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I know it depends on the type of snow (dry or wet) and the rougness of the surface. I'm looking for an approximate rule of thumb answer. Assuming a reasonably smooth surface, at what angle it's likely to be free from snow?

Google results only discuss snow and slopes in context of skiing and roofs not caving in. Obviously, a snow safe roof (60+ degrees) will still accumulate a few inches of snow.

For example, imagine a sandwitch board sign. What's the minimum "uprightness" for it to not get obscured by snow and stay readable throughout the year?

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    $\begingroup$ A vertical surface can still be snowcovered. $\endgroup$ – gerrit Jun 5 '14 at 15:04
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    $\begingroup$ Are you trying to find a engineer an object that will stay "completely" free of snow? If so, this seems more like a engineering question that might reference applied physics & chemistry than anything directly related to Earth Science other than snow itself. $\endgroup$ – blunders Jun 8 '14 at 23:42
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An object on an angular surface will fall off, when forces exerted along the slope (gravitational, tangent to the surface: gt) are greater then those pressing the object to the slope (gravitational, perpendicular to the slope, gp) or working in the opposite direction on the slope (friction, $fr$).

As an example, $10\ \mathrm{cm}$ of snow, on a 0.5-by-1-meter surface:

  • $0.1 \times 0.5 \times 1 = 0.05\ \mathrm{m^3}$ of snow, weighing approximately $0.05 \ \mathrm{m^3}\times 300\ \mathrm{\dfrac{kg}{m^3}}$ (density of snow = $300\ \mathrm{\dfrac{kg}{m^3}}$) = $15\ \mathrm{kg}$.
  • Gravitational force exerted = $9.81\ \mathrm{\dfrac{m}{s^2}} \times 15\ \mathrm{kg} = 147\ \mathrm{N}$.
  • Frictional force $fr = 147\ \mathrm{N}\times 0.53$ (as a lower bound guess, see link) = $77.91\ \mathrm{N}$.

The snow will fall off if $gp > fr$, and this occurs at an angle of $\arcsin{\dfrac{fr}{gp}} = 32^{\circ}$.

Recalculating for the high bound friction coefficient ($1.76$), results in the snow not even falling off if the surface were vertical.

I think you already pointed out the biggest problem: any estimate is heavily dependent on the contact surface, type of snow and other conditions (temperature, wind, etc.). I haven't done a thorough looking around, but from the friction coefficients listed in this one article, I can only guess that you'll have a wide variety of ranges, making a rule of thumb difficult.

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What's the minimum "uprightness" for it to not get obscured by snow and stay readable throughout the year?

The answer to this problem is not in the angle at all. No amount of math will give you an angle at which you can reasonably presume your surface will remain clear. Snow can (and does) accumulate on vertical and even negatively sloped surfaces. Instead of thinking about the angle, you should be thinking about the materials and conditions involved.

Snow covered vertical signage

There are lots of factors that determine whether snow will stick, but a few that stand out most are:

  1. Temperature differentials between a surface and the atmosphere:

    This is particularly noticeable on surfaces of enclosures such as vehicles or buildings where the internal temperature causes the surface to be out of sync with the environment. You might notice snot collecting on the vertical glass surface of car side windows. This will be more pronounced in cars that were parked full of warm air when it started snowing. The warm glass would have melted the initial snow flakes. The wet surface will eventually freeze giving a textured surface for new snow flakes to sit on instead of sliding off.

    The same is true for objects with thermal mass that retain heat or cold after a sudden change in air temperature.

  2. Surface properties such as roughness.

    If even a few snow flakes find purchase on a surface, more will find a way to build on that and obscure your surface. This is far more important than angle.

  3. Atmospheric conditions such as humidity and the nature of the snow fakes formed.

To effectively keep a surface clear it should be ① smooth, ② quickly transfer heat to adapt to the current air temperature, and ③ be sheltered from being in contact with falling snow in the first place. Even highway signs with are smooth, vertical and thin metal are only so effective at staying clear and in some regions will have eaves protecting them from some snowfall.

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    $\begingroup$ +1 For a sign, guessing that some sort of snowmelt system would be the most cost effective solution. $\endgroup$ – blunders Jun 11 '14 at 18:13

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