• Lecture 1. units
  • 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?
  • 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,  i with hat on top comma space j with hat on top comma space k with hat on top.
      • 2. Describe the position of a point that is located at an angle  30 degree from the horizontal, and has a length of 2 m, using conventional static unit vectors,  i with hat on top comma space j with hat on top comma space k with hat on top.
    • II. Coordinate System Nomenclature
      • 1. If  stack e subscript 1 with overbrace on top equals i with hat on top comma space stack e subscript 3 with hat on top equals j with hat on top comma space stack e subscript 3 with hat on top equals k with hat on top, rewrite question 1 in part 1, using the  e with overbrace on top base vectors.
      • 2. If  stack e subscript 1 with overbrace on topmakes an angle of  30 degreewith the horizontal, rewrite question 2 in part 1 using the  e with hat on top base vectors.
    • III. Derivative of position (one-dimensional)
      • If r with rightwards harpoon with barb upwards on top equals open parentheses 3 space m over s close parentheses space cos open parentheses open parentheses 5 space fraction numerator r a d over denominator s end fraction close parentheses t close parentheses space i with hat on top, what are  v with rightwards harpoon with barb upwards on top and  a with rightwards harpoon with barb upwards on top?
      • If null. what are  v with rightwards harpoon with barb upwards on top and  a with rightwards harpoon with barb upwards on top?
      • If  r with rightwards harpoon with barb upwards on top equals space open parentheses negative 3 space m over s squared close parentheses space t squared space j with hat on top, what are  v with rightwards harpoon with barb upwards on top and  a with rightwards harpoon with barb upwards on top?
    • 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,  g space equals space 9.8 space m over s, 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 x open parentheses t close parentheses equals sin space open parentheses t close parentheses, what is  fraction numerator d x over denominator d t end fraction?
      • If  y open parentheses t close parentheses equals 2, what is  fraction numerator d y over denominator d t end fraction?
      • Evaluate  a with rightwards harpoon with barb upwards on top equals space fraction numerator d over denominator d t end fraction open parentheses fraction numerator d x over denominator d t end fraction i with hat on top plus fraction numerator d y over denominator d t end fraction j with hat on top close parentheses
    • 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  t subscript f
      • 7. For the values,  h space equals space 10 space m,  stack v subscript 0 with rightwards harpoon with barb upwards on top equals 2 space m over s i with hat on top plus 4 space m over s j with hat on top (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, t subscript f.
      • 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
  • 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, theta?
    • What is  fraction numerator d open square brackets A close square brackets over denominator d t end fraction for that matrix?
    • What is  fraction numerator d open square brackets A close square brackets over denominator d t end fraction open square brackets A close square brackets to the power of T for that matrix (the G matrix)?  (Don’t forget the  theta with dot on top.)
    • What is the angular velocity vector, omega with rightwards harpoon with barb upwards on top, 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  cos space open parentheses pi over 2 plus phi close parentheses 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, theta subscript 1, about the z axis from the inertial, open square brackets i with hat on top comma space j with hat on top comma space k with hat on top close square brackets, coordinate system to the rotating, open square brackets stack e subscript 1 with hat on top comma space stack e subscript 2 with hat on top comma space stack e subscript 3 with hat on top close square brackets, coordinate system?
    • 2. What is the coordinate transformation matrix for a rotation through the angle, theta subscript 2, about the z axis from the rotating, open square brackets stack e subscript 1 with hat on top comma stack e subscript 2 with hat on top comma stack e subscript 3 with hat on top close square brackets, coordinate system to the rotating, open square brackets stack e apostrophe subscript 1 with hat on top comma space stack e apostrophe subscript 2 with hat on top comma space stack e apostrophe subscript 3 with hat on top close square brackets, coordinate system?
    • 3. What is the coordinate transformation matrix for the combined rotations described in questions 1 and 2?
    • 4. Compute  fraction numerator d open square brackets A close square brackets over denominator d t end fraction open square brackets A close square brackets to the power of T for the CTM in problem 1.
    • 5. Compute  fraction numerator d open square brackets A close square brackets over denominator d t end fraction open square brackets A close square brackets to the power of T for the CTM in problem 2.
    • 6. Compute  fraction numerator d open square brackets A close square brackets over denominator d t end fraction open square brackets A close square brackets to the power of T for the combined CTM in problem 3.
    • 7. Show that  omega with rightwards harpoon with barb upwards on top equals stack theta subscript 1 with dot on top stack e subscript 3 with hat on top plus stack theta subscript 2 with dot on top stack e apostrophe subscript 3 with hat on top 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  open parentheses open square brackets A close square brackets minus open square brackets I close square brackets close parentheses open square brackets x close square brackets equals open square brackets 0 close square brackets for the CTM in Lecture 1, part 1 for  open square brackets x close square brackets and show that the normalized vector is the  stack e subscript 3 with hat on topvector.
    • NOTES [PDF1][PDF2][PDF3]
    • Examples: 
    • Double pendulum [ video
    • Governor [  video]
  • Lecture 8. Straight line dynamics
  • 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  sum for i of m subscript i stack a subscript i with rightwards harpoon with barb upwards on top 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  sum for i of stack r subscript i with rightwards harpoon with barb upwards on top cross times m subscript i stack a subscript i with rightwards harpoon with barb upwards on top
    • 2. Compute  sum for i of stack r subscript i with rightwards harpoon with barb upwards on top cross times stack F subscript i with rightwards harpoon with barb upwards on top
    • 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, r with rightwards harpoon with barb upwards on top, (computed in part one, 1) to compute the acceleration of the center of mass. HINT:  stack a subscript i with rightwards harpoon with barb upwards on top equals a with rightwards harpoon with barb upwards on top plus fraction numerator d squared stack R subscript i with rightwards harpoon with barb upwards on top over denominator d t squared end fraction
    • 2. Use the center of mass (computed in part on, 1) and the answer to part Three, 1 to compute  r with rightwards harpoon with barb upwards on top cross times sum for i of m subscript i fraction numerator d squared stack R subscript i with rightwards harpoon with barb upwards on top over denominator d t squared end fraction
    • 3. Compute  sum for i of stack R subscript i with rightwards harpoon with barb upwards on top cross times stack F subscript i with rightwards harpoon with barb upwards on top .
    • 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  u with rightwards harpoon with barb upwards on top equals 2 space i with hat on top space plus space 3 space j with hat on top comma space v with rightwards harpoon with barb upwards on top space equals space 3 space i with hat on top space plus thin space 4 stack space j with hat on top comma space w with rightwards harpoon with barb upwards on top space equals space minus i with hat on top space plus space 4 space j with hat on top
    • 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