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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 = 44 m from the center of rotation. The turbine rotates with a frequency of f = 13 rpm.
a. Calculate the total moment of inertia of the wind turbine about its axis, in units of kilogram meters squared.
b. Calculate the angular momentum of the wind turbine, in units of kilogram meters squared per second.

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Rau7star

Answer:

a)106.48 x 10⁵ kg.m²

b)144.97 x 10⁵  kgm² s⁻¹  

Explanation:

a)Given

m = 5500 kg

l = 44 m

Moment of inertia of one blade

[tex]I[/tex]= 1/3 x m l²

where m is mass of the blade

l is length of each blade.

Putting all the required values, moment of inertia of one blade will be

[tex]I[/tex]= 1/3 x 5500 x 44²  

[tex]I[/tex]= 35.49 x 10⁵ kg.m²

Moment of inertia of 3 blades

[tex]I[/tex]= 3 x 35.49 x 10⁵ kg.m²

[tex]I[/tex]= 106.48 x 10⁵ kg.m²

b) Angular momentum 'L' is given by

L =[tex]I[/tex] x ω

where,

[tex]I[/tex]= moment of inertia of turbine i.e  106.48 x 10⁵ kg.m²

ω=angular velocity =2π f

f is frequency of rotation of blade i.e  13 rpm

f = 13 rpm=>= 13 / 60 revolution per second

ω = 2π f =>  2π  x  13 / 60 rad / s

L=[tex]I[/tex] x ω =>106.48 x 10⁵ x   2π  x  13 / 60

  = 144.97 x 10⁵  kgm² s⁻¹    

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