Mathematics and Snow

card1tinyJust before Christmas 2005, a radio I’d left playing in a back room shorted out and started a fire. Fortunately, I was home. Though most of the damage was caused by smoke, the room needed to be gutted, which exposed a large beam hewn in 1792 when the farmhouse was built. We’d been living in the house for five years and had seen two and three feet of snow accumulate on the roof. The house never complained.

Building codes have changed since the house was built, and our contractor was required to have an inspector approve the framing before drywall was hung. He came with a tape measure and clipboard and passed everything except an unsupported span of original beam longer than twenty feet. It had been unsupported since 1792. A support meant placing an upright in the middle of the room. He argued there was no way to test the beam’s strength. I believed two hundred years was an adequate test, but code is code.

I thought it fair to assume that a building standing after two centuries was proven strong, but then a neighbor’s barn collapsed. He told me he knew the barn was close to failing when his horses refused to enter it. I have new respect for horses. I also have a new and unwelcome nervousness when snow begins to collect on my roof. Do Jack and Gilbert want to get out of the house because they’re cats and want to act like cats instead of lap warmers or do they know something I don’t know? Is that a normal creak and groan or is my house about to come tumbling down?

My house sits on a hilltop. Past owners planted trees on the south and west sides to block the wind, which hits 40+ mph several times a year and often gusts much higher than that. On a clear day I can see a wind farm on a ridge to the east. Power outages are not uncommon. My dairy farmer neighbor has a generator. Cows must be milked. I have an oil lamp and candles and woodstoves. The trees block the wind all too well. Snow piles up on the roof instead of being scoured off. I was ambivalent about the trees until I saw a neighbor’s snow-free roof stripped of many of its shingles in one particularly strong wind. I decided then that snow was preferable to crawling on my roof in winter trying to nail down shingles. But I long for the days of ignorance when a roof, like the sky, was once simply there.

Chicken Little couldn’t reassure himself by calculating sky loads, but K-12 students can measure snowfall and water content, which determines weight, and calculate snow loads.

Anyone who has ever shoveled new-fallen snow understands that wet snow weighs more than dry snow. Wet snow has higher water content. Snow to Liquid Equivalent explains how atmospheric and ground conditions affect the water content of snow and consequently its weight. Another factor affecting weight is the density of snowpack. Again, anyone who has returned from vacation to clear a driveway understands that the weight of accumulated snow increases with its density. Sometimes this can be a benefit. Inuits build snow shelters with sawn blocks of dense snowpack. They couldn’t do so with the dry powder that makes for good tobogganing.

Snow to water equivalent (SWE) complicates the calculation of snow load. Estimates or arbitrarily assigned values can be used with younger students. Older students can calculate the ratio in a number of ways presented in some of the following resources.

The Snow Booklet (92 pgs, 25.1 MB) is a good general resource for all grades. It describes how snow forms, how it falls, how it changes, how it melts, and how it affects us. The booklet provides detailed instructions on how to measure snow accurately and consistently under different weather conditions. Accurate snowfall measurement will result in accurate SWE and snow load calculations.

One method to calculate SWE requires a shovel, ruler, and five gallon bucket. See How to Calculate the Weight of Snow.

Another method uses the weight of a shovel-full of snow to calculate weight per square unit. (Depth is not considered in this method. Using cubic units would probably improve accuracy.) The difficulty guaranteeing consistent shovel-loads is not addressed either but will obviously affect accuracy. How do I calculate how much snow I shoveled? requires only a snow shovel and a bathroom scale.

Older students will gain insight with Snow Calculations Made Simple, Or Not. This exercise demonstrates how much easier it is to make calculations using the metric rather than the imperial system (UK). The UK and US share the same linear system but volume calculations are in imperial gallons in the UK. (1 imperial gallon = approximately 1.2 US gallons.) And when weight is converted to stone (14 pounds), it becomes even more cumbersome. The calculations involved with these conversions are best described as “harassing and training aids.” The Marine Corps always had a way with words.

How to Calculate Roof Snow Loads suggests one way to determine how much weight a roof is supporting, but the previous methods can suggest variations, some more accurate than others.

And finally, here are a two online volume to weight conversion tools, one for freshly fallen snow and a second for packed snow. These tools can be used to provide estimates and to judge calculations using other methods.

Comments (1)

  1. Thanks for your post! Great applications and things to stimulate students’ thinking!

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