Series
Classical Mechanics I Studied
27 posts
- #1
Vectors and Polar Coordinates
A breezy walkthrough of cross products, the BAC-CAB rule, coordinate transformation matrices, and polar coords — all the vector stuff that actually felt new in university physics.
· 7 min read - #2
Work-Energy Theorem and Conservative Forces
Sneakily deriving the Work-Energy Theorem from Newton's second law via the chain rule, then defining potential energy and unpacking what conservative forces actually are!
· 5 min read - #3
Oscillations and Damped Harmonic Motion
Zoom in on literally any potential with a Taylor expansion and BAM — it's a spring, which is why oscillations pop up absolutely everywhere in physics.
· 6 min read - #4
Forced Harmonic Oscillation and Resonance
What happens when the driving frequency hits just right? We work through forced harmonic oscillation — no damping first, then with resistance — to see why resonance is so wild.
· 4 min read - #5
2D and 3D Isotropic Harmonic Oscillators and Lissajous Figures
Jumping into the 2D/3D isotropic harmonic oscillator, separating equations of motion and eliminating t to uncover the elliptical paths and Lissajous figures hiding inside!
· 6 min read - #6
Inertial and Non-Inertial Frames and the Galilean Transformation
Turns out Newton's first law isn't obvious at all — it's secretly guaranteeing that inertial frames exist, and that's the whole setup for the Galilean transformation.
· 8 min read - #7
Rotating Reference Frames: Coriolis, Transverse, and Centrifugal Forces
We figure out the discrepancy between a stationary frame and a purely-rotating one — and that's exactly where Coriolis, centrifugal, and transverse forces come from!
· 11 min read - #8
Kepler's Laws: Ellipse Law, Equal-Area Law, and Harmonic Law (Part 1)
A breezy Newton sidebar on universal gravitation sets the stage for Kepler's laws — including a quick sanity check on whether treating planets as point masses is actually legit, heh.
· 10 min read - #9
Kepler's Laws: Ellipse Law, Equal-Area Law, and Harmonic Law (Part 2)
We derive the polar equation of an ellipse straight from its definition, then match it to the orbital equation — and yeah, gravity really does give you ellipses!!!
· 4 min read - #10
The Orbit Equation via Energy Methods
We re-derive the orbital equation using energy conservation instead of angular momentum — plus a quick vibe check on why gravitational potential energy is negative.
· 7 min read - #11
Orbital Stability
Why does Earth just keep cruising in its orbit without getting knocked off track? Turns out there's a sneaky spring constant hiding in the radial equation of motion!
· 3 min read - #12
Series Quick-Reference Overview
A handy visual quick-reference overview for the series, packed with everything you need at a glance.
· 1 min read - #13
Center of Mass (CM)
Breaking down the center of mass — turns out it's just a mass-weighted average of particle positions, and yeah, once you see it that way it actually makes sense.
· 2 min read - #14
Momentum of a System of Particles
We sum momenta across a whole system of particles, watch internal forces cancel out via Newton's third law, and land smack on conservation of linear momentum.
· 2 min read - #15
Angular Momentum of a System of Particles
We decompose total angular momentum into an orbital piece (system as one lump at the CM) and a spin piece (particles wiggling around it) — and show why those cross terms vanish.
· 5 min read - #16
Rigid Body Dynamics and the Center of Mass of a Rigid Body
We swap the sigma for an integral and hunt down the center of mass of a rigid body — basically a particle system with locked spacing — then crunch through a solid hemisphere and a shell.
· 3 min read - #17
Moment of Inertia: Perpendicular Axis Theorem and Parallel Axis Theorem
A casual, build-it-up walkthrough of moment of inertia for flat rigid bodies, covering why it acts like rotational mass and how the perpendicular and parallel axis theorems let you shift between axes.
· 6 min read - #18
Radius of Gyration
Breaking down what the radius of gyration k actually is — why we square distances, what dividing by mass really does, and how k is just the mean of squared distances laid bare.
· 2 min read - #19
Physical Pendulum and Center of Oscillation
A casual walkthrough of how a rigid body swings as a physical pendulum, and why its period maps neatly onto the classic simple pendulum formula via the radius of gyration.
· 4 min read - #20
Rigid Body in Planar Motion
Fixed-axis rotation is so last chapter — now the axis itself moves, and we figure out exactly when that messy extra torque term thankfully drops to zero.
· 3 min read - #21
Three-Dimensional Motion of a Rigid Body
We level up from flat pancakes to sweet potatoes — spinning a 3D rigid body on an arbitrary axis and cooking up the full inertia tensor from direction cosines.
· 4 min read - #22
The Inertia Tensor
A casual, honest walkthrough of finally grokking the moment of inertia tensor — what tensors actually are, why they're 'absolute', and how the matrix form works.
· 8 min read - #23
Principal Axes of a Rigid Body
A fun intro to principal axes of a rigid body — why they matter, how products of inertia vanish, and all the notation conventions that come with them.
· 5 min read - #24
Euler's Equations of Motion for a Rigid Body
We finally dig into full-on 3D rigid body rotation — inertial frames, rotating frames, and how Euler's equations of motion fall out of it all. lol.
· 4 min read - #25
Euler Angles
Breaking down Euler angles — theta, phi, and psi — and how three coordinate systems team up to fully describe a spinning top's orientation in space.
· 7 min read - #26
Euler Angles (Part 2)
Breaking down the angular velocity vector into spin, nutation, and precession — yes, I hate just swallowing equations, but here we go anyway.
· 3 min read - #27
Angle ψ Between the Ellipse Axis and the x-Axis
Finally worked out the actual derivation for the angle ψ the ellipse's major axis makes with the x-axis — something my textbook just handed me without proof and I never questioned until now.
· 1 min read