Classical Mechanics — Daily Schedule Term 3

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Week 8 — Rotational/Internal Kinetic Energy — This Finishes Conservation Laws

Thursday, Oct. 30 — Study Chapter C11 — The derivation culminating in Eq. C11.14 is the central result showing that we can divide up the total kinetic energy into C.o.M. K.E. and internal K.E. — The latter is very often, but not necessarily, rotational kinetic energy — It can also be pulsations or any other internal motion — Problem Set 11: 1. Re-do Problem 3 on Exam 1 where a rod was modeled as 20 equal masses, but this time compute the kinetic energy 2. C11B.6; 3. C11B.8; 4. C11M.7

Week 9 — Newton’s Laws — Forces From Motion (By Taking Derivatives of Position)

Monday, Nov. 3 — Study Chapter N1 — Problem Set 12: 1. N1M.3; 2. N1M.4; 3. N1D.2 (you can use the result of N1D.1); 4. N1R.3 — I find that my diagram for circular motion scaling is both more exact and less complex than Moore’s Fig. N1.6
Thursday, Nov. 6 — Study Chapter N2 — Problem Set 13: 1. N2B.4; 2. N2B.8; 3. N2M.5; 4. N2M.8; 5. N2R.3

Week 10 — Motion from Forces (By Doing Integrals of Acceleration) — Statics

Monday, Nov. 10 — Review (if needed) Appendix NB on integral calculus, and then study Chapter N3 — Problem Set 14: 1. N3M.6; 2. N3D.4; 3. Use the same setup as N3D.4, except this time find a_x(t); 4. N3R.1; 5. N3R.2
Presentations for Monday:
- Grace: Newton had astronomical data for the average Earth-Moon distance and the period of the Moon’s orbit around the Earth which is 27.3 days (not the usually-quoted 29.5 days) — Demonstrate, as Newton did, that there is consistency between the downward acceleration of objects near the Earth’s surface, and the centripetal acceleration of the Moon as it proceeds around its approximately circular orbit (note, they are not identical, and also, you need equation N2.4a to demonstrate consistency) — Reference and explain the famous thought experiment that Newton included in The Principia — Grace found this fun simulation of Newton’s Cannon
- Sam: Can you lead us through something like Example 3.5, but for several steps of the elliptical orbit of Comet C/2025 R2 (SWAN)? — Start of a worksheet is here
Thursday, Nov. 13 — Study Chapter N4, which won’t be on Monday, Nov 17th’s exam — You can bring any review topics you want to discuss from Chapters C8-C11 and N1-N3 and/or I can be available for Q&A at multiple times on Saturday — Problem Set 15 on Chapter N4: 1. N4X.3 (try not to look at Moore’s answer until you have one of your own); 2. N4B.5; 3. N4M.7 (give everything symbolic names before plugging in, m for the mass of the paint can, c for the length of the ladder leg, 2a for the separation of the rollers, and 2b for the length of the tie rod); 4. N4R.2

Week 11 — Exam 2 — Linearly-Constrained Motion, Static Friction, and Kinetic Friction

Monday, Nov. 17 — Exam 2, covering Moore Chapters C8-C11 and N1-N3
Wednesday, Nov. 19 — Chapter N5 — Problem Set 16: 1. N5T.10 (this is much easier to handle as a change in momentum problem rather than as a change in kinetic energy problem); 2. N5B.6 (be efficient and use that this is a proportional reasoning problem and you’ll see that the units drop out); 3. N5B.7 (a lot like N5T.10 except this time he wants the distance traveled, not the change in speed); 4. N5M.10; 5. N5M.14 (instead of plugging in numbers, don’t even bother with that, and just give good names to the variables, especially the ones you are solving for, which we can call μ_s and d); 6. N5D.3

Week 12 — Coupled Objects

Monday, Nov. 24 — Study Chapter N6 — Coupled objects are really important to build intuition for — We draw a free-body diagram for each object in the system — We write down Newton’s Laws for each object in the system — The equations for each object depend on the others (maybe not all the others, often just the nearest neighbors) — The coupled equations are often challenging to solve, but when solved, they often have surprising and lovely properties — Problem Set 17: 1. N6B.2; 2. N6B.6 Part (a) only; 3. N6M.4; 4. N6M.10; 5. N6R.3; 6. N6A.1
Presentations for Monday:
- Grace: Redo N6M.10 that goes with Figure N6.7, except instead of a single pulley, imagine using the block-and-tackle shown below. The hook holds the worker’s chair. The rope that isn’t connected to anything goes to the worker’s hands. With this block-and-tackle system, now how much rope tension does the worker have to hang on to to keep in position?
- Sam: If the worker pulls the rope in by 1m meter, how much does the worker’s chair rise in Problem N6M.10? How much does the worker rise in Grace’s version of the problem (which has a block-and-tackle)? Make a work and potential energy argument that shows that the work done is in agreement with the potential energy increase?
- Sasha: There are two types of solutions to the situation pictured below, and they are called “normal modes.” In the easier of the two modes to analyze, the two objects move together. Begin by drawing free-body diagrams for each cart. The gravity and normal forces hardly matter (because they must balance), but include them for completeness. The important forces are the forces exerted by the three springs on the two carts. Each cart has two springs acting on it. If the two springs move together (that is, if x₁=x₂), what is the whopping simplification in the equations?
- Brian: Do the other normal mode for Sasha’s Problem.

Week 13 — Non-Uniform Circular Motion — Non-Inertial Frames and Fictitious Forces

Monday, Dec. 1 — Study Chapter N7 — You have already studied circular motion and know the fundamental result, but there are common and important additional examples of circular motion that are the focus of this chapter — As one of the many additional examples, Moore shows you how to deal with circular motion on a banked corner — Problem Set 18: 1. N7M.1; 2. N7M.4; 3. N7M.5; 4. N7M.9 (for N7M.9 you need to apply Eq. N7.18, including combining the two parts of N7.18 vectorially to find the length of the resulting acceleration vector)
- Presentations for Monday:
  - Grace: This is a slo-mo video of you on a swing — Use some equally-spaced frames in the video to measure some successive angles, and from the angles and an estimate of the length of the rope (two meters?), you can make an accurate motion diagram and thereby reasonably-accurately deduce the acceleration vectors at any point in the arc — For definiteness and for agreement with Sasha’s presentation, strive for the part-way point to be the point where you were 60° from vertical
  - Sam: Explain the proof/theory that culminates in Eq. N7.18
  - Sasha: Analyze the Grace-on-the-swing problem theoretically when she is part-way down at an angle 60° from vertical, assuming the maximum point on her arc was 75° from vertical, and the length of the swing is two meters, apply N7.18 to the analysis
  - Brian: Fictitious forces derivation (the theory for Section N8.5 of the coming chapter) for the Coriolis and centrifugal forces, and I will simplify by focusing only on a flat rotating coordinating system, such as a merry-go-round:
Thursday, Dec. 4 — Study Chapter N8, especially Sections N8.1, N8.2, and N8.3 which are quite important — Sections N8.4 and N8.5 may help build your intuition — Section N8.6 on fictitious forces is really pretty advanced, and except for one question about the Coriolis force on Problem Set 19, we will not consider fictitious forces any further
- Problem Set 19 for Thursday: 1. Do these five two-minute problems, N8T.1, N8T.3, N8T.5, N8T.6, N8T.7; 2. N8M.4; 3. N8M.9; 4. N8R.3; 5. At a point not too far from the North Pole, where we can treat the Earth as a flat merry-go-round spinning around the North Pole, Santa is rushing north at speed, v_Y=50 m/s to go pick up another batch of presents (a) Taking east (around the merry-go-round) as the X direction and north (towards the center of the merry-go-round) as the Y direction, compute the Coriolis force on Santa and his sleigh (for this part, don’t plug in any numbers and just assume Santa and his sleigh have mass m) (b) You should have found only an X-component in the Coriolis force in Part (a), and setting this equal to m a_X, what is the X-component (eastward component) of Santa’s acceleration as he rushes to the north? (still do not plug in numbers) (c) Assuming this eastward acceleration is constant and acts for a duration of 15 minutes, how much is Santa deflected to the east (now plug in numbers, using ω for the rotating Earth, and round your answer to the nearest number of kilometers)
- Presentations for Thursday: Each person pick one additional problem (please coordinate with one another) from the modeling, derivations, or rich-context Problems from Chapter 8 to present. I will Do N8D.2 (a taste of how Einstein approached the bending of light).

Week 14 — Projectile Motion — Oscillatory Motion

Monday, Dec. 8 — Chapter N9
Thursday, Dec. 11 — Chapter N10

Week 15 — Exam 3 — Kepler’s Laws

Monday, Dec. 15 — Exam 3, covering Moore Chapters N4-10
Thursday, Dec. 18 — Kepler’s Laws — Following Chapter N11 of Moore — Or perhaps use “Feynman’s Lost Lecture” — The goal is to see how Kepler’s Laws can be explained by Newton’s Laws, which is one of Newton’s definitive triumphs — Others in Newton’s day had conjectures and incomplete arguments about a gravitational force law — Newton pushed his “Law of Universal Gravitation” through to an unambiguous conclusion that explained Brahe’s, Kepler’s and Galileo’s work in a unifying sweep that has set the gold standard for all fields of science ever since
This presentation postponed, and will likely be modified to do an elliptical orbit instead of a linear acceleration with drag:
- I will provide you with a numerical integration example for a problem involving a constant accelerating force and a drag force that increases with velocity (such problems can often only be solved numerically, so only in special cases will you find them on problem sets or exams) — Here is an HTML version of a Jupyter notebook I built for another class: Home Run with Air Resistance — If you want the Jupyter notebook instead of the HTML version of the notebook, just knock the .html extension off of the URL