GEOL 5470

Homework/Labwork 3^rd week Humphrey

Read the Note on Models, especially the part on ABSTRACTION. We will discuss in class.

Our starting problem this week is to investigate falling bodies (hopefully not people!, but rocks). Basically the question is simple: How does a mass of rock react to gravity. The question ranges from the orbits of planets (since planets are actually falling in the sun’s gravity well), to the trajectory of magmatic ejecta, to the falling of a single rock on a cliff, to the settling of a grain of sand in the wind or in water, all the way down to the slow downward drift of a dead radiolarian. Although the forces are simple, just gravity and drag, the details are complex. After reading the ‘note on what is a model’, we realize that we need to specify some small part of the problem, and even then abstract only the main parts of the problem.

So let’s look at probably the simplest of the above problems: The fall of one smallish rock straight down from a cliff. Basically we want to duplicate Galileo’s experiment of dropping a rock off the tower of Pisa, but we want to know what happened during the fall. We will discuss the physics in class. But after the discussion we will be left with the abstraction that the motion can be approximated by a = g – C_drag * rho * V² *A/M , where a is the acceleration, g is gravity, C is the drag coefficient (assumed a constant), rho is density of air (or fluid), V is velocity and A is cross-sectional area. M is the mass of the rock, which is approximately rho_rock*4*pi*r³/3. If we recognize that in this problem the motion is downward (z direction) with time (t), and that the acceleration can be written as the time derivative of V_z, then we can turn the above equation in 2 unknowns (a and V) into one differential equation in 1 unknown (V): dV_z(t)/dt = g – C_drag * rho * V_z² *A /(M).

So, once we have done all the thinking that went into abstracting the problem, the easy part is turning this into a Numerical model to produce plots of the approximate results of a Pisa experiments. That’s for you to do.

1. Make a model of a falling rock. Include gravity and a 1^st cut at drag assuming a constant drag coefficient (which it isn’t). Have the rock start at zero velocity and let it fall until terminal velocity is approximately reached. You can let gravity be 9.8 m/s². Assume the density of air is 1kg/m³, density of rock is 2700kg/m³, the drag Coeff is assumed to be a reasonable 0.75, and the size of the rock is assumed to be about 1cm, then C_drag * 3*rho /(4*rho_r*D ) will be about 0.02, and we are solving ): dV_z(t)/dt = g – 0.02* V_z²

A Use the scipy solver ‘odeint( your derivative function, start value for V, a vector of times to solve at)’

B Use your own solver, we will discuss in class.

If you try your code for different rock sizes, you will get errors away from what is realistic; much smaller than 1cm you start getting significant viscous effects, and larger than 10cm you start running into compressibility and trans-sonic effects as you approach terminal velocity (neither size is easy to treat).