For the next few weeks, we'll be discussing the dynamics of how systems change through time. This means that we'll be incorporating some physics into the class, starting with one of the most fundamental physical equations:

Force = mass * acceleration

This equation means that any time an object has a net force on it, it will accelerate at a rate that depends on how massive the object is. The more massive an object, the more slowly it will accelerate for a given force. You should also note that forces are additive and that their direction matters. For example, if an object is traveling at a constant velocity (i.e., acceleration = 0), then all of the forces acting on the object balance each other.

Terminal velocity: We can see a force balance in action when we consider an object's terminal velocity as it moves through a fluid. When that object is no longer accelerating (it has reached its terminal velocity), the force moving the object forward is balanced by the drag force slowing the object down. At that point, there is no net force on the object.

Stokes Law: Putting this into action, we can consider how spherical particles fall or rise as a function of gravity. At what rate do bubbles rise and sand grains settle? Stokes Law expresses this velocity assuming 1) the particle is a sphere, 2) the particle is not feeling any "edge effects," 3) the particle has reached its terminal velocity, and 4) the fluid has a lower Reynolds number. The law is:

v = 1/18*[(ρ_p-ρ_f)/μ]gD²

where v is the particle's velocity, ρ_p is the particle's density, ρ_f is the fluid's density, μ is the fluid's viscosity, g is the acceleration due to gravity, and D is the particle's diameter. This comes from the balance of the particle's buoyant force and its drag force.

F_b: The buoyant force is a function of the particle's mass in the fluid and gravity. Since F=ma, we have

F_b = [(ρ_p-ρ_f)(πD³/6)]g

F_d: The drag force is a function of the energy used to move the fluid out of the way of the particle. It looks similar to kinetic energy (E=½mv²) divided by D, with an additional term called the drag coefficient (C_d) that takes into account the effects of turbulence.

F_d = ½ρ_f(πD²/4)v²C_d

C_d and Re: The drag coefficient is a function of fluid turbulence, which is described by the Reynolds number. The Reynolds number is

Re = ρ_fvD/μ

When Re>2000, the flow is turbulent; when Re<500, the flow is laminar; intermediate values represent a transition from laminar to turbulent flow. At Re<10, C_d=24/Re. So

C_d = 24μ/(ρ_fvD)

Plugging this back into the equation for drag force, we have

F_d = ½ρ_f(πD²/4)v²[24μ/(ρ_fvD)] = 3(πD)μv

Stokes Law can now be derived by setting F_b=F_d. Whew!!