- Lecture 1. units
- I. The lecture on units (part one). The SI system of units and notes
- What agreement defines the SI systems of units?
- What is the date of the latest agreement?
- What are the important mechanical base units?
- What law defines the derived unit, the newton?

- II. The lecture on units (part two). naming, abbreviations, exponents, and prefixes
- Example 1. exponents and prefixes
- What is the correct way to write the name of the unit named after Sir Isaac Newton? What is the correct abbreviation for this unit?
- What is the correct way to write the name of the unit named after Michael Faraday? What is the correct abbreviation for this unit?
- If you were to have a unit named after you, what would you want this unit to be called (write it correctly)? What would you want the abbreviation to be?
- Write using a prefix.
- Write using a prefix.
- Write Mg using a prefix (mega-grams).
- Write pm using a prefix (pico-meters).

- III. The lecture on units (part three). derived units
- Example 2. convert from basic to derived
- Multiply and leave the answer in an appropriate derived unit. .

- IV. The lecture on units (part four). US Customary units
- Example 3. US Customary to SI
- What empire provides the origin for US Customary units?
- Is the United States officially metric?
- What is a remnant of the 1970s push for adoption of the metric system? Look around and see if you can find a remnant other than mine. If possible, take a picture.
- Unit Conversion Examples – Notes

- I. The lecture on units (part one). The SI system of units and notes
- Lecture 2. gravitation
- I. Newton’s Laws [video]
- If an unbalanced force that points to the right is acting on a particle, what direction does the particle accelerate?
- If a particle is moving to the left at 1 m/s and an unbalanced force that points to the right acts on the particle, what happens to the velocity? (a. increases to the left, b. decreases to the left, c. stays the same, d. flips to the right)
- If one particle exerts a force of 1 N acting to the left on another particle, what is the force that the second particle exerts on the first?

- II. Newton’s Law of Gravitation [video]
- What is the value of the universal gravitation constant (include units)?
- What is the force exerted by two 1 kg masses separated by 1 m?

- III. Acceleration due to gravity [video]
- Mass of earth is 5.972 × 10^24 kg.
- Mass of mars is 6.39 × 10^23 kg.
- Radius of earth (average) is 6378 km.
- Radius of mars (average) is 3,390 km.
- Using Earth’s geoid [https://rechneronline.de/earth-radius/] to find the radius of Earth for latitude, 45 degrees, and longitude, 22 degrees, what is the value of acceleration due to gravity at that location?
- What is the acceleration due to gravity for a mass that is 1 km above the surface of the earth at that latitude and longitude?
- What is the approximate acceleration due to gravity on earth at average radius?
- What is the acceleration due to gravity on Mars?

- I. Newton’s Laws [video]
- Lecture 3. Straight line motion (Dynamics Lecture 1)
- Straight line notes [PDF]
- 1. Description of a point
- 1. Describe the position of a point that is located 4 m in the x direction, 2 m in the y direction, and 3 m in the z direction, using conventional static unit vectors, .
- 2. Describe the position of a point that is located at an angle from the horizontal, and has a length of 2 m, using conventional static unit vectors, .

- II. Coordinate System Nomenclature
- 1. If , rewrite question 1 in part 1, using the base vectors.
- 2. If makes an angle of with the horizontal, rewrite question 2 in part 1 using the base vectors.

- III. Derivative of position (one-dimensional)
- If , what are and ?
- If null. what are and ?
- If , what are and ?

- IV. Aside
- There are no questions for this part.

- V. A mass accelerating under gravity (dropping)
- 1. A 1 kg mass is dropping in a uniform gravity field, , from an initial height, 10 m. How long does the mass take to hit the ground? What velocity is the mass travelling when it hits the ground?
- 2. A 2 kg mass is dropping in a uniform gravity field. If the mass takes 1 s to hit the ground, how high was it initially? What is the velocity of the mass when it hits the ground?

- VI. Curvilinear Motion (2D)
- If , what is ?
- If , what is ?
- Evaluate

- VII. Projectile Motion
- 1. What is the vector differential equation for projectile motion?
- 2. What are the two scalar differential equations for projectile motion?
- 3. What are the equations for x and y for projectile motion?
- 4. What are the equations for x and y velocity for project motion?
- 5. What is the equation for range?
- 6. Solve the quadratic equation to get an expression for
- 7. For the values, , (initial velocity in x and y), what is the range, R, for a 1 kg projectile?
- 8. For the projectile in 7, what is the time when the projectile hits the ground, .
- 9. For the projectile in 7, what is the y component of the velocity as a function of time?

- Lecture 4. Matrices and Coordinate Transformations
- Matrices [ NOTES]
- Coordinate Transformations [ NOTES]
- Example of transforming a vector
- Dr. Wright’s Coordinate Transformation Notes [ PDF] pages 1-4.
- PYTHON: How to use the matrices [ script].
- PYTHON: How to use the coordinate transformation matrix [ script].
- NOTE: python files are zipped. I cannot upload .py files to blackboard for security reasons. You will have to download, unzip, and then open in python to run the scripts.

- Lecture 5. Angular Velocity
- Dr. Wright’s Coordinate Transformation Notes [PDF] (Note: the last pages relate to lecture material that will be included in the next lecture.)
- Angular Velocity [NOTES] (This is an “old school” derivation lecture. Towards the end, you get the important formulas.)
- What is the coordinate transformation matrix for a rotation about z through the angle, ?
- What is for that matrix?
- What is for that matrix (the G matrix)? (Don’t forget the .)
- What is the angular velocity vector, , for this matrix? (NOTE: I cannot find upper case Omega in this equation editor.)
- Example: rotation about Y
- 1. What is the G matrix for the rotation about y?
- 2. Convert to a simpler expression using the angle addition formula.
- 3. What is the angular velocity for a rotation about y?

- Lecture 6. Velocity and Acceleration Formulas
- dof = degree-of-freedom
- Example 1. Simple Pendulum (1 dof) [video]
- Example 2. Wheel (2 dof) [video]
- Example 3. Ball (Spherical Coordinates, 3 dof) [video]
- Example 4. Double Pendulum (2 dof, planar, simple pendulum + simple pendulum) [video]
- Example 5. Airplane propeller with turning airplane (2 dof – wheel + wheel) [video]
- Example 6. Robot arm (3 dof, wheel + double pendulum) [video] NOTE: there are a few “fix ups” in the middle. These were added after the video was done.)
- Example 7. Governor (2 dof, wheel + simple pendulum) [done in class, and in next Lecture (Euler angles)]
- Example 8. 4-bar linkage (1 dof, simple pendulum + simple pendulum + simple pendulum) (delayed)
- Example 9. Slider-crank (1 dof, simple pendulum + simple pendulum) (delayed)
- Example 10. Gravitron (2 dof, wheel + simple pendulum) (delayed – should be somewhere else as I put this on tests more than once)
- If you wonder why I stop at 3 dof, mechanical complexity goes up steeply with each dof. 3 is pushing the limit for most dynamics problems.
- —
- Videos explaining how to use the python script
- http://calliope.us/wp-content/uploads/2021/10/accel_formula_symbol_howto_1.mov
- http://calliope.us/wp-content/uploads/2021/10/Screen-Recording-2021-10-15-at-11.37.40-AM.mov
- Python script (remember to unzip before you try to use)
- http://calliope.us/wp-content/uploads/2021/10/accel_formula_symbolic.py_.zip
- Compute the coordinate transformation matrices and angular velocities for the following situations following situations – Alternative Formats . Use the CTM formula (manually) and double check with the python script. Copy/paste the answers into a word-processed document.
- —
- Use the velocity and acceleration formulas to calculate the velocity and acceleration of the various mechanisms.
- This homework is extremely important in working through the method that I use to calculate velocity and acceleration. I follow a mechanistic approach, whereas the book follows a “learn through a million examples” approach. Please review the videos to see how I do this methodically. If you can figure out the method from these basic examples, the more general problems will not be all that hard. However, if you follow the approach in most books, each problem can be very challenging.
- Once we work through this by hand, you will be able to see how to follow the mechanistic approach in python. Once you get to that point, you can use python to produce solutions to very complicated problems without any extra effort than getting it to produce answers to simple problems. One of the challenges with kinematics is how mistake prone the algebra can be. If you set the problem up and let the solver do the algebra for you, you eliminate that source of mistakes.
- Of course, if you set the problem up incorrectly, you get gibberish. So, you need to understand the “do it by hand” method to assist in building an intuition on what look right or wrong when you get an answer from the solver.
- Example (video)
- Example (video)
- Example (video)
- Notes
- Problem 1 worked (video)
- Problem 2 worked (video)
- Problem 8 worked (video)
- Fall 2021 new videos with python script output
- Example (double pendulum – part 1) [video]
- Example (double pendulum – part 2) [video]
- Example (governor) [video]
- Table template [ png ][ pdf table_template.pdf – Alternative Formats ]
- Problem 17.8 and 17.9 [ part 1][ part 2][ part 3][ part 4 – working the problem with python]
- —
- Example (video)
- Example (video)
- Example (video)
- Notes
- Problem 1 worked (video)
- Problem 2 worked (video)
- Problem 8 worked (video)
- Fall 2021 new videos with python script output
- Example (double pendulum – part 1) [ video]
- Example (double pendulum – part 2) [ video]
- Example (governor) [ video]

- Lecture 7. Euler angles
- Read section 20.3 of Bedford and Fowler. Skip the stuff about “moment equation.” Focus on the stuff about angles.
- Lecture 1. chaining coordinate systems and angular velocity
- 1. What is the coordinate transformation matrix for a rotation through the angle, , about the z axis from the inertial, , coordinate system to the rotating, , coordinate system?
- 2. What is the coordinate transformation matrix for a rotation through the angle, , about the z axis from the rotating, , coordinate system to the rotating, , coordinate system?
- 3. What is the coordinate transformation matrix for the combined rotations described in questions 1 and 2?
- 4. Compute for the CTM in problem 1.
- 5. Compute for the CTM in problem 2.
- 6. Compute for the combined CTM in problem 3.
- 7. Show that for the combined CTM of problem 3.
- Lecture 2. more on angular velocity
- This lecture is a little out-of-order. Questions on this content will come later.
- Lecture 3. Euler’s theorem
- 1. Evaluate the determinant of the CTM in Lecture 1, part 1.
- 2. Solve for the CTM in Lecture 1, part 1 for and show that the normalized vector is the vector.
- NOTES [PDF1][PDF2][PDF3]
- Examples:
- Double pendulum [ video]
- Governor [ video]

- Lecture 8. Straight line dynamics
- Tools to work problems:
- 1. python script to comute acceleration formulas [zip-file]
- 2. formula sheet to manually compute acceleration formulas [pdf-file]
- velocity formula, part 1 [notes]
- acceleration formula, part 1 [notes]
- full velocity and acceleration formulas [notes]
- multiple rotating coordinate systems [notes1] [notes2]
- Use the velocity and acceleration formulas to calculate the velocity and acceleration of the various mechanisms. Use the table provided in “pdf-file” (attached to the assignment.
- Problems:
- .————-
- 2023, worked in class
- prob1
- prob2
- Read Bedford and Fowler, chapter 14
- notes
- part 1 (video)
- part 2 (video)
- part 3 (video)
- part 4 (video)
- —
- worked problems
- Ex14-3
- 14-3
- 14-4
- 14-6
- 14-10
- 14-11
- 14-36/14-37

- Lecture 9. Planar rigid body motion and Moment of inertia
- This is a double lecture. Due date for response to lecture will be extra long.
- Part One – System of Particles – Newton’s 2nd [video]
- The following questions apply to a 3 particle system.
- 1. What is the center of mass of these particles?
- 2. What is for this system?
- 3. What is the rigid body assumption?
- Part Two – rigid body and angular momentum [video]
- The following external forces are acting on the system of particles in Part One.
- 1. Compute
- 2. Compute
- NOTE: this is a made-up problem and the result of 1 will not equal the result of 2. In real life, they would.
- Part Three – Euler Equation [video]
- 1. Use the center of mass, , (computed in part one, 1) to compute the acceleration of the center of mass. HINT:
- 2. Use the center of mass (computed in part on, 1) and the answer to part Three, 1 to compute
- 3. Compute .
- Part Three – Euler’s moment equation for a planar rigid body [video]
- 1. What is a body fixed coordinate system?
- 2. What is the time derivative of the components of position vectors referenced to the center of mass of a rigid body in body fixed coordinates?
- 3. Show that the vector identity is true for the vectors
- 4. For a wheel rolling in a plane, where the z-axis is the hub of the wheel, what is the moment of inertia about z (look it up)?
- 5. For a slender rod serving as a pendulum arm, what is the moment of inertia about z (the axis of rotation) (look it up)?
- 6. What are the Newton-Euler equations for a planar rigid body (last equations in the lecture)?

- Lecture 10. Planetary Gear System
- Lecture 11. Using Python to Solve dynamics problems

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