The Horns Rev wind farm is 15 miles off the west coast of the Danish peninsula, near Ejsberg. You can see a movie of the wind farm here: http://www.youtube.com/watch?v=BOnfbTyMnzA. There are 171 wind turbines like this spread out over the shallow North Sea floor, some of which (the Horns Rev 2 field) can generate 6 MW of power. As we discussed in class (and also in the 4th edition) the work done by a torque is (torque x rotation angle in radians). To get the power from the wind then it is (torque x angular velocity in radians per second). Assuming no losses, the 6 MW electrical power generated must correspond to an equivalent 6 MW of power done by work from the wind on the turbines. Since there are three blades, then each blade must have 2 MW of power done on it from the wind. A typical rotation speed is 20 rotations per minute, which corresponds to 40pi or 120 radians per minute, or 2 radians per second. That means the TORQUE provided by the wind must be 2 MW/2 radians per second = 1 million Newton-meters of torque.

These turbines are absolutely gigantic.  Look here: http://en.wikipedia.org/wiki/Wind_turbines (see "Record-Holding turbines" at the bottom). The Enercon 6 MW E-126 is 198 meters (650 ft!) tall, and the turbine is 126 meters (413 ft!) in diameter. That means each propeller blade is 63 m (206 ft!) long. If we assume that the wind force on the turbine blade acts at its center, 31.5 meters from the rotation axis, then the 1 million Newton-meters of torque from the wind, corresponds to a force from the wind on the turbine of 32,000 Newtons which is equivalent to the weight of a mass of 3200 kg, which is 6400 pounds or a little over 3 tons on each of the three propeller (turbine) blades!

In this short essay, consider that there is an extremely large force from the wind on the turbine which provides a torque which does work (and generates a lot of power). Answer the following questions in your essay:

1.  What is the net torque on the turbine? How do you know? List all the torques on the turbine which make this happen.

2.  Considering that there is a giant force from the wind ON the turbine blade, how is Newton's 3rd law satisfied? How could you tell that Newton's 3rd law is satisfied?

First, we should carefully define the components of our interacting system.

We have the turbine itself, obviously, but let's think about what the turbine actually is. It's the part of the system that looks like a fan—the blades and the hub to which they are attached. It's the part of the system that we can see rotating.

The turbine is attached to the generator, the next part of the system. In this problem, we should consider the turbine and generator to be two distinct things, though it may be more intuitive to consider them as one—don't! Instead, think of it this way: the turbine turns the generator, and the generator resists being turned. Consequently, if the turbine is turning, the generator makes it stop. We'll call this a stopping torque—a torque that causes the turbine to slow down and stop.

The third component is the air in which the turbine is immersed. When the air blows against the turbine, it will cause the turbine to turn. So, if the turbine isn't turning, blowing air makes it start. We'll call this a driving torque—a torque that causes the turbine to start and speed up.

Now, with all the components (and torques on the turbine) established, we may consider the questions being asked. Let us take them one at a time.

1. What is the net torque on the turbine? How do you know? List all the torques on the turbine which make this happen.

Well, whenever the turbine turns, the generator turns, since the generator is attached to the turbine. But remember, the generator resists being turned. Consequently, it causes the turbine to stop (via the stopping torque the generator exerts on the turbine.) However, in the video above, we see the turbines turning at constant speeds, so they are not stopping! What keeps them going? A driving torque caused by a third agent that balances out the stopping torque from the generator, perhaps?

2. Considering that there is a giant force from the wind ON the turbine blade, how is Newton's 3rd law satisfied? How could you tell that Newton's 3rd law is satisfied?

This question specifically asks about the interaction of the moving air and the turbine, but Newton's third law applies whenever any two physical objects interact with each other. So, for example, the generator exerts a stopping torque on the turbine because the turbine exerts a driving torque on the generator. Similarly, the turbine exerts a stopping torque on the moving air, because the air exerts a driving torque on the turbine. Both of those are examples of Newton's third law in action. So, in the case of the turbine vs. the air, how can you tell that Newton's third law is being satisfied? Perhaps by measuring some change in the wind?

I hope this helps,

Aaron