Question

Modern wind turbines generate electricity from wind power. The large, massive blades have a large moment of inertia and carry a great amount of angular momentum when rotating. A wind turbine has a total of 3 blades. Each blade has a mass of m = 5500 kg distributed uniformly along its length and extends a distance r = 45 m from the center of rotation. The turbine rotates with a frequency of f = 11 rpm.
(a) Enter an expression for the total moment of inertia of the wind turbine about its axis of rotation, in terms of the defined quantities.
(b) Calculate the total moment of inertia of the wind turbine about its axis, in units of kilogram meters squared.
(c) Enter an expression for the angular momentum of the wind turbine, in terms of the defined quantities. sig.gif?tid=0N86-9A-18-43-94ED-17253
(d) Calculate the angular momentum of the wind turbine, in units of kilogram meters squared per second.

Answer 1

Answer:

Explanation:

a )

Each blade is in the form of rod with axis near one end of the rod

Moment of inertia of one blade

= 1/3 x m l²

where m is mass of the blade

l is length of each blade.

Total moment of moment of 3 blades

= 3 x$\frac{1}{3}$  x m l²

ml²

2 )

Given

m = 5500 kg

l = 45 m

Putting these values we get

moment of inertia of one blade

= 1/3 x 5500 x 45 x 45

= 37.125 x 10⁵ kg.m²

Moment of inertia of 3 blades

= 3 x 37.125 x 10⁵ kg.m²

= 111 .375 x 10⁵ kg.m²

c )

Angular momentum

= I x ω

I is moment of inertia of turbine

ω is angular velocity

ω = 2π f

f is frequency of rotation of blade

d )

I = 111 .375 x 10⁵ kg.m² ( Calculated )

f = 11 rpm ( revolution per minute )

= 11 / 60 revolution per second

ω = 2π f

=  2π  x  11 / 60 rad / s

Angular momentum

= I x ω

111 .375 x 10⁵ kg.m² x  2π  x  11 / 60 rad / s

= 128.23 x 10⁵  kgm² s⁻¹ .