Series
21 posts
A casual walkthrough of university-level vector concepts — cross products, the BAC-CAB rule, and coordinate transformation matrices — with encouragement to derive it yourself!
Deriving the Work-Energy Theorem and potential energy straight from F=ma using the chain rule — turns out it's shockingly simple and kinda blew my mind!!!
Springs are literally everywhere — turns out any potential looks like a spring up close, and that's why oscillations are SO important in physics!!
A step-by-step walkthrough of forced harmonic oscillation — what happens when the driving frequency meets the oscillator's natural frequency and resonance kicks in.
We're already at chapter 4 — diving into isotropic harmonic oscillators in 2D and 3D, where separating x and y equations leads to super cool Lissajous figures!
Struggling through chapter 5 of General Mechanics I, we dig into what inertial and non-inertial frames actually mean and why Newton's 1st law is sneakier than it looks.
We figure out what happens when your coordinate system won't stop spinning — hello Coriolis, centrifugal, and transverse forces!
We rederive the orbit equation using energy conservation — kinetic plus potential — then sneak in angular momentum and the u=1/r substitution to nail the same result.
Center of mass is basically lumping all your particles into one spot — a weighted average of positions that makes multi-particle dynamics way less of a headache!
We walk through how a system's total momentum equals m·v_cm, and why zero external force means momentum stays constant — aka conservation of linear momentum!
Spinning into rotational mechanics — we break down angular momentum for particle systems, torque, conservation laws, and how to re-express everything around the center of mass!
We dig into what a rigid body actually is (spoiler: it's an idealized fake lol) and grind through center of mass integrals for 3D solids and 2D plates, hemisphere included.
We spin up a planar rigid body, derive the moment of inertia from scratch, then work through the perpendicular and parallel axis theorems step by step!
A casual breakdown of the radius of gyration — what it really means to divide moment of inertia by total mass and slap a root on it to get an average distance feel.
We level up from fixed-axis rotation to planar motion where the axis moves too, and figure out that sneaky extra term that pops up in the torque equation T_T
We level up from planar rigid bodies to full 3D — think spinning a sweet potato on an arbitrary axis — and grind through the inertia tensor using direction cosines and cross products.
Still fuzzy on what a tensor really *is*, but studying the moment-of-inertia tensor this time actually made things click — let's muddle through it together!
We finally dig into principal axes — the special axes on a rigid body where all the products of inertia vanish, plus all the notation conventions that come with them!
We finally put inertia tensors and principal axes to work deriving Euler's equations of motion for a rigid body, with a spinning sweet potato thrown in for good measure.
Breaking down Euler angles — those three tricky angles (Θ, Φ, ψ) that link a fixed coordinate system to a rotating one — using a spinning top as our guide.
Picking up right where we left off — breaking down what ψ̇, θ̇, and φ̇ actually mean physically for a spinning top, then working out their vector directions step by step.