We have had a lot of rain in Minnesota this summer. As I sit here staring at the rain falling hard outside, I occurs to me that a cloud must be a very heavy thing in order to drop this much rain. It seems to me that figuring out the weight of a rain cloud would be a good Fermi problem. This cloud also dropped quite a bit of rain – how much of the cloud mass was lost as rain? Let's dig in …
I need to round up some facts about clouds.
- Clouds float for the same reason a balloon floats – cloud material weighs less than air.
- The density of air at a typical altitude and temperature is about 1.007 kg/m3.
This is an interesting number. Air is heavier than I thought. Think about it – a cubic meter of air weighs 1.0 kg, which is 2.2 lbs. For some reason that seems like a lot to me.
- The density of cloud droplets is about 1.003 kg/m3.
Given these typical characteristics, lets try a specific example.
A Cloud Example
The cloud that hangs over my home can be modeled as a cuboid.
- The cloud height (top to bottom) is 2 km (hCloud= 2 km)
- The cloud length is 10 km (lCloud= 10 km)
- The cloud width is 5 km (wCloud= 5 km)
- This cloud dropped 2 inches of rain (lRain= 2 inches)
A quick bit of Mathcad work gives me my estimate for cloud weight and the percentage of weight that rain represents.
I would have never guessed that a cloud weighed that much. I also am surprised that a major rainfall represents such a small portion of the weight.