# AP Physics C: Mechanics Rotational Dynamics

Section 7: Rotational Motion: Lecture 4 | 56:58 min

This is basically a culmination video with no new concepts or math, we simply want to review what we’ve learned and bring everything together. Before watching, be sure you’re comfortable with torque and using the moment of inertia. This will serve as a guide to bring together all the concepts and math involved with rotational motion, and will make sure you’re comfortable using everything that you’re used thus far. Next we’ll learn the last of the rotational motion topics— angular momentum.

Table of Contents

7m 12s

- Intro0:00
- Objectives0:11
- What is Physics?0:27
- Why?0:50
- Physics Answers the 'Why' Question0:51
- Matter1:27
- Matter1:28
- Mass1:43
- Inertial Mass1:50
- Gravitational Mass2:13
- A Spacecraft's Mass3:03
- What is the Mass of the Spacecraft?3:05
- Energy3:37
- Energy3:38
- Work3:45
- Putting Energy and Work Together3:50
- Mass-Energy Equivalence4:15
- Relationship between Mass & Energy: E = mc²4:16
- Source of Energy on Earth4:47
- The Study of Everything5:00
- Physics is the Study of Everything5:01
- Mechanics5:29
- Topics Covered5:30
- Topics Not Covered6:07
- Next Steps6:44
- Three Things You'd Like to Learn About in Physics6:45

1h 51s

- Intro0:00
- Objectives0:10
- Vectors and Scalars1:06
- Scalars1:07
- Vectors1:27
- Vector Representations2:00
- Vector Representations2:01
- Graphical Vector Addition2:54
- Graphical Vector Addition2:55
- Graphical Vector Subtraction5:36
- Graphical Vector Subtraction5:37
- Vector Components7:12
- Vector Components7:13
- Angle of a Vector8:56
- tan θ9:04
- sin θ9:25
- cos θ9:46
- Vector Notation10:10
- Vector Notation 110:11
- Vector Notation 212:59
- Example I: Magnitude of the Horizontal & Vertical Component16:08
- Example II: Magnitude of the Plane's Eastward Velocity17:59
- Example III: Magnitude of Displacement19:33
- Example IV: Total Displacement from Starting Position21:51
- Example V: Find the Angle Theta Depicted by the Diagram26:35
- Vector Notation, cont.27:07
- Unit Vector Notation27:08
- Vector Component Notation27:25
- Vector Multiplication28:39
- Dot Product28:40
- Cross Product28:54
- Dot Product29:03
- Dot Product29:04
- Defining the Dot Product29:26
- Defining the Dot Product29:27
- Calculating the Dot Product29:42
- Unit Vector Notation29:43
- Vector Component Notation30:58
- Example VI: Calculating a Dot Product31:45
- Example VI: Part 1 - Find the Dot Product of the Following Vectors31:46
- Example VI: Part 2 - What is the Angle Between A and B?32:20
- Special Dot Products33:52
- Dot Product of Perpendicular Vectors33:53
- Dot Product of Parallel Vectors34:03
- Dot Product Properties34:51
- Commutative34:52
- Associative35:05
- Derivative of A * B35:24
- Example VII: Perpendicular Vectors35:47
- Cross Product36:42
- Cross Product of Two Vectors36:43
- Direction Using the Right-hand Rule37:32
- Cross Product of Parallel Vectors38:04
- Defining the Cross Product38:13
- Defining the Cross Product38:14
- Calculating the Cross Product Unit Vector Notation38:41
- Calculating the Cross Product Unit Vector Notation38:42
- Calculating the Cross Product Matrix Notation39:18
- Calculating the Cross Product Matrix Notation39:19
- Example VII: Find the Cross Product of the Following Vectors42:09
- Cross Product Properties45:16
- Cross Product Properties45:17
- Units46:41
- Fundamental Units46:42
- Derived units47:13
- Example IX: Dimensional Analysis47:21
- Calculus49:05
- Calculus49:06
- Differential Calculus49:49
- Differentiation & Derivative49:50
- Example X: Derivatives51:21
- Integral Calculus53:03
- Integration53:04
- Integral53:11
- Integration & Derivation are Inverse Functions53:16
- Determine the Original Function53:37
- Common Integrations54:45
- Common Integrations54:46
- Example XI: Integrals55:17
- Example XII: Calculus Applications58:32

23m 47s

- Intro0:00
- Objectives0:10
- Position / Displacement0:39
- Object's Position0:40
- Position Vector0:45
- Displacement0:56
- Position & Displacement are Vectors1:05
- Position & Displacement in 1 Dimension1:11
- Example I: Distance & Displacement1:21
- Average Speed2:14
- Average Speed2:15
- Average Speed is Scalar2:27
- Average Velocity2:39
- Average Velocity2:40
- Average Velocity is a Vector2:57
- Example II: Speed vs. Velocity3:16
- Example II: Deer's Average Speed3:17
- Example II: Deer's Average Velocity3:48
- Example III: Chuck the Hungry Squirrel4:21
- Example III: Chuck's Distance Traveled4:22
- Example III: Chuck's Displacement4:43
- Example III: Chuck's Average Speed5:25
- Example III: Chuck's Average Velocity5:39
- Acceleration6:11
- Acceleration: Definition & Equation6:12
- Acceleration: Units6:19
- Relationship of Acceleration to Velocity6:52
- Example IV: Acceleration Problem7:05
- The Position Vector7:39
- The Position Vector7:40
- Average Velocity9:35
- Average Velocity9:36
- Instantaneous Velocity11:20
- Instantaneous Velocity11:21
- Instantaneous Velocity is the Derivative of Position with Respect to Time11:35
- Area Under the Velocity-time Graph12:08
- Acceleration12:36
- More on Acceleration12:37
- Average Acceleration13:11
- Velocity vs. Time Graph13:14
- Graph Transformations13:59
- Graphical Analysis of Motion14:00
- Velocity and acceleration in 2D14:35
- Velocity Vector in 2D14:39
- Acceleration Vector in 2D15:26
- Polynomial Derivatives16:10
- Polynomial Derivatives16:11
- Example V: Polynomial Kinematics16:31
- Example VI: Velocity Function17:54
- Example VI: Part A - Determine the Acceleration at t=1 Second17:55
- Example VI: Part B - Determine the Displacement between t=0 and t=5 Seconds18:33
- Example VII: Tortoise and Hare20:14
- Example VIII: d-t Graphs22:40

36m 47s

- Intro0:00
- Objectives0:09
- Special Case: Constant Acceleration0:31
- Constant Acceleration & Kinematic Equations0:32
- Deriving the Kinematic Equations1:28
- V = V₀ + at1:39
- ∆x = V₀t +(1/2)at²2:03
- V² = V₀² +2a∆x4:05
- Problem Solving Steps7:02
- Step 17:13
- Step 27:18
- Step 37:27
- Step 47:30
- Step 57:31
- Example IX: Horizontal Kinematics7:38
- Example X: Vertical Kinematics9:45
- Example XI: 2 Step Problem11:23
- Example XII: Acceleration Problem15:01
- Example XIII: Particle Diagrams15:57
- Example XIV: Particle Diagrams17:36
- Example XV: Quadratic Solution18:46
- Free Fall22:56
- Free Fall22:57
- Air Resistance23:24
- Air Resistance23:25
- Acceleration Due to Gravity23:48
- Acceleration Due to Gravity23:49
- Objects Falling From Rest24:18
- Objects Falling From Rest24:19
- Example XVI: Falling Objects24:55
- Objects Launched Upward26:01
- Objects Launched Upward26:02
- Example XVII: Ball Thrown Upward27:16
- Example XVIII: Height of a Jump27:48
- Example XIX: Ball Thrown Downward31:10
- Example XX: Maximum Height32:27
- Example XXI: Catch-Up Problem33:53
- Example XXII: Ranking Max Height35:52

30m 34s

- Intro0:00
- Objectives0:07
- What is a Projectile?0:28
- What is a Projectile?0:29
- Path of a Projectile0:58
- Path of a Projectile0:59
- Independence of Motion2:45
- Vertical & Horizontal Motion2:46
- Example I: Horizontal Launch3:14
- Example II: Parabolic Path7:20
- Angled Projectiles8:01
- Angled Projectiles8:02
- Example III: Human Cannonball10:05
- Example IV: Motion Graphs14:39
- Graphing Projectile Motion19:05
- Horizontal Equation19:06
- Vertical Equation19:46
- Example V: Arrow Fired from Tower21:28
- Example VI: Arrow Fired from Tower24:10
- Example VII: Launch from a Height24:40
- Example VIII: Acceleration of a Projectile29:49

30m 24s

- Intro0:00
- Objectives0:08
- Radians and Degrees0:32
- Degrees0:35
- Radians0:40
- Example I: Radians and Degrees1:08
- Example I: Part A - Convert 90 Degrees to Radians1:09
- Example I: Part B - Convert 6 Radians to Degrees2:08
- Linear vs. Angular Displacement2:38
- Linear Displacement2:39
- Angular Displacement2:52
- Linear vs. Angular Velocity3:18
- Linear Velocity3:19
- Angular Velocity3:25
- Direction of Angular Velocity4:36
- Direction of Angular Velocity4:37
- Converting Linear to Angular Velocity5:05
- Converting Linear to Angular Velocity5:06
- Example II: Earth's Angular Velocity6:12
- Linear vs. Angular Acceleration7:26
- Linear Acceleration7:27
- Angular Acceleration7:32
- Centripetal Acceleration8:05
- Expressing Position Vector in Terms of Unit Vectors8:06
- Velocity10:00
- Centripetal Acceleration11:14
- Magnitude of Centripetal Acceleration13:24
- Example III: Angular Velocity & Centripetal Acceleration14:02
- Example IV: Moon's Orbit15:03
- Reference Frames17:44
- Reference Frames17:45
- Laws of Physics18:00
- Motion at Rest vs. Motion at a Constant Velocity18:21
- Motion is Relative19:20
- Reference Frame: Sitting in a Lawn Chair19:21
- Reference Frame: Sitting on a Train19:56
- Calculating Relative Velocities20:19
- Calculating Relative Velocities20:20
- Example: Calculating Relative Velocities20:57
- Example V: Man on a Train23:19
- Example VI: Airspeed24:56
- Example VII: 2-D Relative Motion26:12
- Example VIII: Relative Velocity w/ Direction28:32

23m 57s

- Intro0:00
- Objectives0:11
- Newton's 1st Law of Motion0:28
- Newton's 1st Law of Motion0:29
- Force1:16
- Definition of Force1:17
- Units of Force1:20
- How Much is a Newton?1:25
- Contact Forces1:47
- Field Forces2:32
- What is a Net Force?2:53
- What is a Net Force?2:54
- What Does It Mean?4:35
- What Does It Mean?4:36
- Objects at Rest4:52
- Objects at Rest4:53
- Objects in Motion5:12
- Objects in Motion5:13
- Equilibrium6:03
- Static Equilibrium6:04
- Mechanical Equilibrium6:22
- Translational Equilibrium6:38
- Inertia6:48
- Inertia6:49
- Inertial Mass6:58
- Gravitational Mass7:11
- Example I: Inertia7:40
- Example II: Inertia8:03
- Example III: Translational Equilibrium8:25
- Example IV: Net Force9:19
- Free Body Diagrams10:34
- Free Body Diagrams Overview10:35
- Falling Elephant: Free Body Diagram10:53
- Free Body Diagram Neglecting Air Resistance10:54
- Free Body Diagram Including Air Resistance11:22
- Soda on Table11:54
- Free Body Diagram for a Glass of Soda Sitting on a Table11:55
- Free Body Diagram for Box on Ramp13:38
- Free Body Diagram for Box on Ramp13:39
- Pseudo- Free Body Diagram15:26
- Example V: Translational Equilibrium18:35

23m 57s

- Intro0:00
- Objectives0:09
- Newton's 2nd Law of Motion0:36
- Newton's 2nd Law of Motion0:37
- Applying Newton's 2nd Law1:12
- Step 11:13
- Step 21:18
- Step 31:27
- Step 41:36
- Example I: Block on a Surface1:42
- Example II: Concurrent Forces2:42
- Mass vs. Weight4:09
- Mass4:10
- Weight4:28
- Example III: Mass vs. Weight4:45
- Example IV: Translational Equilibrium6:43
- Example V: Translational Equilibrium8:23
- Example VI: Determining Acceleration10:13
- Example VII: Stopping a Baseball12:38
- Example VIII: Steel Beams14:11
- Example IX: Tension Between Blocks17:03
- Example X: Banked Curves18:57
- Example XI: Tension in Cords24:03
- Example XII: Graphical Interpretation27:13
- Example XIII: Force from Velocity28:12
- Newton's 3rd Law29:16
- Newton's 3rd Law29:17
- Examples - Newton's 3rd Law30:01
- Examples - Newton's 3rd Law30:02
- Action-Reaction Pairs30:40
- Girl Kicking Soccer Ball30:41
- Rocket Ship in Space31:02
- Gravity on You31:23
- Example XIV: Force of Gravity32:11
- Example XV: Sailboat32:38
- Example XVI: Hammer and Nail33:18
- Example XVII: Net Force33:47

20m 41s

- Intro0:00
- Objectives0:06
- Coefficient of Friction0:21
- Coefficient of Friction0:22
- Approximate Coefficients of Friction0:44
- Kinetic or Static?1:21
- Sled Sliding Down a Snowy Hill1:22
- Refrigerator at Rest that You Want to Move1:32
- Car with Tires Rolling Freely1:49
- Car Skidding Across Pavement2:01
- Example I: Car Sliding2:21
- Example II: Block on Incline3:04
- Calculating the Force of Friction3:33
- Calculating the Force of Friction3:34
- Example III: Finding the Frictional Force4:02
- Example IV: Box on Wood Surface5:34
- Example V: Static vs. Kinetic Friction7:35
- Example VI: Drag Force on Airplane7:58
- Example VII: Pulling a Sled8:41
- Example VIII: AP-C 2007 FR113:23
- Example VIII: Part A13:24
- Example VIII: Part B14:40
- Example VIII: Part C15:19
- Example VIII: Part D17:08
- Example VIII: Part E18:24

32m 10s

- Intro0:00
- Objectives0:07
- Retarding Forces0:41
- Retarding Forces0:42
- The Skydiver1:30
- Drag Forces on a Free-falling Object1:31
- Velocity as a Function of Time5:31
- Velocity as a Function of Time5:32
- Velocity as a Function of Time, cont.12:27
- Acceleration12:28
- Velocity as a Function of Time, cont.15:16
- Graph: Acceleration vs. Time16:06
- Graph: Velocity vs. Time16:40
- Graph: Displacement vs. Time17:04
- Example I: AP-C 2005 FR117:43
- Example I: Part A17:44
- Example I: Part B19:17
- Example I: Part C20:17
- Example I: Part D21:09
- Example I: Part E22:42
- Example II: AP-C 2013 FR224:26
- Example II: Part A24:27
- Example II: Part B25:25
- Example II: Part C26:22
- Example II: Part D27:04
- Example II: Part E30:50

20m 31s

- Intro0:00
- Objectives0:06
- Drawing Free Body Diagrams for Ramps0:32
- Step 1: Choose the Object & Draw It as a Dot or Box0:33
- Step 2: Draw and Label all the External Forces0:39
- Step 3: Sketch a Coordinate System0:42
- Example: Object on a Ramp0:52
- Pseudo-Free Body Diagrams2:06
- Pseudo-Free Body Diagrams2:07
- Redraw Diagram with All Forces Parallel to Axes2:18
- Box on a Ramp4:08
- Free Body Diagram for Box on a Ramp4:09
- Pseudo-Free Body Diagram for Box on a Ramp4:54
- Example I: Box at Rest6:13
- Example II: Box Held By Force6:35
- Example III: Truck on a Hill8:46
- Example IV: Force Up a Ramp9:29
- Example V: Acceleration Down a Ramp12:01
- Example VI: Able of Repose13:59
- Example VII: Sledding17:03

24m 58s

- Intro0:00
- Objectives0:07
- What is an Atwood Machine?0:25
- What is an Atwood Machine?0:26
- Properties of Atwood Machines1:03
- Ideal Pulleys are Frictionless and Massless1:04
- Tension is Constant1:14
- Setup for Atwood Machines1:26
- Setup for Atwood Machines1:27
- Solving Atwood Machine Problems1:52
- Solving Atwood Machine Problems1:53
- Alternate Solution5:24
- Analyze the System as a Whole5:25
- Example I: Basic Atwood Machine7:31
- Example II: Moving Masses9:59
- Example III: Masses and Pulley on a Table13:32
- Example IV: Mass and Pulley on a Ramp15:47
- Example V: Ranking Atwood Machines19:50

37m 34s

- Intro0:00
- Objectives0:07
- What is Work?0:36
- What is Work?0:37
- Units of Work1:09
- Work in One Dimension1:31
- Work in One Dimension1:32
- Examples of Work2:19
- Stuntman in a Jet Pack2:20
- A Girl Struggles to Push Her Stalled Car2:50
- A Child in a Ghost Costume Carries a Bag of Halloween Candy Across the Yard3:24
- Example I: Moving a Refrigerator4:03
- Example II: Liberating a Car4:53
- Example III: Lifting Box5:30
- Example IV: Pulling a Wagon6:13
- Example V: Ranking Work on Carts7:13
- Non-Constant Forces12:21
- Non-Constant Forces12:22
- Force vs. Displacement Graphs13:49
- Force vs. Displacement Graphs13:50
- Hooke's Law14:41
- Hooke's Law14:42
- Determining the Spring Constant15:38
- Slope of the Graph Gives the Spring Constant, k15:39
- Work Done in Compressing the Spring16:34
- Find the Work Done in Compressing the String16:35
- Example VI: Finding Spring Constant17:21
- Example VII: Calculating Spring Constant19:48
- Example VIII: Hooke's Law20:30
- Example IX: Non-Linear Spring22:18
- Work in Multiple Dimensions23:52
- Work in Multiple Dimensions23:53
- Work-Energy Theorem25:25
- Work-Energy Theorem25:26
- Example X: Work-Energy Theorem28:35
- Example XI: Work Done on Moving Carts30:46
- Example XII: Velocity from an F-d Graph35:01

28m 4s

- Intro0:00
- Objectives0:08
- Energy Transformations0:31
- Energy Transformations0:32
- Work-Energy Theorem0:57
- Kinetic Energy1:12
- Kinetic Energy: Definition1:13
- Kinetic Energy: Equation1:55
- Example I: Frog-O-Cycle2:07
- Potential Energy2:46
- Types of Potential Energy2:47
- A Potential Energy Requires an Interaction between Objects3:29
- Internal energy3:50
- Internal Energy3:51
- Types of Energy4:37
- Types of Potential & Kinetic Energy4:38
- Gravitational Potential Energy5:42
- Gravitational Potential Energy5:43
- Example II: Potential Energy7:27
- Example III: Kinetic and Potential Energy8:16
- Example IV: Pendulum9:09
- Conservative Forces11:37
- Conservative Forces Overview11:38
- Type of Conservative Forces12:42
- Types of Non-conservative Forces13:02
- Work Done by Conservative Forces13:28
- Work Done by Conservative Forces13:29
- Newton's Law of Universal Gravitation14:18
- Gravitational Force of Attraction between Any Two Objects with Mass14:19
- Gravitational Potential Energy15:27
- Gravitational Potential Energy15:28
- Elastic Potential Energy17:36
- Elastic Potential Energy17:37
- Force from Potential Energy18:51
- Force from Potential Energy18:52
- Gravitational Force from the Gravitational Potential Energy20:46
- Gravitational Force from the Gravitational Potential Energy20:47
- Hooke's Law from Potential Energy22:04
- Hooke's Law from Potential Energy22:05
- Summary23:16
- Summary23:17
- Example V: Kinetic Energy of a Mass24:40
- Example VI: Force from Potential Energy25:48
- Example VII: Work on a Spinning Disc26:54

54m 56s

- Intro0:00
- Objectives0:09
- Conservation of Mechanical Energy0:32
- Consider a Single Conservative Force Doing Work on a Closed System0:33
- Non-Conservative Forces1:40
- Non-Conservative Forces1:41
- Work Done by a Non-conservative Force1:47
- Formula: Total Energy1:54
- Formula: Total Mechanical Energy2:04
- Example I: Falling Mass2:15
- Example II: Law of Conservation of Energy4:07
- Example III: The Pendulum6:34
- Example IV: Cart Compressing a Spring10:12
- Example V: Cart Compressing a Spring11:12
- Example V: Part A - Potential Energy Stored in the Compressed Spring11:13
- Example V: Part B - Maximum Vertical Height12:01
- Example VI: Car Skidding to a Stop13:05
- Example VII: Block on Ramp14:22
- Example VIII: Energy Transfers16:15
- Example IX: Roller Coaster20:04
- Example X: Bungee Jumper23:32
- Example X: Part A - Speed of the Jumper at a Height of 15 Meters Above the Ground24:48
- Example X: Part B - Speed of the Jumper at a Height of 30 Meters Above the Ground26:53
- Example X: Part C - How Close Does the Jumper Get to the Ground?28:28
- Example XI: AP-C 2002 FR330:28
- Example XI: Part A30:59
- Example XI: Part B31:54
- Example XI: Part C32:50
- Example XI: Part D & E33:52
- Example XII: AP-C 2007 FR335:24
- Example XII: Part A35:52
- Example XII: Part B36:27
- Example XII: Part C37:48
- Example XII: Part D39:32
- Example XIII: AP-C 2010 FR141:07
- Example XIII: Part A41:34
- Example XIII: Part B43:05
- Example XIII: Part C45:24
- Example XIII: Part D47:18
- Example XIV: AP-C 2013 FR148:25
- Example XIV: Part A48:50
- Example XIV: Part B49:31
- Example XIV: Part C51:27
- Example XIV: Part D52:46
- Example XIV: Part E53:25

16m 44s

- Intro0:00
- Objectives0:06
- Defining Power0:20
- Definition of Power0:21
- Units of Power0:27
- Average Power0:43
- Instantaneous Power1:03
- Instantaneous Power1:04
- Example I: Horizontal Box2:07
- Example II: Accelerating Truck4:48
- Example III: Motors Delivering Power6:00
- Example IV: Power Up a Ramp7:00
- Example V: Power from Position Function8:51
- Example VI: Motorcycle Stopping10:48
- Example VII: AP-C 2003 FR111:52
- Example VII: Part A11:53
- Example VII: Part B12:50
- Example VII: Part C14:36
- Example VII: Part D15:52

13m 9s

- Intro0:00
- Objectives0:07
- Momentum0:39
- Definition of Momentum0:40
- Total Momentum1:00
- Formula for Momentum1:05
- Units of Momentum1:11
- Example I: Changing Momentum1:18
- Impulse2:27
- Impulse2:28
- Example II: Impulse2:41
- Relationship Between Force and ∆p (Impulse)3:36
- Relationship Between Force and ∆p (Impulse)3:37
- Example III: Force from Momentum4:37
- Impulse-Momentum Theorem5:14
- Impulse-Momentum Theorem5:15
- Example IV: Impulse-Momentum6:26
- Example V: Water Gun & Horizontal Force7:56
- Impulse from F-t Graphs8:53
- Impulse from F-t Graphs8:54
- Example VI: Non-constant Forces9:16
- Example VII: F-t Graph10:01
- Example VIII: Impulse from Force11:19

46m 30s

- Intro0:00
- Objectives0:08
- Conservation of Linear Momentum0:28
- In an Isolated System0:29
- In Any Closed System0:37
- Direct Outcome of Newton's 3rd Law of Motion0:47
- Collisions and Explosions1:07
- Collisions and Explosions1:08
- The Law of Conservation of Linear Momentum1:25
- Solving Momentum Problems1:35
- Solving Momentum Problems1:36
- Types of Collisions2:08
- Elastic Collision2:09
- Inelastic Collision2:34
- Example I: Traffic Collision3:00
- Example II: Collision of Two Moving Objects6:55
- Example III: Recoil Velocity9:47
- Example IV: Atomic Collision12:12
- Example V: Collision in Multiple Dimensions18:11
- Example VI: AP-C 2001 FR125:16
- Example VI: Part A25:33
- Example VI: Part B26:44
- Example VI: Part C28:17
- Example VI: Part D28:58
- Example VII: AP-C 2002 FR130:10
- Example VII: Part A30:20
- Example VII: Part B32:14
- Example VII: Part C34:25
- Example VII: Part D36:17
- Example VIII: AP-C 2014 FR138:55
- Example VIII: Part A39:28
- Example VIII: Part B41:00
- Example VIII: Part C42:57
- Example VIII: Part D44:20

28m 26s

- Intro0:00
- Objectives0:07
- Center of Mass0:45
- Center of Mass0:46
- Finding Center of Mass by Inspection1:25
- For Uniform Density Objects1:26
- For Objects with Multiple Parts1:36
- For Irregular Objects1:44
- Example I: Center of Mass by Inspection2:06
- Calculating Center of Mass for Systems of Particles2:25
- Calculating Center of Mass for Systems of Particles2:26
- Example II: Center of Mass (1D)3:15
- Example III: Center of Mass of Continuous System4:29
- Example IV: Center of Mass (2D)6:00
- Finding Center of Mass by Integration7:38
- Finding Center of Mass by Integration7:39
- Example V: Center of Mass of a Uniform Rod8:10
- Example VI: Center of Mass of a Non-Uniform Rod11:40
- Center of Mass Relationships14:44
- Center of Mass Relationships14:45
- Center of Gravity17:36
- Center of Gravity17:37
- Uniform Gravitational Field vs. Non-uniform Gravitational Field17:53
- Example VII: AP-C 2004 FR118:26
- Example VII: Part A18:45
- Example VII: Part B19:38
- Example VII: Part C21:03
- Example VII: Part D22:04
- Example VII: Part E24:52

21m 36s

- Intro0:00
- Objectives0:08
- Uniform Circular Motion0:42
- Distance Around the Circle for Objects Traveling in a Circular Path at Constant Speed0:51
- Average Speed for Objects Traveling in a Circular Path at Constant Speed1:15
- Frequency1:42
- Definition of Frequency1:43
- Symbol of Frequency1:46
- Units of Frequency1:49
- Period2:04
- Period2:05
- Frequency and Period2:19
- Frequency and Period2:20
- Example I: Race Car2:32
- Example II: Toy Train3:22
- Example III: Round-A-Bout4:07
- Example III: Part A - Period of the Motion4:08
- Example III: Part B- Frequency of the Motion4:43
- Example III: Part C- Speed at Which Alan Revolves4:58
- Uniform Circular Motion5:28
- Is an Object Undergoing Uniform Circular Motion Accelerating?5:29
- Direction of Centripetal Acceleration6:21
- Direction of Centripetal Acceleration6:22
- Magnitude of Centripetal Acceleration8:23
- Magnitude of Centripetal Acceleration8:24
- Example IV: Car on a Track8:39
- Centripetal Force10:14
- Centripetal Force10:15
- Calculating Centripetal Force11:47
- Calculating Centripetal Force11:48
- Example V: Acceleration12:41
- Example VI: Direction of Centripetal Acceleration13:44
- Example VII: Loss of Centripetal Force14:03
- Example VIII: Bucket in Horizontal Circle14:44
- Example IX: Bucket in Vertical Circle15:24
- Example X: Demon Drop17:38
- Example X: Question 118:02
- Example X: Question 218:25
- Example X: Question 319:22
- Example X: Question 420:13

32m 52s

- Intro0:00
- Objectives0:07
- Radians and Degrees0:35
- Once Around a Circle: In Degrees0:36
- Once Around a Circle: In Radians0:48
- Measurement of Radian0:51
- Example I: Radian and Degrees1:08
- Example I: Convert 90° to Radians1:09
- Example I: Convert 6 Radians to Degree1:23
- Linear vs. Angular Displacement1:43
- Linear Displacement1:44
- Angular Displacement1:51
- Linear vs. Angular Velocity2:04
- Linear Velocity2:05
- Angular Velocity2:10
- Direction of Angular Velocity2:28
- Direction of Angular Velocity2:29
- Converting Linear to Angular Velocity2:58
- Converting Linear to Angular Velocity2:59
- Example II: Angular Velocity of Earth3:51
- Linear vs. Angular Acceleration4:35
- Linear Acceleration4:36
- Angular Acceleration4:42
- Example III: Angular Acceleration5:09
- Kinematic Variable Parallels6:30
- Kinematic Variable Parallels: Translational & Angular6:31
- Variable Translations7:00
- Variable Translations: Translational & Angular7:01
- Kinematic Equation Parallels7:38
- Kinematic Equation Parallels: Translational & Rotational7:39
- Example IV: Deriving Centripetal Acceleration8:29
- Example V: Angular Velocity13:24
- Example V: Part A13:25
- Example V: Part B14:15
- Example VI: Wheel in Motion14:39
- Example VII: AP-C 2003 FR316:23
- Example VII: Part A16:38
- Example VII: Part B17:34
- Example VII: Part C24:02
- Example VIII: AP-C 2014 FR225:35
- Example VIII: Part A25:47
- Example VIII: Part B26:28
- Example VIII: Part C27:48
- Example VIII: Part D28:26
- Example VIII: Part E29:16

24m

- Intro0:00
- Objectives0:07
- Types of Inertia0:34
- Inertial Mass0:35
- Moment of Inertia0:44
- Kinetic Energy of a Rotating Disc1:25
- Kinetic Energy of a Rotating Disc1:26
- Calculating Moment of Inertia (I)5:32
- Calculating Moment of Inertia (I)5:33
- Moment of Inertia for Common Objects5:49
- Moment of Inertia for Common Objects5:50
- Example I: Point Masses6:46
- Example II: Uniform Rod9:09
- Example III: Solid Cylinder13:07
- Parallel Axis Theorem (PAT)17:33
- Parallel Axis Theorem (PAT)17:34
- Example IV: Calculating I Using the Parallel Axis Theorem18:39
- Example V: Hollow Sphere20:18
- Example VI: Long Thin Rod20:55
- Example VII: Ranking Moment of Inertia21:50
- Example VIII: Adjusting Moment of Inertia22:39

26m 9s

- Intro0:00
- Objectives0:06
- Torque0:18
- Definition of Torque0:19
- Torque & Rotation0:26
- Lever Arm ( r )0:30
- Example: Wrench0:39
- Direction of the Torque Vector1:45
- Direction of the Torque Vector1:46
- Finding Direction Using the Right-hand Rule1:53
- Newton's 2nd Law: Translational vs. Rotational2:20
- Newton's 2nd Law: Translational vs. Rotational2:21
- Equilibrium3:17
- Static Equilibrium3:18
- Dynamic Equilibrium3:30
- Example I: See-Saw Problem3:46
- Example II: Beam Problem7:12
- Example III: Pulley with Mass10:34
- Example IV: Net Torque13:46
- Example V: Ranking Torque15:29
- Example VI: Ranking Angular Acceleration16:25
- Example VII: Café Sign17:19
- Example VIII: AP-C 2008 FR219:44
- Example VIII: Part A20:12
- Example VIII: Part B21:08
- Example VIII: Part C22:36
- Example VIII: Part D24:37

56m 58s

- Intro0:00
- Objectives0:08
- Conservation of Energy0:48
- Translational Kinetic Energy0:49
- Rotational Kinetic Energy0:54
- Total Kinetic Energy1:03
- Example I: Disc Rolling Down an Incline1:10
- Rotational Dynamics4:25
- Rotational Dynamics4:26
- Example II: Strings with Massive Pulleys4:37
- Example III: Rolling without Slipping9:13
- Example IV: Rolling with Slipping13:45
- Example V: Amusement Park Swing22:49
- Example VI: AP-C 2002 FR226:27
- Example VI: Part A26:48
- Example VI: Part B27:30
- Example VI: Part C29:51
- Example VI: Part D30:50
- Example VII: AP-C 2006 FR331:39
- Example VII: Part A31:49
- Example VII: Part B36:20
- Example VII: Part C37:14
- Example VII: Part D38:48
- Example VIII: AP-C 2010 FR239:40
- Example VIII: Part A39:46
- Example VIII: Part B40:44
- Example VIII: Part C44:31
- Example VIII: Part D46:44
- Example IX: AP-C 2013 FR348:27
- Example IX: Part A48:47
- Example IX: Part B50:33
- Example IX: Part C53:28
- Example IX: Part D54:15
- Example IX: Part E56:20

33m 2s

- Intro0:00
- Objectives0:09
- Linear Momentum0:44
- Definition of Linear Momentum0:45
- Total Angular Momentum0:52
- p = mv0:59
- Angular Momentum1:08
- Definition of Angular Momentum1:09
- Total Angular Momentum1:21
- A Mass with Velocity v Moving at Some Position r1:29
- Calculating Angular Momentum1:44
- Calculating Angular Momentum1:45
- Spin Angular Momentum4:17
- Spin Angular Momentum4:18
- Example I: Object in Circular Orbit4:51
- Example II: Angular Momentum of a Point Particle6:34
- Angular Momentum and Net Torque9:03
- Angular Momentum and Net Torque9:04
- Conservation of Angular Momentum11:53
- Conservation of Angular Momentum11:54
- Example III: Ice Skater Problem12:20
- Example IV: Combining Spinning Discs13:52
- Example V: Catching While Rotating15:13
- Example VI: Changes in Angular Momentum16:47
- Example VII: AP-C 2005 FR317:37
- Example VII: Part A18:12
- Example VII: Part B18:32
- Example VII: Part C19:53
- Example VII: Part D21:52
- Example VIII: AP-C 2014 FR324:23
- Example VIII: Part A24:31
- Example VIII: Part B25:33
- Example VIII: Part C26:58
- Example VIII: Part D28:24
- Example VIII: Part E30:42

1h 1m 12s

- Intro0:00
- Objectives0:08
- Simple Harmonic Motion0:45
- Simple Harmonic Motion0:46
- Circular Motion vs. Simple Harmonic Motion (SHM)1:39
- Circular Motion vs. Simple Harmonic Motion (SHM)1:40
- Position, Velocity, & Acceleration4:55
- Position4:56
- Velocity5:12
- Acceleration5:49
- Frequency and Period6:37
- Frequency6:42
- Period6:49
- Angular Frequency7:05
- Angular Frequency7:06
- Example I: Oscillating System7:37
- Example I: Determine the Object's Angular Frequency7:38
- Example I: What is the Object's Position at Time t = 10s?8:16
- Example I: At What Time is the Object at x = 0.1m?9:10
- Mass on a Spring10:17
- Mass on a Spring10:18
- Example II: Analysis of Spring-Block System11:34
- Example III: Spring-Block ranking12:53
- General Form of Simple Harmonic Motion14:41
- General Form of Simple Harmonic Motion14:42
- Graphing Simple Harmonic Motion (SHM)15:22
- Graphing Simple Harmonic Motion (SHM)15:23
- Energy of Simple Harmonic Motion (SHM)15:49
- Energy of Simple Harmonic Motion (SHM)15:50
- Horizontal Spring Oscillator19:24
- Horizontal Spring Oscillator19:25
- Vertical Spring Oscillator20:58
- Vertical Spring Oscillator20:59
- Springs in Series23:30
- Springs in Series23:31
- Springs in Parallel26:08
- Springs in Parallel26:09
- The Pendulum26:59
- The Pendulum27:00
- Energy and the Simple Pendulum27:46
- Energy and the Simple Pendulum27:47
- Frequency and Period of a Pendulum30:16
- Frequency and Period of a Pendulum30:17
- Example IV: Deriving Period of a Simple Pendulum31:42
- Example V: Deriving Period of a Physical Pendulum35:20
- Example VI: Summary of Spring-Block System38:16
- Example VII: Harmonic Oscillator Analysis44:14
- Example VII: Spring Constant44:24
- Example VII: Total Energy44:45
- Example VII: Speed at the Equilibrium Position45:05
- Example VII: Speed at x = 0.30 Meters45:37
- Example VII: Speed at x = -0.40 Meter46:46
- Example VII: Acceleration at the Equilibrium Position47:21
- Example VII: Magnitude of Acceleration at x = 0.50 Meters47:35
- Example VII: Net Force at the Equilibrium Position48:04
- Example VII: Net Force at x = 0.25 Meter48:20
- Example VII: Where does Kinetic Energy = Potential Energy?48:33
- Example VIII: Ranking Spring Systems49:35
- Example IX: Vertical Spring Block Oscillator51:45
- Example X: Ranking Period of Pendulum53:50
- Example XI: AP-C 2009 FR254:50
- Example XI: Part A54:58
- Example XI: Part B57:57
- Example XI: Part C59:11
- Example XII: AP-C 2010 FR31:00:18
- Example XII: Part A1:00:49
- Example XII: Part B1:02:47
- Example XII: Part C1:04:30
- Example XII: Part D1:05:53
- Example XII: Part E1:08:13

34m 59s

- Intro0:00
- Objectives0:07
- Newton's Law of Universal Gravitation0:45
- Newton's Law of Universal Gravitation0:46
- Example I: Gravitational Force Between Earth and Sun2:24
- Example II: Two Satellites3:39
- Gravitational Field Strength4:23
- Gravitational Field Strength4:24
- Example III: Weight on Another Planet6:22
- Example IV: Gravitational Field of a Hollow Shell7:31
- Example V: Gravitational Field Inside a Solid Sphere8:33
- Velocity in Circular Orbit12:05
- Velocity in Circular Orbit12:06
- Period and Frequency for Circular Orbits13:56
- Period and Frequency for Circular Orbits13:57
- Mechanical Energy for Circular Orbits16:11
- Mechanical Energy for Circular Orbits16:12
- Escape Velocity17:48
- Escape Velocity17:49
- Kepler's 1st Law of Planetary Motion19:41
- Keller's 1st Law of Planetary Motion19:42
- Kepler's 2nd Law of Planetary Motion20:05
- Keller's 2nd Law of Planetary Motion20:06
- Kepler's 3rd Law of Planetary Motion20:57
- Ratio of the Squares of the Periods of Two Planets20:58
- Ratio of the Squares of the Periods to the Cubes of Their Semi-major Axes21:41
- Total Mechanical Energy for an Elliptical Orbit21:57
- Total Mechanical Energy for an Elliptical Orbit21:58
- Velocity and Radius for an Elliptical Orbit22:35
- Velocity and Radius for an Elliptical Orbit22:36
- Example VI: Rocket Launched Vertically24:26
- Example VII: AP-C 2007 FR228:16
- Example VII: Part A28:35
- Example VII: Part B29:51
- Example VII: Part C31:14
- Example VII: Part D32:23
- Example VII: Part E33:16

28m 11s

- Intro0:00
- Problem 10:30
- Problem 20:51
- Problem 31:25
- Problem 42:00
- Problem 53:05
- Problem 64:19
- Problem 74:48
- Problem 85:18
- Problem 95:38
- Problem 106:26
- Problem 117:21
- Problem 128:08
- Problem 138:35
- Problem 149:20
- Problem 1510:09
- Problem 1610:25
- Problem 1711:30
- Problem 1812:27
- Problem 1913:00
- Problem 2014:40
- Problem 2115:44
- Problem 2216:42
- Problem 2317:35
- Problem 2417:54
- Problem 2518:32
- Problem 2619:08
- Problem 2720:56
- Problem 2822:19
- Problem 2922:36
- Problem 3023:18
- Problem 3124:06
- Problem 3224:40

28m 11s

- Intro0:00
- Question 10:15
- Part A: I0:16
- Part A: II0:46
- Part A: III1:13
- Part B1:40
- Part C2:49
- Part D: I4:46
- Part D: II5:15
- Question 25:46
- Part A: I6:13
- Part A: II7:05
- Part B: I7:48
- Part B: II8:42
- Part B: III9:03
- Part B: IV9:26
- Part B: V11:32
- Question 313:30
- Part A: I13:50
- Part A: II14:16
- Part A: III14:38
- Part A: IV14:56
- Part A: V15:36
- Part B16:11
- Part C17:00
- Part D: I19:56
- Part D: II21:08

For more information, please see full course syllabus of AP Physics C: Mechanics

1 answer

Last reply by: Professor Dan Fullerton

Tue Dec 1, 2015 8:20 AM

Post by Akilah Miller on November 29, 2015

Hello Professor Fullerton,

I hope you are having a wonderful day.

I have a question: In example 2 of rotational dynamics, you stated that the normal force of a rotating object is at the point of contact; yet, I have encountered problems where that does not seem to be the case.

For example, according to my text book a light, nonstretching cable wrapped around a solid cylinder, of a certain mass and diameter, which rotates in frictionless bearings about a stationary horizontal axis has a normal force that is at an angle above the horizontal once we pull the string and set the cylinder in motion.

Could you explain why this is?

Thank you very much for your concern for this matter and I apologize for the length of my question.

1 answer

Last reply by: Professor Dan Fullerton

Mon Mar 2, 2015 9:38 PM

Post by Bob Jones on March 2, 2015

In the amusement park problem, how are you allowed to just divide the two equations?

2 answers

Last reply by: Professor Dan Fullerton

Tue Nov 11, 2014 10:06 AM

Post by Scott Beck on November 11, 2014

Example IX: AP-C 2013 FR3

Part E

You said that I does not effect linear acceleration, so why does I effect the linear velocity in Example VII: AP-C 2006 FR3 Part D.

(Also, college board says part E is less than?)