Modeling Shadows

Today we turned our attention to shadows, noticing first that we could get shadows of different sizes and compositions (some are really dark, some are really light, some have a dark nougaty core surrounded by a lighter halo...). 

How could we predict what kind of shadow we'll get?

We listed variables that might help determine the shadow size and/or composition:

• Bulb/ball distance (our objects were balls)
• Ball/wall distance (the shadows were cast on walls)
• Bulb size
• Ball size
• Bulb brightness
• Ball shape
• Bulb shape
• Alignment of ball/bulb/wall

This is by far the biggest list of variables that we've had so far, so they started to narrow down a few:

• They decided to stick with the "normal" incandescent bulbs that we had for the moment
• We'll stick with the 2.5" styrofoam balls on little wire stands for now
• They decided that the bulb brightness wouldn't really do much, and that we didn't really have a way to vary it anyway 
• We'll stick with a linear arrangement - ball, bulb, wall in a line, level with each other

That basically left us with the ball/bulb and ball/wall distances - we needed to determine their effects on the shadow size and composition.  How do we do that?  How many experiments will we need?  They landed on four experiments, controlling one distance and varying the other, and measuring/describing either the size or composition.  It's a nice number, since we have four groups.  Each group took data and presented their results to the rest of the groups:

Varying Ball/Bulb distance, examining shadow size

 

Varying Ball/Wall distance, examining shadow size 

Varying Ball/Wall distance, examining shadow composition 

 

Varying Ball/Bulb distance, examining shadow composition 

There are some great observations here, chief among them that the relationships aren't linear. With so many variables and nonlinear relationships, we went to a digrammatic method to try to make the predictions.  I didn't picture any of these, but each group was able to realize:

• Light travels in straight lines
• Shadow happens because (at least some of) the light from the bulb is blocked from reaching some location on the wall
• Not all of the light has to be blocked to make a shadow, but if it is, it'll be that dark part of the shadow
• Finding the dark part of the shadow (which we then named the umbra) wasn't too terribly difficult
• Finding where the edge of the shadow was (we called the light outer part the penumbra) was much more difficult - at this point, we had a billion rays going every which way!
• We needed a better method - one that required drawing fewer rays, for sure.  That's our motivation for doing our reading tonight.