Question

Commercial wind turbines consists of three 6400-kg blades, each 35.0 m long. It generates 1.0 MW of electrical power when it rotates with very little friction at a constant speed in the usually steady wind. However, when the wind suddenly stops,the turbine spins to a stop in 100 s. From this information, estimate the following for the situation when the wind is blowing steadily:

(a) the rotational kinetic energy of the turbine,

(b) its rotational speed, and

(c) the magnitude of its angular momentum.

Answer 1

The mass of each blade is M=6400kg, length of each blade is L=35m. The moment of inertia of a single blade rotating about its end is

The moment of inertia of the turbine is equal to the moment of inertia of three blades

For the turbine to move at constant speed, the net torque on the turbine is zero. In steady state, the torque due to wind completely overcomes the torque due to friction.

Let be the steady state angular speed of the turbine. The power supplied by the wind is

Using (1)

If there was no wind, the turbine will come to rest in time t=100 seconds. The angular acceleration of the turbine due to frictional force is

When no wind the initial angular speed is and the final angular speed is 0

The torque applied by friction is

Substituting into (3)

(a)

The rotational kinetic energy of the turbine is

Using (4)

Substituting P=1MW=10⁶W, and t=100s we get

(b)

Using (1) in (4) we get

Solving steady state rotational speed

Substituting values

(c)

Substituting values