speed of the pendulum at the equilibrium position.3093

V = square root of 2GH if we want to use conservation of energy, which is square root of 2gL, 1 - cos(θ)...3097

...we have derived that a couple of times at this point, or the square root of 2 × 10 (g), (l) 0.3 × 1 - cos(12 degrees) or about 0.632 m/s.3107

Carrying this one a little further -- Find the restoring force at maximum displacement.3130

All right. The force at maximum displacement -- we just said was mg-sin(θ), the component of the weight pulling it back down -- mg-sin(θ), which is going to be its mass of 1 kg × 10 m/s2 (g)-sin(12 degrees), or about 2.08 N.3136

And how about the tension in the string at the equilibrium position?3158

Well at that point we can make our FBD.3162

There is our tension; there is our weight -- if it is in equilibrium, those must match -- or net force in the centripetal direction is mAC which implies that (t), which is in the direction toward the center of the circle...3166

...tension - mg = mv2/r, which implies then that the tension is mg + mv2/r.3181

Therefore, the tension must be (mass) 1 × 10 (g) + 1 (mass).3193

Our velocity we found was about 0.362 m/s2 over our radius (0.3 m) or about 10.44 N of tension in that string.3201

How long should the pendulum be in order to keep perfect time with a period of 1 second?3220

Well let us start there and we are going to assume again that it is an ideal pendulum, no friction and everything is perfect, and no mass in the string.3227

Period is 2π square root L/g and we want a time, a period of 1 s, so we are solving for (L).3235

Let us take t2 = 4π2 (L/g).3247

Therefore L = gt2/4π2, which implies then that the length (L) should be G (10 m/s2)...3254

...with a period of 1 s2/4π2 or about 0.253 m, a quarter of a meter.3270

How long should the pendulum be if the period is to be half a second?3285

Well, same thing -- let us just plug in a couple of different values here.3289

L = gt2, so (g) × 0.52/4π2, where g = 10 m/s2, which gives us about 0.633 m.3293

A lot shorter. A lot shorter -- one-fourth.3308

Let us take a look then at a pendulum on the moon.3313

How long must a pendulum of period 1 second be on the moon if the acceleration due to gravity on the moon is about 1.6 m/s2?3316

Well we can use our same formula, L = gt2/4π2, where (g) = 1.6 m/s2 on the moon × our period of 1 s2)/4π2, which is about 0.405 m.3325

All right. Doing great. Hang in there. One last sample problem.3355

Mass (m) is placed on a horizontal frictionless surface and attached to a spring with spring constant (k).3360

The mass is pulled back a distance (x) and released to oscillate horizontally.3368

What is the kinetic energy and potential energy of the mass at a displacement halfway between the equilibrium position and maximum displacement?3373

Well, let us draw a picture first of what our situation is going to look like.3382

Horizontal spring-block oscillator -- let us color that in there.3386

We will put our mass over here (m), some spring with spring constant (k), and we will start this at some displacement 0.3391

Here is our (x) and it is somewhere in there we are going to have a point (A), and what we know is at the maximum energy is 1/2 kx2.3405

The potential energy due to the spring at (A) is going to be 1/2 × (k) -- well (A) is halfway between these two, so that is going to be at x/22.3420

That will be 1/2 k × x2/4 or 1/8 kx2, which is 1/4 × 1/2 kx2.3438

Why did I write it that way?3451

Well, 1/2 ks2 is our maximum spring potential energy, so this then says that UA must equal 1/4 of UMax.3454

So at (A) it has 1/4 of its maximum energy, so where is that other 3/4 of the energy?3471

That has to be kinetic energy at (A), so that must be 3/4 of the maximum spring potential energy.3476

Notice here that halfway between the two, the spring potential energy and the kinetic energy are not the same -- they are not equal.3485

You have to go through the steps to go solve for points in between those two.3495

Do not take shortcuts.3499

Hopefully this is a good start for simple harmonic motion.3501

Thank you so much for your time and for watching Educator.com. Make it a great day everyone!3504

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