Thursday, December 27, 2012

Blog with the Wind

It's been pretty windy today, and I'm reminded of a problem I tried to work out when Hurricane Sandy came through in October. As the wind howled relentlessly, thumping against windows and bending trees, I wondered: how strong a gust of wind would it take to move my car? Unsurprisingly, this idle thought did nothing to ease my fears as Frankenstorm barreled down on my house. But occupying myself with the calculations did somewhat lessen my cabin fever.

So the first question is, what's keeping my car from moving? Gravity clearly plays a role, as does friction. Because it's a much simpler calculation, I first looked at this from the standpoint of friction. This is asking how strong a gust of wind it takes to slide my car across the ground.

To answer this, we need to know how strong the friction force is that prevents my car from sliding. Friction is pretty complicated, but it can be approximated simply. Friction = N&#956, where N is the normal force, and &#956 is the coefficient of friction, a measure of how hard it is to move an object composed of material A across a surface composed of material B. &#956 is determined experimentally, but it's easy to find estimates for common materials. Between the rubber of my tires and the pavement of my driveway, a decent estimate I found is 0.9. Now, the normal force pushing up against my car is just its weight. My Mazda3 comes in at about 1400 kg, which works out to 13,720 N. Multiply this by &#956 and the total force required to slide my car is something like 12,000 N, since it's probably a little wet out there anyway.

Okay, now how many newtons is a gust of wind? We want to know the pressure the wind exerts on my car. It turns out that we've seen wind pressure before in the form of the air resistance that kept our pets from becoming apocalyptically fast. We had a formula for terminal velocity, which was v = √(2mg)/(&#961AC). How does that help us? If we look at things from the perspective of the animal, then the animal is stationary and the wind is moving past it at a speed equal to the terminal velocity. We know terminal velocity means that the wind and gravity balance each other, so the force of that wind is equal to the weight of the animal. This means we have found an equation to determine the speed of a gust of wind based on the force of that wind (and vice versa).

Plugging in the numbers (my car's profile is 6.6 m2, and its C is 1-ish), we come out with an answer of 54 m/s, or about a 120 mph gust of wind. Whew. But what about gravity? It was at this point that a friend of mine on Facebook pointed out that, according to a "reliable source," an 80 mph wind can roll over a 6-wheel army truck. Why such a disparity between the two answers? Because two different questions are being asked. When we try to overcome gravity rather than friction, we're attempting to roll the car, a task that requires torque.

This turns out to be a more difficult question to answer because at least one hidden assumption we made earlier becomes an important unknown now. The new variable is the duration of a gust of wind. We need the duration of the gust of wind because we're attempting to move the car a certain angular distance, which takes time. Specifically, we need to tip the car such that the center of mass is no longer over the wheel-base. This requires tipping the car to about 57 degrees. For now, we'll estimate that a gust of wind lasts a single second.

Now we need to figure out the torque of a gust of wind. The torque is equal to the force of the wind multiplied by the moment arm, which is the vertical distance from that force to the car's axis of rotation. The most logical axis of rotation is the tire on the side of the car opposite the wind. But where is the wind blowing? We can assume that the cumulative effect of an evenly spread out gust of wind acts at half the height of the car. If my car is 1.46 m tall, then this length is .73 m. In order to lift the car any distance, this torque must exceed the torque due to gravity. Gravity acts at the car's center of mass, which is a vertical distance of roughly .88 m from the axis of rotation, and has a force equal to the weight of the car. This means the force of the wind must be 1.2 (the ratio of the weight's moment arm to the wind's moment arm) times the weight of the car, or about 16,500 N.

But that amount of force will take approximately forever to tip my car. What matters is the force in excess of 16,500 N. To get that, we need to know the angular acceleration (&#945) required to move a particular angular distance (&#952). The equation is &#952 = &frac12&#945t2. t is 1 second, and it turns out that 57 degrees is about 1 radian, so the angular acceleration is 2 rad/s2. We multiply this by the moment of inertia of my car, which is a measure of how the mass is distributed about the axis of rotation, and we get a net torque of 5000 Nm. Adding this onto the minimum needed to move the car, and we're at 23,000 N, which is a wind speed of 73 m/s, or about 163 mph.

Wow, that's twice my friend's estimate of 80 mph, and also significantly different from an earlier calculation I did. I'm inclined to believe my previous calculation is incorrect (or at least more incorrect) because I did it earlier in the semester before actually getting to torque. But what explains the disparity between my result and my friend's reliable source? My friend's source could be wrong, but a quick perusal of Wikipedia shows that 163 mph is enough to demolish a house. My best guess is that a 1 second gust of wind is too short to be a good average.

How long would an 80 mph gust have to blow to roll my car? Well, an 80 mph gust of wind only produces 5,300 N of force, so it's never able to lift my car. The minimum 16,500 N is a 141 mph gust. A 145 mph gust would flip my car in 3 or 4 seconds. Again, Wikipedia indicates 111-135 mph is enough to lift cars off the ground. I don't know where its data come from, but that means my rough calculation is in the right neighborhood, even if a little high.

The conclusion? Unless you're dealing with tornadoes or very powerful hurricanes, your car should be safe from the wind. Unless, you know, I'm wrong.

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