Mechanics

Problems first

Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.

Prerequisites

• AP Precalculus
• Single-Variable Calculus

What this actually is

Newtonian mechanics is the claim that a force changes momentum, together with the models of the forces you actually meet: gravity near the earth, gravity between point masses, springs, and friction. The subject becomes mathematics the moment you write \(\mathbf F=d\mathbf p/dt\) as a differential equation. Energy and momentum are not extra laws. They are the consequences of Newton’s laws when the forces are what they are: momentum is conserved when the total external force vanishes, and mechanical energy is conserved when the forces are gradients of a potential.

OpenStax University Physics Volume 1 is free, calculus-based, and the working book. Halliday, Resnick, and Walker is the traditional paid text; no edition was verified. Kleppner and Kolenkow, and Morin’s Introduction to Classical Mechanics (ISBN 978-0-521-87622-3), are the harder books. Morin’s ISBN was confirmed in the Open Library catalog. MIT 8.01 is the course with exams. Irodov is the ceiling. Feynman is the preview, and Feynman is a bad place to learn to solve problems.

A free-body diagram that does not name the body is a decoration. Draw the body, then the arrows that act on it, then the equation.

Core results

Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.

1. Newton. In an inertial frame, \(\mathbf F_{\mathrm{net}}=d\mathbf p/dt\). For constant mass this is \(m\mathbf a\). Superposition: the net force is the vector sum. The third law, in the form of equal-and-opposite central forces, is what makes internal forces cancel in the momentum balance of a system.

   I wrote a proof. Reveal.

2. Energy. The work–energy theorem: the work of the net force equals the change in kinetic energy. If every force is conservative, mechanical energy is conserved. Gravity near Earth has potential \(mgy\). A spring has potential \(\tfrac12 kx^2\).

   I wrote a proof. Reveal.

3. Momentum and rotation. External impulse equals change in momentum. For a rigid body about a fixed axis, \(\tau=I\alpha\), and \(I\) depends on the axis by the parallel-axis theorem. Angular momentum about a point fixed in an inertial frame, or the center of mass, satisfies \(\boldsymbol\tau=d\mathbf L/dt\) under the usual hypotheses.

   I wrote a proof. Reveal.

4. Orbits. An inverse-square central force conserves angular momentum and energy, and the orbit is a conic. For gravity, closed orbits are ellipses with the center of force at a focus.

   I wrote a proof. Reveal.

5. Oscillations. \(m\ddot x=-kx\) has solutions of period \(2\pi\sqrt{m/k}\). Small oscillations about a stable equilibrium are harmonic, with \(k\) equal to the second derivative of the potential at the minimum. Damping and driving add the standard linear terms.

   I wrote a proof. Reveal.

The ladder

Four rungs, in order. Do not skip a rung because the next one looks more serious.

Tier 0 / First contact

Watch or read once. Do not study from this.

The Feynman Lectures on Physics, Volume I, opened from the official site
Read once, do not study from this. Read the table of contents of Volume I and one lecture if you want the voice. Feynman will not teach you to finish Irodov. Close it.

Tier 1 / Main text

The thing you work through.

OpenStax University Physics Volume 1, and MIT 8.01SC
Free calculus-based text. The course with exams is https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/ . Halliday is the paid traditional book; no edition was verified. OpenStax plus 8.01 is the better pair because you can start tonight and sit a real exam.

Tier 2 / Hard problems

The problem book.

Morin, Introduction to Classical Mechanics, ISBN 978-0-521-87622-3, and Kleppner & Kolenkow
Both books are paid. Morin’s ISBN was catalogued. Kleppner’s ISBN was not, so it is unnamed. The link is 8.01’s problem sets, which are the hard layer you can do before you buy either book.

Tier 3 / The ceiling

The hardest questions that exist at this level.

Irodov, Problems in General Physics, and Krotov
Irodov is the ceiling. Krotov’s Aptitude Test Problems in Physics is the olympiad companion: https://archive.org/details/ptitudetestproblemsinphysics . Use the mechanics parts. The answers in the back of Irodov are a check after the writeup.

• Tier 0 done once, then closed.
• Main text worked, exercises written.
• Hard-problem set written up.
• Ceiling problems attempted under the timer.
• Mastery gate passed.

Hardest questions

Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.

1. Irodov, the mechanics problems, in order, especially the later ones in each section
   The early items are drills. The later items add a constraint, a noninertial frame, or an energy argument you must notice.
2. Krotov, mechanics
   Olympiad flavor: the setup is the problem, and the physics is short once you see it.
3. MIT 8.01 exams
   Timed, with free-body diagrams that are part of the grade.
4. Morin, the problems on oscillations and central forces
   Morin is paid and unlinked as a file. These 8.01 topics are the public stand-in. Morin’s problems are hard because they ask for a clever frame or a conserved quantity.
5. Derive the range formula and the 45° result from Newton’s laws
   If this is not clean, Irodov will be misery for no good reason.

Mastery gate

Pass: one 8.01 exam, timed, plus ten Irodov mechanics problems with free-body diagrams and a sentence naming the conserved quantity if there is one. Fail if a solution uses a formula you cannot derive from \(\mathbf F=m\mathbf a\) in one line.

I passed this gate with written solutions.

Diagrams

Escalating sets

Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.

Set 1. Newton

1. Incline, frictionless.
2. Incline with friction \(\mu\), find the acceleration and the condition for sliding.
3. Atwood machine.
4. Two blocks in contact, a force on the first. Find the contact force.
5. A mass on a string in a horizontal circle. Tension.

Set 2. Energy and momentum

1. Prove the work–energy theorem in one dimension.
2. A spring launches a mass. Find the speed.
3. Elastic collision in one dimension, equal masses.
4. Inelastic collision, the combined speed.
5. Ballistic pendulum: combine the two conservation laws and say which applies to which stage.

Set 3. Rotation

1. Parallel-axis theorem for a point mass.
2. A pulley of nonzero moment of inertia in an Atwood machine. Write the equations. You need \(\tau=I\alpha\).
3. Rolling without slipping: \(v=\omega R\). Where does it come from?
4. Angular momentum of a particle, \(\mathbf r\times\mathbf p\).
5. A force through the center of mass does not torque about the center of mass. Prove it.

Set 4. Orbits and oscillators

1. Period of a spring.
2. Physical pendulum, small angles, period \(2\pi\sqrt{I/mgd}\). Derive it.
3. Circular orbit speed for gravity, from \(GMm/r^2=mv^2/r\).
4. Escape speed.
5. Projectile range, the attack, written cleanly.

Set 5. Ceiling

1. Ten Irodov mechanics problems.
2. Three Krotov problems.
3. One 8.01 exam.
4. Derive Kepler’s equal-area law from angular momentum.
5. Oral free-body diagram of the incline.

Return bank

Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.

1. Frictionless incline. Log a writeup
2. Atwood. Log a writeup
3. Projectile range at 45°. Log a writeup
4. Work–energy theorem. Log a writeup
5. Ballistic pendulum. Log a writeup
6. Spring period. Log a writeup
7. Escape speed. Log a writeup
8. Rolling condition \(v=\omega R\). Log a writeup
9. One Irodov problem that took over an hour. Log a writeup
10. Kepler equal areas. Log a writeup

Oral pass

State Newton’s second law, derive the incline acceleration, and explain when mechanical energy is conserved. No notes.

I explained this from a blank page, out loud, with no notes.