A typical propeller of a turbine used to generate electricity from the wind consists of three blades as in the figure below. Each blade has a length of L = 33 m and a mass of m = 415 kg. The propeller rotates at the rate of 21 rev/min. 120degree angle between blades
a) Convert the angular speed of the propeller to units of rad/s.
(b) Find the moment of inertia of the propeller about the axis of rotation.___J · s2
(c) Find the total kinetic energy of the propeller.______ J
(A) w = (21rev/min) * (1min/60sec) * (2pi rad/1rev) = 2.2 rad/s
(B)
I = MR^2
I = (415kg)(33m)^2
I = 4.52E5 (I for one propeller)
I for all three propellers = (4.52E5) * 3 = 1.36E6
(C) KE = (1/2) * Iw^2
= (1/2)(4.52E5)(2.2)^2
=1.1E6 J
The book gives solutions for I and KE using one propeller. My thought were to multiply I *3....a little confusing. Thoughts...?
(a) To convert the angular speed of the propeller in revolutions per minute (rev/min) to radians per second (rad/s), we need to use the conversion factor:
1 rev = 2π rad
1 min = 60 s
So, the angular speed in radians per second can be found by:
Angular speed in rad/s = (Angular speed in rev/min) × (2π rad/1 rev) × (1 min/60 s)
Given that the propeller rotates at the rate of 21 rev/min, we can substitute this value into the formula:
Angular speed in rad/s = (21 rev/min) × (2π rad/1 rev) × (1 min/60 s)
Simplifying the equation:
Angular speed in rad/s = (21 × 2π) / (60)
Calculating the value:
Angular speed in rad/s ≈ 2.199 rad/s
Therefore, the angular speed of the propeller is approximately 2.199 rad/s.
(b) The moment of inertia of the propeller is given by the formula:
Moment of inertia (I) = (1/3) × m × L²
Given that the length of each blade is L = 33 m and the mass of each blade is m = 415 kg, we can substitute these values into the formula:
Moment of inertia (I) = (1/3) × (415 kg) × (33 m)²
Calculating the value:
Moment of inertia (I) ≈ 1.264 × 10⁶ kg.m²
Therefore, the moment of inertia of the propeller about the axis of rotation is approximately 1.264 × 10⁶ kg.m².
(c) The total kinetic energy of the propeller can be found using the formula:
Kinetic energy = (1/2) × I × ω²
Given that the moment of inertia (I) is approximately 1.264 × 10⁶ kg.m² and the angular speed (ω) is approximately 2.199 rad/s, we can substitute these values into the formula:
Kinetic energy = (1/2) × (1.264 × 10⁶ kg.m²) × (2.199 rad/s)²
Calculating the value:
Kinetic energy ≈ 3.43 × 10⁶ J
Therefore, the total kinetic energy of the propeller is approximately 3.43 × 10⁶ J.
(a) To convert the angular speed of the propeller from revolutions per minute (rev/min) to radians per second (rad/s), we need to use the conversion factor:
1 rev = 2π rad
1 min = 60 s
First, let's convert the revolutions per minute to revolutions per second:
21 rev/min × (1 min/60 s) = 0.35 rev/s
Next, we convert the revolutions to radians:
0.35 rev/s × (2π rad/1 rev) = 2.2π rad/s
Therefore, the angular speed of the propeller is 2.2π rad/s.
(b) The moment of inertia of an object is a measure of its resistance to rotational motion. To find the moment of inertia of the propeller about the axis of rotation, we need to know the shape and distribution of the mass.
Since the propeller consists of three blades with a 120-degree angle between them, we can approximate the moment of inertia as that of a solid disk. The moment of inertia of a solid disk rotating about its central axis is given by the formula:
I = (1/2) MR^2
where I is the moment of inertia, M is the mass of the disk, and R is the radius of the disk.
In this case, each blade of the propeller can be considered as a solid disk. Therefore, the total moment of inertia of the propeller is three times the moment of inertia of one blade.
First, let's find the radius of each blade. Since the length of each blade is given as L = 33 m, the radius can be calculated as half of the length:
R = L/2 = 16.5 m
Next, we can calculate the moment of inertia of one blade:
I_blade = (1/2) m R^2
Substituting the given values:
I_blade = (1/2) (415 kg) (16.5 m)^2 = 552112.5 kg·m^2
Finally, we can find the moment of inertia of the propeller:
I_propeller = 3 I_blade = 3 * 552112.5 kg·m^2 = 1656337.5 kg·m^2
Therefore, the moment of inertia of the propeller about the axis of rotation is approximately 1656337.5 kg·m^2.
(c) The total kinetic energy of the propeller can be calculated using the formula:
KE = (1/2) I ω^2
where KE is the kinetic energy, I is the moment of inertia, and ω is the angular speed.
Substituting the given values:
KE = (1/2) (1656337.5 kg·m^2) (2.2π rad/s)^2
Simplifying the equation:
KE ≈ 2.2π^2 * 1656337.5 kg·m^2 ≈ 22817936.1 J
Therefore, the total kinetic energy of the propeller is approximately 22,817,936.1 J.
This is pretty standard question. What is it you do not understand?
For the moment of inertial figure each blade rotating about its end, then multiply by three.
KE=1/2 Itotal*w^2