Exploring Mechanics / 0. Toolkit
This section establishes the language used throughout the repository. Drag things, drop things, drag sliders — every idea below is meant to be played with, not just read.
Length scales linearly, surface area scales with the square of length, and volume (so mass, for a fixed density) scales with the cube of length. Drag the slider and watch why a scaled-up elephant can't just be a scaled-up mouse — its legs would need to grow far thicker than its body just to keep up with its own weight. The two shapes below scale by the same factor k at the same time, on purpose: the ratios don't care whether the object is a cube or a sphere.
At k = 1.0, both objects are identical: everything scales together.
For a shape with a characteristic length L (a cube's side, a sphere's radius, an animal's height — anything with consistent proportions), surface area scales like L², because area is always a product of two lengths, and volume scales like L³, because volume is a product of three lengths. Scaling that shape by a factor k (so L → kL) therefore sends:
Area → k²·(original area), Volume → k³·(original volume).
A sphere's surface area is 4πr² and its volume is (4/3)πr³ — different constants out front than a cube's 6L² and L³, but the same k² and k³ scaling with size. That's the whole point of dimensional analysis: the exponents on length are a property of geometry (area = length × length, volume = length × length × length), not of the particular shape.
Drag the tip of vector A (blue). The dashed lines show its horizontal and vertical components. Turn on vector B to see how two vectors add tip-to-tail into a resultant R.
The same point has Cartesian coordinates (x, y) and polar coordinates (r, θ). Drag the point and switch grids to see why polar coordinates make circular motion look simple, while Cartesian coordinates make straight-line motion look simple.
There is one walker, taking steps at a steady pace on a moving airport-style walkway. Two observers watch that same walker and, if you asked each of them "how fast are they going?", they would give you two different — and equally correct — numbers, because "how fast" always means fast relative to something. Below, the top strip is what a person standing on solid ground next to the walkway sees; the bottom strip is what a person riding the walkway itself (moving along with the belt) sees. Same walker, same instant, two viewpoints.
u is how fast the walkway (the belt) itself moves, measured by the ground observer. w is how fast the walker is stepping, measured against the walkway surface under their feet — i.e. how fast they'd be moving if the belt were switched off.
Push or pull on a box at an angle, and that one push doesn't act in just one direction — it splits. Part of it acts horizontally (actually sliding the box along) and part acts vertically (pressing the box down into the ground, or lifting it up off of it). A free-body diagram is a picture of exactly how much of each force is "felt" in each direction, all at once, so you can add them up. Drag the applied force's angle slider below and watch the blue arrow's horizontal and vertical reach change — and watch the green (normal) and red (friction) arrows react to it.
Toggle forces on and off. The dashed dark arrow is the net force — the sum of every active force's horizontal parts and every active force's vertical parts, added separately. Un-check the normal force to see the classic student mistake: forgetting it produces a net force straight down through a box that should just be sitting still.