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Wednesday, January 28, 2015

"Snowmageddon" Snowmen

Two children making a snowman complete with 'hair' in Finsbury Park, London, circa 1935
A 1935 photo of two children building a two-section snowman.
source: article

Greetings from the vestiges of New England! Yesterday the Worldwide Center of Math closed our office to work from home as one of the biggest blizzards Boston has ever seen blew through. Many of the area's universities and high schools have another snow day today. While movies and hot chocolate are always fun, nothing makes a snow day like a snowman.

To make a perfect snowman, you must start with snow. With Winter Storm Juno on her way out, that should be no problem for anyone on the northern east coast this week. The interns at the Center of Math can't wait for the snow to become a little more compactable so we can get building. What are the other conditions for a perfect snowman? We need a stable platform, two or three snowballs of varying size to stack, and decorations. However, we don't want just any snowman. We want an ideally shaped, mathematically supported snowman. We did a bit of research and extrapolated some information from this 2003 journal from West Virginia State College through our university library. Then we applied our own math to come up with explainations and instructions for our readers.

Luckily, we didn't need to cross-country ski into work yesterday.

An impressive snowman will have to be about as tall as a person. Here we’ll apply some mathematics to make sure we have the most stable sizes for the individual sections. We know that a loose handful of snow weighs next to nothing. However, when snow is packed tightly, it becomes a compact material that is perfect for building snow-people. An estimate that we’ll use (from the Weather Channel online) is that fluffy, powdery snow weighs 7 pounds per cubic foot when uncompacted and 20 pounds when compacted tightly.When we roll up spheres to make a snowman, we’re compacting that snow. We need to be sure that our base can withstand the weight of the two spheres above it. To do this, we’ll use the diameters suggested in the above article.

Our base sphere will have a diameter of 3 feet, half the height of our snowman. We use the equation for the volume of a sphere to get a volume of 14.14 cubic feet, and multiply this by 20 pounds of compacted wet snow to find that our base will weigh a staggering 282.8 pounds. Our middle snowball will have a diameter two-thirds of the length of the first, or 2 feet. Following the same formulas, we will see that the volume of this sphere is 4.19 cubic feet and weighs 83.8 pounds. You’ll have to have a strong friend in your snowman-making group to lift this section onto the stationary base! Lastly, we have to roll a head for the snowman. This section should have one-third the diameter of our base, or 1 foot. It will have a volume of 0.524 cubic feet, and a weight of approximately 10.5 pounds. Once this section is lifted onto the middle, the snowman is approximately 6 feet tall, stable, and ready to decorate.

If you play around with the numbers we have now, you’ll find a nice agreement in our ratios. We worked in thirds to find the diameters of the two upper spheres. If we add the masses of the two smaller spheres (83.8 and 10.5 lbs), the total is 94.3 lbs, approximately one-third of the mass of the largest snowman section.

Here's the snowman (and snow cat!) Tori and friends created earlier this week

What if you’re in an area that didn’t get a lot of snow because of Juno? The snow will likely pull up a lot of dirt or dead grass as you build it up into spheres. If you don’t have a lot to work with without getting the snow grassy or dirty, you’ll want to minimize the surface area that shows. This works best if you’re not worried about stability or want a more exciting, less classical snowman, because the shape will end up a lot different than the scenario we calculated above. In order to find this kind of snowman, we’ll use optimization, a method found in Worldwide Differential Calculus and Worldwide AP Calculus.
Let’s imagine we want to minimize the surface area, but have a two-sectioned snowman that is six feet tall like the one we talked about previously. Instead of typing out the thought process, math intern Tori has written out the problem on paper here:

Click the picture to enlargen!

As we can see, the radius of both spheres must be 1.5 feet. This will result in two spheres the size of the base of the first snowman. It will take a lot of strength to lift one section on top of the other! If you're confused about any of the math here, we invite you to watch one of the Worldwide Center of Math video lectures on optimization by clicking here.

As long as you have compactable snow, friends to help you lift the heavy middle (or top) section, and the ability to estimate the diameter of a sphere, you have a recipe for building a successful snowman this winter. If you build one using our guide, make sure to share a picture with us in the comments or tag us in your post.

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