Lesson 1: Scaling 1

Spherical Cow Approximation (SCA)

Given SCA, we can predict what will happen. For our sphere, volume is related to radius through the equation V = 4/3(pi)r³. If you double the size of r, the volume increases by a factor of 2³, or 8. We can predict how the surface area (A_surface= 4(pi)r²) changes. Doubling the size will increase surface area by a factor of 2², or 4. Ratio of mass (volume to surface area) changes like size³/size²

Surface area vs. mass

A_surface is proportional to M^2/3

Convergent Evolution

When physical laws present a difficult problem for life, there are sometimes very few workable solution so the same solutions are often arrived at over and over again, completely independent, during the course of evolution.

Diffusion

As a result of random thermal motion, molecules spread out evenly due to atoms and molecules moving with speeds that depend on their temperature.

Oxygen Diffusion

The amount of oxygen something needs depends on its total volume, which changes like size³. The amount of oxygen it can absorb through diffusion depends on surface area, which changes like size².

Isomorphic

Keeps the same shape with size change

How has evolution overcome the size limitation?

By growing complex, spongy lungs, your body increases the available surface area for oxygen diffusion by a factor of 100, easily enough to let you grow bid.

Human - Lungs

Insects - Tracheal Tubes

Fish - Gills

Volume and Surface Area

Cube of edge length L: SA (6L²) V (L³)

Sphere of radius R : SA (4(pi)R²) V (4(pi)R³/3)

Cylinder with Radius R and length L: SA (2(pi)R²+2(pi)RL) V (L³)

Trigonometry and the Pythagorean theorem

A²+B²=C²

sin = B/C = opposite/hypotenuse

cos = A/C = adjacent/hypothenuse

tan = B/A opposite/adjacent

Time

seconds - defined as the time required for a Cesium atom to vibrate 9,192,631,170 times

Distance

meters - distance traveled in 1/299,752,458 second about, 3.28 feet

Mass

kilograms - mass of a little cylinder kept in Paris, about 2.2 pounds familiar units

Electric charge

coulombs - the amount of electric charge contained by about 6.25×10¹⁸

Dimensions of measurement in physics

Length: L

Time: T

Mass: M

Charge: Q

Tera

10¹² a trillion

Giga

10¹⁹ a billion

Mega

10⁶ - million

Kilo

10³ - a thousand

Centi

10^-2 1/100

Milli

10^-3 1/1000

Micro

10^-6 1/1 million

Nano

10^-9 1/1 billion

Pico

10^-12 1/1 trillion

Scalars

Physical properties that can be represented by just a single number. Mass, diameter, temperature, density, pressure, metabolic rate, pH, age, and cost. All scalar properties can be fully described by just one number. We sometimes say that scalars are properties of things that have only a magnitude.

Vectors

Things that require us to specify both a magnitude and a direction to give a complete description. Force, displacement, velocity, magnetic, and gravitational fields