Tuesday, November 13, 2007

Why is Snow so Slippery?

Snow sports are a blast. I prefer snowboarding, but I would never miss out on a toboggan ride or a turn on one of those old fashioned snow saucers (What genius conceived of those anyway? Is there anything more insane that cruising down a hill on a disk with no obvious way to steer or stop? Of course, I love it anyway.) As embarrassing as it is, I've even been known to to ride an inner tube now and then.

The key to all those sports - as well as skiing, snowlerblades, ski bikes, and cafeteria lunch trays - is that things slide well on snow. You could leave it at that and simply go outside to have fun, but I just had to know a bit more.

If you hunt around, you'll see that there are at least two possible reasons why snow is slippery. One common explanation is that the melting point of ice rises as you squeeze it. Water ice is unusual in that way. This could explain why ice skates work. When you stand on skates, it creates very high pressures under the sharp blades. The pressure raises the melting point of the ice until it creates a thin layer of water, which is very slippery. (This is, however, not a universally accepted explanation.)

Pressure might work for ice skaters, but it's not much help for sleds, skis, snowboards, or any other device that slides on a large, flat surface. Because your weight is distributed over a big board, the pressures are very low. At best, you might raise the melting point of the ice by a degree or so, but if the temperature of the air and snow outside is more than a few degrees below freezing, you won't actually melt any snow with pressure.

Another possibility I've heard on occasion is that the friction of the board sliding on the snow creates heat, which melts some of the snow and creates a thin lubricating layer of water. Now, this explanation sounded just absurd to me. But I didn't want to dismiss it until a estimated just how much heating you might get from snowboarding down a hill.

You know what? I was stunned -- STUNNED -- to find out that cruising down a hill creates lots of heat. At a relatively modest 36 kilometers an hour (22 mph) down an equally modest hill (imagine a 15 degree slope or so), someone about my size (75 kilograms) creates about 300 watts of heat power due to friction between the snowboard and the snow. That's three times as much power as put out by a typical light bulb. If you've ever touched a lit incandescent bulb, you know that's hot. Theoretically, friction could heat the bottom of the board by nearly 40 degrees Celsius (about 100 degrees Fahrenheit)!

In the real world, the bottom of your board will never get that hot. It will only warm up to the point that it melts the snow. It takes energy to melt snow, and the melting ends up using the energy that would heat your board to any higher temperature than 0 degrees C (32 degrees Fahrenheit).

But suppose you were snowboarding on a day when temperatures went as low as at -40 degrees C, even moving slowly would generate enough heat from friction to melt the snow and provide a very thin layer of water for you to slide on. (Of course, -40 C is just about the lowest temperature ever recorded in Alaska, so you'd probably get an awful case of frostbite before you got to the top of the lift.)

So there you have it - it's friction, not pressure, that melts snow below your board and let's you whiz down the hill at a breakneck pace. Wacky.

For those of you who like the details, the math that led me to this conclusion is below.

The Math and Physics

You can change the parameters of the problem to model any sliding sport you like, but I'm using a snowboard in my analysis.

The first thing to do is figure out how much work is done when you slide on a snowboard. Work is defined as force times distance. Specifically, we're interested in the frictional force multiplied by the distance traveled.

The force of friction is just the force pushing the snowboard down onto the snow multiplied by the coefficient of friction (u).

Because the snowboard is moving downhill, the force pushing against the snow due to the mass of the snowboard and rider is reduced by the cosine of the hill slope.

Friction force = m g u cosine(theta)

= rider's mass (I'm using 75 kilograms)
g = acceleration due to gravity (0.8 meters per second squared)
u = coefficient of friction for a waxed board sliding on snow (about 0.04, according to Wikipedia)
theta = slope of the hill (I'm using a modest 15 degrees)

If you multiply this by the distance traveled on the hill, you get the total energy expended on the trip. If instead, you multiply the force by the velocity of the snowboard, you get the work per unit time. That's the same as the power (watts in SI units).

Power = m g u cosine(theta)* v

v = velocity (for this problem, I'm using 10 meters/second, which is about 36 kph, or 22.5 mph)

Plug all that in, and you find a power output of about 284 watts to slide down the hill.

The equation for heat conduction through a slab of material (such as a snowboard deck) is

Power = dQ/dt = k A (T2-T1)/L

k = thermal conductivity (about 0.25 Watts/(meters*degrees Kelvin))
A = area of board touching the ground (about .25 square meters for a typical board)
T2 = temperature on the hot side of the board (in Kelvin)
T1 = the temperature on the cold side of the board
L = the thickness of the board (I'm using one centimeter, 0.01 meters)

When I rearrange this to solve for the temperature difference between the two sides of the board (T2-T1) and plug in the numbers, I get temperature difference of about45 degrees.

As I mentioned above, that's the maximum temperature difference between the two sides, but the hot side should never get above the freezing point of water because the heat generated by friction would have to melt all the snow before it could lead to higher temperatures. That's because of the phase transition from ice to water that occurs at the freezing point (It's the same reason that water with any ice in it at all will have a temperature of exactly 0 degrees Celsius. You can confirm this by putting a pot of snow on the stove and turning up the heat. The temperature will rise to 0 degrees C and stay there until all the snow is melted.).

So, if you're out snowboarding when the air temperature (and top of your board) are at -5 degrees Celsius, the temperature of the bottom of your board be about 0 degrees Celsius when you're moving along at 10 meters/second. In fact, it will always be at about 0 degrees Celsius if you're moving at almost any reasonable speed, and there will be a very thin layer of water under it due to all the frictional energy you're generating by sliding down the mountain.