# Classical Mechanics (Fall 2016)

Massachusetts Institute of Technology via MIT OpenCourseWare

## Overview

## Course Features

- Video lectures
- Captions/transcript
- Online textbooks
- Assignments: problem sets (no solutions)

## Course Description

This first course in the physics curriculum introduces classical mechanics. Historically, a set of core concepts—space, time, mass, force, momentum, torque, and angular momentum—were introduced in classical mechanics in order to solve the most famous physics problem, the motion of the planets.

The principles of mechanics successfully described many other phenomena encountered in the world. Conservation laws involving energy, momentum and angular momentum provided a second parallel approach to solving many of the same problems. In this course, we will investigate both approaches: Force and conservation laws.

Our goal is to develop a conceptual understanding of the core concepts, a familiarity with the experimental verification of our theoretical laws, and an ability to apply the theoretical framework to describe and predict the motions of bodies.

## Syllabus

8.01SC Classical Mechanics Introduction.

0.1 Vectors vs. Scalars.

0.2 Vector Operators.

0.3 Coordinate Systems and Unit Vectors.

0.4 Vectors - Magnitude and Direction.

0.5 Vector Decomposition into components.

0.6 Going Between Representations.

1.0 Week 1 Introduction.

1.1 Coordinate Systems and Unit Vectors in 1D.

1.2 Position Vector in 1D.

1.3 Displacement Vector in 1D.

1.4 Average Velocity in 1D.

1.5 Instantaneous Velocity in 1D.

1.7 Worked Example: Derivatives in Kinematics.

2.1 Introduction to Acceleration.

2.2 Acceleration in 1D.

2.3 Worked Example: Acceleration from Position.

2.4 Integration.

3.1 Coordinate System and Position Vector in 2D.

3.2 Instantaneous Velocity in 2D.

3.3 Instantaneous Acceleration in 2D.

3.4 Projectile Motion.

3.5 Demo: Shooting an Apple.

3.5 Demo: Relative Motion Gun.

PS.1.1 Three Questions Before Starting.

PS.1.2 Shooting the apple solution.

P.1.3 Worked Example: Braking Car.

P.1.4 Sketch the Motion.

P.1.5 Worked Example: Pedestrian and Bike at Intersection.

4.0 Week 2 Introduction.

4.1 Newton's First and Second Laws.

4.2 Newton's Third Law.

4.3 Reference Frames.

4.4 Non-inertial Reference Frames.

5.1 Universal Law of Gravitation.

5.2 Worked Example: Gravity - Superposition.

5.3 Gravity at the surface of the Earth: The value of g..

6.1 Contact Forces.

6.2 Static Friction.

7.1 Pushing Pulling and Tension.

7.2 Ideal Rope.

7.3 Solving Pulley Systems.

7.4 Hooke's Law.

DD.1.1 Friction at the Nanoscale.

PS.2.1 Worked Example - Sliding Block.

PS.2.2 Worked Example - Stacked Blocks - Free Body Diagrams and Applying Newtons 2nd Law.

PS.2.2 Worked Example - Stacked Blocks - Solve for the Maximum Force.

PS.2.2 Worked Example - Stacked Blocks - Choosing the System of 2 Blocks Together.

PS.2.3 Window Washer Free Body Diagrams.

PS.2.3 Window Washer Solution.

Newton's 3rd Law Pairs.

Internal and External Forces.

Applying Newton's 2nd Law.

8.0 Week 3 Introduction.

8.1 Polar Coordinates.

8.2 Circular Motion: Position and Velocity Vectors.

8.3 Angular Velocity.

9.1 Uniform Circular Motion.

9.2 Uniform Circular Motion: Direction of the Acceleration.

10.1 Circular Motion - Acceleration.

10.2 Angular Acceleration.

10.3 Worked Example - Angular position from angular acceleration..

11.1 Newton's 2nd Law and Circular Motion.

11.2 Worked Example - Car on a Banked Turn.

11.3 Demo: Rotating Bucket.

PS.3.1 Worked Example - Orbital Circular Motion - Radius.

PS.3.1 Worked Example - Orbital Circular Motion - Velocity.

PS.3.1 Worked Example - Orbital Circular Motion - Period.

12.0 Week 4 Introduction.

12.1 Pulley Problems.

12.2 Constraint Condition.

12.3 Virtual Displacement.

12.4 Solve the System of Equations.

12.5 Worked Example: 2 Blocks and 2 Pulleys.

13.1 Rope Hanging Between Trees.

13.2 Differential Analysis of a Massive Rope.

13.3 Differential Elements.

13.4 Density.

13.5 Demo: Wrapping Friction.

13.6 Summary for Differential Analysis.

14.1 Intro to resistive forces.

14.2 Resistive forces - low speed case.

14.3 Resistive forces - high speed case.

15.0 Week 5 Introduction.

15.1 Momentum and Impulse.

15.2 Impulse is a Vector.

15.3 Worked Example - Bouncing Ball.

15.4 Momentum of a System of Point Particles.

15.5 Force on a System of Particles.

16.1 Cases of Constant Momentum.

16.2 Momentum Diagrams.

17.1 Definition of the Center of Mass.

17.2 Worked Example - Center of Mass of 3 Objects.

17.3 Center of Mass of a Continuous System.

17.5 Worked Example - Center of Mass of a Uniform Rod.

17.6 Velocity and Acceleration of the Center of Mass.

17.7 Reduction of a System to a Point Particle.

18.0 Week 6 Introduction.

18.1 Relative Velocity.

18.2 Set up a Recoil Problem.

18.3 Solve for Velocity in the Ground Frame.

18.4 Solve for Velocity in the Moving Frame.

19.1 Rocket Problem 1 - Set up the Problem.

19.2 Rocket Problem 2 - Momentum Diagrams.

19.3 Rocket Problem 3 - Mass Relations.

19.4 Rocket Problem 4 - Solution.

19.5 Rocket Problem 5 - Thrust and External Forces.

19.6 Rocket Problem 6 - Solution for No External Forces.

19.7 Rocket Problem 7 - Solution with External Forces.

PS.6.1 Rocket Sled - Differential Equation.

PS.6.1 Rocket Sled - Integrate the Rocket Equation.

PS.6.1 Rocket Sled - Solve for Initial Velocity.

PS.6.2 Snowplow Problem.

20.0 Week 7 Introduction.

20.1 Kinetic Energy.

20.2 Work by a Constant Force.

20.3 Work by a Non-Constant Force.

20.4 Integrate adt and adx.

20.5 Work-Kinetic Energy Theorem.

20.6 Power.

21.1 Scalar Product Properties.

21.2 Scalar Product in Cartesian Coordinates.

21.3 Kinetic Energy as a Scalar Product.

21.4 Work in 2D and 3D.

21.5 Work-Kinetic Energy Theorem in 2D and 3D.

21.6 Worked Example: Block Going Down a Ramp.

22.1 Path Independence - Gravity.

22.2 Path Dependence - Friction.

22.3 Conservative Forces.

22.4 Non-conservative Forces.

22.5 Summary of Work and Kinetic Energy.

PS.7.1 Worked Example - Collision and Sliding on a Rough Surface.

23.0 Week 8 Introduction.

23.1 Introduction to Potential Energy.

23.2 Potential Energy of Gravity near the Surface of the Earth.

23.3 Potential Energy Reference State.

23.4 Potential Energy of a Spring.

23.5 Potential Energy of Gravitation.

24.1 Mechanical Energy and Energy Conservation.

24.2 Energy State Diagrams.

24.3 Worked Example - Block Sliding Down Circular Slope.

24.4 Newton's 2nd Law and Energy Conservation.

25.1 Force is the Derivative of Potential.

25.2 Stable and Unstable Equilibrium Points.

25.3 Reading Potential Energy Diagrams.

26.0 Week 9 Introduction.

26.1 Momentum in Collisions.

26.2 Kinetic Energy in Collisions.

26.3 Totally Inelastic Collisions.

27.1 Worked Example: Elastic 1D Collision.

27.2 Relative Velocity in 1D.

27.3 Kinetic Energy and Momentum Equation.

27.4 Worked Example: Elastic 1D Collision Again.

27.5 Worked Example: Gravitational Slingshot.

27.6 2D Collisions.

DD.2.1 Position in the CM Frame.

DD.2.2 Relative Velocity is Independent of Reference Frame.

DD.2.3 1D Elastic Collision Velocities in CM Frame.

DD.2.4 Worked Example: 1D Elastic Collision in CM Frame.

DD.2.5 Kinetic Energy in Different Reference Frames.

DD.2.6 Kinetic Energy in the CM Frame.

DD.2.7 Change in the Kinetic Energy.

28.0 Week 10 Introduction.

28.1 Rigid Bodies.

28.2 Introduction to Translation and Rotation.

28.3 Review of Angular Velocity and Acceleration.

29.1 Kinetic Energy of Rotation.

29.2 Moment of Inertia of a Rod.

29.3 Moment of Inertia of a Disc.

29.4 Parallel Axis Theorem.

29.5 Deep Dive - Moment of Inertia of a Sphere.

29.6 Deep Dive - Derivation of the Parallel Axis Theorem.

30.1 Introduction to Torque and Rotational Dynamics.

30.2 Cross Product.

30.3 Cross Product in Cartesian Coordinates.

30.4 Torque.

30.5 Torque from Gravity.

31.1 Relationship between Torque and Angular Acceleration.

31.2 Internal Torques Cancel in Pairs.

31.3 Worked Example - Find the Moment of Inertia of a Disc from a Falling Mass.

31.4 Worked Example - Atwood Machine.

31.5 Massive Pulley Problems.

31.7 Worked Example - Two Blocks and a Pulley Using Energy.

PS.10.1 Worked Example - Blocks with Friction and Massive Pulley.

32.0 Week 11 Introduction.

32.1 Angular Momentum for a Point Particle.

32.2 Calculating Angular Momentum.

32.3 Worked Example - Angular Momentum About Different Points.

32.4 Angular Momentum of Circular Motion.

33.1 Worked Example - Angular Momentum of 2 Rotating Point Particles.

33.2 Angular Momentum of a Symmetric Object.

33.4 If Momentum is Zero then Angular Momentum is Independent of Origin.

33.5 Kinetic Energy of a Symmetric Object.

34.1 Torque Causes Angular Momentum to Change - Point Particle.

34.2 Torque Causes Angular Momentum to Change - System of Particles.

34.3 Angular Impulse.

34.4 Demo: Bicycle Wheel Demo.

34.5 Worked Example - Particle Hits Pivoted Ring.

35.0 Week 12 Introduction.

35.1 Translation and Rotation of a Wheel.

35.2 Rolling Wheel in the Center of Mass Frame.

35.3 Rolling Wheel in the Ground Frame.

35.4 Rolling Without Slipping Slipping and Skidding.

35.5 Contact Point of a Wheel Rolling Without Slipping.

36.1 Friction on a Rolling Wheel.

36.2 Worked Example - Wheel Rolling Without Slipping Down Inclined Plane - Torque Method.

36.3 Demo: Spool Demo.

36.4 Worked Example - Yoyo Pulled Along the Ground.

36.5 Analyze Force and Torque in Translation and Rotation Problems.

37.1 Kinetic Energy of Translation and Rotation.

37.2 Worked Example - Wheel Rolling Without Slipping Down Inclined Plane.

37.3 Angular Momentum of Translation and Rotation.

DD.3.1 Deep Dive - Gyroscopes - Free Body Diagrams, Torque, and Rotating Vectors.

DD.3.2 Deep Dive - Gyroscopes - Precessional Angular Velocity and Titled Gyroscopes.

DD.3.3 Deep Dive - Gyroscopes - Nutation and Total Angular Momentum.

### Taught by

Prof. Deepto Chakrabarty , Dr. Peter Dourmashkin , Dr. Michelle Tomasik , Prof. Anna Frebel and Prof. Vladan Vuletic