Physics 105 — Road to the Superegg

How to use this index
You don't follow this like a schedule — you use it like a map. Each time you sit down, find where you are, pick the next chapter or problem set, and run one focused session. Read Taylor to understand, then read the matching Sengupta lecture to see how Berkeley 105 actually framed and tested the material. When you finish a session, write the next question in your Notes. That's your re-entry point next time, even if next time is three weeks away.

Session Protocol — Every Subject, Every Time

Before · 5 min

Journal First

What do you already think you know? Where did you get stuck last time? Write without looking anything up.

During · 20–40 min

One Problem

Work it fully. If stuck after 10 min, write exactly where — not "I didn't know it" but which specific step broke.

After · 15 min

Synthesize

Write one rule in your own words. "For X, always Y because Z." Then write the next question in your index note.

For Concepts

Taylor → Sengupta

Read Taylor chapter first for understanding. Then matching Sengupta lecture to see what 105 emphasized and tested.

Your three tools, each with one job
Journal — messy thinking before and after every session. Apple Books — reading and annotating Taylor and Sengupta. Flag confusing passages only; don't take notes inside the book. Apple Notes — reference sheets (one per chapter) and session synthesis. This is where you compress and keep. Laptop — your calculation and problem-solving surface. This is where you actually work problems: archived problem sets as PDFs, Wolfram Alpha when an integral gets ugly, and the Sengupta lecture open alongside Taylor.

Mprereq

Math Refresh · Do This First

Linear Algebra & ODEs

~3–4 sessions · Targeted refresh, not a course

Linear Algebra — What 105 Actually Uses

LA·1Matrices — multiplication, transpose, inverse
LA·2Determinants — 2×2 and 3×3, geometric meaning
LA·3Dot and cross products — coordinate-free definitions
LA·4Eigenvalues and eigenvectors — finding them, physical meaning
LA·5Diagonalization — principal axes are diagonalization in disguise
LA·6Index notation & Einstein summation — Sengupta L1 uses this throughout

LA Goal

Eigenvalues feel routine. You can multiply 3×3 matrices confidently and know what a determinant means geometrically. Index notation doesn't scare you — it's how Sengupta writes everything in Lectures 1–2.

ODEs — What 105 Actually Uses

ODE·12nd order linear ODEs with constant coefficients
ODE·2The characteristic equation — roots and solutions
ODE·3Complex exponentials — e^(iωt) = cos + i·sin
ODE·4Real vs. imaginary parts — extracting physical solutions
ODE·5Coupled ODEs — writing as a matrix system

ODE Goal

Given ẍ + 2βẋ + ω²x = 0, you write the solution without thinking. Complex exponentials feel like tools, not tricks.

VIDEO3Blue1Brown "Essence of Linear Algebra" — eigenvalues episode especially
VIDEO3Blue1Brown "Differential Equations" series — first 4 episodes
SENSengupta Lecture 1 (Math Review) — read after the videos
KEY Q"What does an eigenvalue physically tell me about a rotating body?" — answer before Stage 4

0prereq

Physics 7A Brush-Up · Conceptual Only

Newtonian Intuition Rebuilt

~3–5 sessions · Your math is fine — your conceptual picture is rusty

Concepts to Rebuild (not rederive)

7A·1Newton's three laws — what each one actually says, physically
7A·2Free body diagrams — drawing them correctly for any setup
7A·3Work and energy — conservative forces, potential energy surfaces
7A·4Momentum and impulse — when is momentum conserved?
7A·5Rotation basics — torque, angular momentum, moment of inertia
7A·6Equilibrium — when is a system in equilibrium, is it stable?

Stage Goal

Given a physical setup — a pendulum, a block on an incline, a spinning disk — you immediately draw a free body diagram and identify which conservation laws apply. The physics feels physical, not just algebraic.

Practice & Resources

VIDEOWalter Lewin 8.01 — pick 4–5 lectures on Newton's Laws, Work & Energy, Rotation. Don't watch all 35.
PSETBerkeley H7A archived problem sets Ch. 1–4 — 2–3 problems per concept area diagnostically.
KEY Q"What is the physical difference between torque and force?"
KEY Q"Why is angular momentum conserved when there's no external torque?"
KEY Q"When I draw a free body diagram, what forces am I allowed to include and why?"

Sengupta Notes — Stage 0

Skip Sengupta for this stage. The Spring 2019 course assumes 7A is solid and opens immediately with coordinate transformation matrices. Use Lewin and H7A problem sets. K&K (Apple Books) is also excellent here — more physical and intuitive than Taylor for pure Newtonian mechanics.

1

Foundation

Newtonian Mechanics Done Properly

~4–5 sessions · Taylor Chapters 1–4

Chapters

Ch. 1Newton's Laws of Motion
Ch. 2Projectiles & Charged Particles
Ch. 3Momentum & Angular Momentum
Ch. 4Energy — conservative forces, potential energy

Stage Goal

Free body diagrams and energy methods feel completely automatic. You don't have to think about which approach — you just see it. Move faster than you think you should; you've seen all of this in Stage 0.

Sengupta Notes — Stage 1

The Spring 2019 course skips a dedicated Newtonian review. Use H7A problem sets here, not Sengupta. Return to Sengupta at Stage 2 (Lecture 3 onwards).

Practice & Resources

PSETBerkeley H7A archived problem sets — aligned to Ch. 1–4. 2–3 problems per chapter diagnostically.
VIDEOWalter Lewin 8.01 — Newton's Laws and Energy lectures if a concept isn't clicking from Taylor.
NOTEOne Apple Notes reference sheet per chapter — key equations, conditions, worked example skeleton.
DIAGAnnotate Taylor lightly in Apple Books — flag confusing passages only. Synthesize in Notes afterward.
KEY Q"What makes a force conservative, and why does it matter?"
KEY Q"When can I use energy methods instead of Newton's Laws?"
WARNIf stuck for more than 20 min, note where it broke and move on. Stage 1 is consolidation. Save the struggle budget for Stages 2 and 4.

2

Conceptual Heart · Sengupta L3–5

The Lagrangian Framework

~5–6 sessions · Taylor Chapters 6–7

Chapters

Ch. 6Calculus of Variations — E-L equation derived
Ch. 7Lagrange's Equations — constraints, generalized coordinates
—Cyclic coordinates and conserved quantities
—Noether's Theorem (Sengupta L5)
—Introduction to the Hamiltonian

Stage Goal

Given any system, you write L = T − V in generalized coordinates and derive equations of motion via Euler-Lagrange without straining. You understand why this is better than Newton for constrained systems like the superegg.

Practice & Resources

L3Sengupta Lecture 3 — E-L derivation and brachistochrone. Read alongside Taylor Ch. 6.
L4Sengupta Lecture 4 — spherical pendulum and Plateau's problem. Classic 105 exam setups.
L5Sengupta Lecture 5 — Noether's theorem and Hamiltonian intro.
VIDEOSusskind Theoretical Minimum — Classical Mechanics. Watch alongside Ch. 6–7.
PSETBerkeley 105 Problem Sets 1 & 2 — the actual exam preparation for this stage.
PSETTaylor Ch. 6 problems 6.1, 6.2, 6.11, 6.17 — exact problems assigned in Sengupta's course.
KEY Q"Why would I ever not use Newton's Laws?" — write a full paragraph before moving to Stage 3.
KEY Q"What is a generalized coordinate and what constraint does it encode?"
NOTEReference sheet: Euler-Lagrange equation + 3 worked examples (pendulum, bead on wire, Atwood machine).
WARNCh. 6 (calculus of variations) is abstract. Understand the result — the E-L equation — then move to Ch. 7. Don't get stuck in the derivation.

3

Stability · Sengupta L6–8, L17–19

Oscillations & Stability Analysis

~5–6 sessions · Taylor Chapters 5 & 11

Chapters

Ch. 5Oscillations — simple, damped, driven (SHO)
Ch. 11Coupled Oscillators & Normal Modes
—Small oscillations about equilibrium
—Stability — second derivative of potential
—Normal mode frequencies and eigenvectors

Stage Goal

You can determine whether any equilibrium is stable or unstable by examining the potential energy. Normal modes feel like a natural consequence of eigenvalues, not a mysterious technique.

Direct Connection to the Superegg

The superegg question is: perturb the egg from its balanced tip — does it oscillate back or fall over? That is exactly the small-oscillations-about-equilibrium framework from Sengupta L6.

Practice & Resources

L6Sengupta Lecture 6 — pendulum stability analysis. The model for how stability problems are set up. Key for superegg.
L7Sengupta Lecture 7 — linear SHO, phase space, damped SHO. Read alongside Taylor Ch. 5.
L8Sengupta Lecture 8 — forced oscillators, Green functions.
L17Sengupta Lecture 17 — small oscillations revisited with full Lagrangian machinery.
L19Sengupta Lecture 19 — coupled oscillators continued. Alongside Taylor Ch. 11.
PSETBerkeley 105 Problem Set 3 (archived).
VIDEOLewin 8.03 — coupled oscillators. His demonstrations of normal modes are outstanding.
KEY Q"What is the difference between stable, unstable, and neutral equilibrium?" — draw all three potential energy diagrams before moving on.
KEY Q"Why does ω² = ∂²V/∂q² determine stability — what happens when this is negative?"
NOTEReference sheet: stability conditions, ω² formula, potential energy diagrams, normal mode setup.

4

Most Technical · Sengupta L2, L13–16

Rigid Body Dynamics

~5–6 sessions · Taylor Chapter 10

Chapters

Ch. 10Rotational Motion of Rigid Bodies
—Inertia tensor — definition, computation, physical meaning
—Principal axes — the inertia tensor diagonalized
—Euler angles as generalized coordinates
—Euler's equations of motion
—Symmetric top (key worked example before superegg)

Stage Goal

You can set up the Lagrangian for a rigid body with a given symmetry. The inertia tensor and principal axes feel like tools, not mysteries.

Math Refresh Payoff

The inertia tensor is a matrix. Principal axes are its eigenvectors. Diagonalizing the inertia tensor is the same operation as diagonalizing any symmetric matrix. Stage M pays off here.

Practice & Resources

L2Sengupta Lecture 2 — coordinate transforms and angular velocity in index notation. Sets up the language for all rigid body lectures.
L13Sengupta Lecture 13 — rigid body motion intro, inertia tensor definition.
L14Sengupta Lecture 14 (π Day) — explicit inertia tensor calculations. Spend time here.
L15Sengupta Lecture 15 — Euler equations derived and applied.
L16Sengupta Lecture 16 — gravitational torques on rigid bodies. Directly relevant to superegg.
PSETBerkeley 105 Problem Sets 4 & 5 (archived).
VIDEOSusskind Theoretical Minimum — rigid body lectures.
KEY Q"What does the inertia tensor physically tell me — why is it a matrix and not a scalar?"
KEY Q"What is the connection between principal axes and eigenvalues?"
NOTEReference sheet: inertia tensor, principal axes, Euler angles diagram, Euler equations. Most important reference sheet in the plan.
WARNTwo sessions on the inertia tensor alone is completely fine. Do not rush. Every concept from Stage 4 appears directly in Stage 5.

5

Capstone · Everything Comes Together · Sengupta L18

The Superegg

~3–4 sessions · Berkeley 105 Exam Problems

What the superegg actually requires — and which stage built each piece: A superegg is a solid of revolution that balances stably on its tip. To understand why: you set up generalized coordinates describing the superegg rocking on a flat surface (Stage 2), exploit its axial symmetry to simplify the inertia tensor (Stage 4), write the Lagrangian with the no-slip contact constraint built in (Stages 2 & 4), derive equations of motion via Euler-Lagrange (Stage 2), and analyze whether the equilibrium at the tip is stable or unstable via potential energy curvature (Stage 3 — Sengupta L6 and L18). All seven stages converge into one elegant problem.

What to Do

L18Sengupta Lecture 18 — Generalized Lagrangians and Stability Analysis. The most direct preparation. Read it carefully, read it twice.
PSETBerkeley 105 archived midterms — rigid body constraint problems
PSETBerkeley 105 archived finals — stability analysis problems
KEY Q"What geometric property of a surface at the contact point determines whether a resting body is stable?"
KEY Q"How does the superegg's tip curvature differ from a sphere, and why does that change the stability?"
KEY Q"Which symmetries of the superegg reduce the degrees of freedom?"

You've Made It When...

✓You can explain in plain English why the superegg doesn't fall over
✓You can write the Lagrangian for a rocking rigid body on a curved surface
✓You can identify which symmetries reduce the problem's complexity
✓You can carry out the stability analysis and identify the geometric condition for stable balance
✓You understand why Newton's Laws would be a nightmare here — the Lagrangian was necessary
✓Physics 105 no longer feels like a closed door

Sengupta Lecture Map — When to Read Each Lecture

How to use the Sengupta notes
Read Taylor first for understanding, then the matching Sengupta lecture to see how Berkeley 105 framed and tested the material. Sengupta is denser and faster — it's a lecture transcript, not a textbook. Lectures without a stage tag are optional enrichment, not required for the superegg path.

L1Math Review — index notation, coord. transforms. Read during Stage M.

L2Angular Velocity & Rigid Body — rotating frames. Read during Stage 4.

L3Calculus of Variations — E-L derivation, brachistochrone. Stage 2 (Taylor Ch. 6).

L4Lagrangian Applications — spherical pendulum, Plateau's problem. Stage 2 (Ch. 7).

L5Noether's Theorem & Hamiltonian — symmetries, conservation laws. Stage 2.

L6Small Oscillations — pendulum stability, double pendulum. Stage 3. Key for superegg.

L7Oscillations — SHO, phase space, damped SHO. Stage 3 (Taylor Ch. 5).

L8Forced Oscillators — driven SHO, Green functions. Stage 3.

L9Central Force Motion — two-body, effective potential. Optional.

L10Central Force cont. — conic sections, Kepler's laws. Optional.

L11Scattering & Hamiltonian — scattering cross-section, Hamilton's equations. Optional.

L12The Hamiltonian — Legendre transform, phase space. Optional but enriching.

L13Rigid Body Intro — inertia tensor definition. Stage 4.

L14π Day — Inertia Tensor Calcs — explicit computations. Stage 4. Spend time here.

L15Euler Equations & Rotation — Euler equations derived. Stage 4.

L16Gravitational Torques — torques from gravity on rigid bodies. Stage 4. Direct superegg relevance.

L17Small Oscillations Return — with full Lagrangian machinery. Stage 3/5 transition.

L18Generalized Lagrangians & Stability — most directly relevant to superegg. Stage 5. Read twice.

L19Coupled Oscillators cont. — extended normal mode analysis. Stage 3.

L20Perturbation of Normal Modes — perturbation theory. Stage 5 if time allows.

L21Nonlinear Mechanics — beyond small oscillations. Optional enrichment.

L22Nonlinear Dynamical Systems — perturbation theory, chaos. Optional enrichment.

L23Canonical Transformations — infinitesimal transforms. Optional — graduate bridge.

Apple Notes — Copy This Into Your Index Note

Physics 105 — Index

RESOURCES

Taylor, Classical Mechanics (primary) · Kleppner & Kolenkow (Stage 0) · Sengupta Notes Spring 2019 (companion)

WHERE I AM

Stage: ___ Chapter/Lecture: ___ Last session: ___

Next question: ________________________________

STAGE M — Math Refresh

[ ] LA: Matrices, determinants, dot/cross products

[ ] LA: Eigenvalues & eigenvectors

[ ] LA: Index notation (Sengupta L1)

[ ] ODE: 2nd order linear ODEs & characteristic equation

[ ] ODE: Complex exponentials

[ ] ODE: Coupled ODE systems as matrix equations

Resources: 3Blue1Brown LA + DE series · Sengupta L1

STAGE 0 — Physics 7A Brush-Up

[ ] Newton's 3 laws (conceptual) [ ] Free body diagrams [ ] Work & energy

[ ] Momentum and impulse [ ] Rotation basics [ ] Equilibrium

Resources: Lewin 8.01 (select lectures) · H7A PSets Ch.1–4 · K&K Ch. 1–5

STAGE 1 — Newtonian Foundation (Taylor Ch. 1–4)

[ ] Ch. 1 Newton's Laws [ ] Ch. 2 Projectiles [ ] Ch. 3 Momentum [ ] Ch. 4 Energy [ ] H7A PSets

STAGE 2 — Lagrangian Framework (Taylor Ch. 6–7 · Sengupta L3–5)

[ ] Ch. 6 + Sengupta L3 [ ] Ch. 7 + Sengupta L4 [ ] Noether's Theorem L5

[ ] Susskind lectures [ ] 105 Problem Sets 1 & 2

STAGE 3 — Oscillations & Stability (Taylor Ch. 5 & 11 · Sengupta L6–8, L17, L19)

[ ] Ch. 5 + Sengupta L7–8 [ ] Stability analysis + L6 [ ] Ch. 11 + L17, L19 [ ] PSet 3

STAGE 4 — Rigid Body Dynamics (Taylor Ch. 10 · Sengupta L2, L13–16)

[ ] Sengupta L2 [ ] Ch. 10 + Sengupta L13 [ ] Sengupta L14 [ ] L15 [ ] L16 [ ] PSets 4 & 5

STAGE 5 — The Superegg · Capstone

[ ] Sengupta L18 (read twice) [ ] 105 archived midterms [ ] 105 archived finals

[ ] Can explain superegg stability in plain English [ ] Can write the Lagrangian for rocking rigid body

REFERENCE SHEETS (one per chapter, in Apple Notes)

LA Ref · ODE Ref · Ch.1 · Ch.2 · Ch.3 · Ch.4 · Ch.6 · Ch.7 · Ch.5 · Ch.11 · Ch.10