Partnership for Integration of Computation into Undergraduate Physics

1. Go to [climate data](http://www.ncdc.noaa.gov/cdo-web/) and access temperature data for a recent winter for a city in a wintry climate. Download the data as a CSV file. Read the csv file into your computing environment and plot the daily temperature as a function of time, indicating when it is below freezing. Then cut out any temperatures at the beginning that are above freezing (up until the point where there is a long portion of time below freezing). 2. Derive the differential equation for ice thickness, taking into account heat flow through the ice from the water beneath and assuming the water is at the freezing point. As the ice gets thicker the heat flow slows. Answer: $$\frac{dx(t)}{dt}=\frac{\Delta T(t) k}{\rho L x(t)}$$ where x is the thickness of the ice, \(\Delta T(t)\) is the time-dependent temperature difference across the ice, k is the thermal conductivity of the ice, \(\rho\) is the mass density of the water, and L is the heat of fusion for ice. 3. For a constant temperature difference of, say, 10 degrees Celsius, determine the thickness of ice after one month. Note that you can solve the differential equation in (2) analytically for this simplification, but you should also test your numeric differential equation solver to make sure you get the same answer. answer: roughly 0.5 meters. 4. Develop an interpolating function for the data in (1) to be used in your differential equation solver. Apply that function and plot the thickness of ice as a function of time. 5. Building in the appropriate conditional statements to make sure x > 0 at all times, change your code for (4) to go beyond the contiguous freezing temperatures to determine when the ice begins and when it ends.