Math Problem Statement
A school has installed a modestly-sized wind turbine. The 5 blades are 4.0 m long; each blade has a mass of 45 kg. You can assume that the blades are uniform along their lengths. When the blades spin at 237 rpm, what is the kinetic energy of the blade assembly?
Solution
To calculate the kinetic energy of the rotating blade assembly, we will treat each blade as a rotating object. The kinetic energy of a rotating object is given by the formula:
Where:
- is the kinetic energy.
- is the moment of inertia.
- is the angular velocity in radians per second.
Step 1: Moment of Inertia of Each Blade
Each blade can be modeled as a uniform rod rotating about one end. The moment of inertia for a rod rotating about one end is:
Where:
- is the mass of the blade (45 kg),
- is the length of the blade (4.0 m).
So, for each blade:
Since there are 5 blades, the total moment of inertia for the blade assembly is:
Step 2: Convert RPM to Angular Velocity
The angular velocity is related to the rotational speed in rpm (revolutions per minute) by the formula:
Given that the turbine spins at 237 rpm, we can calculate :
Step 3: Calculate the Kinetic Energy
Now, we can use the kinetic energy formula:
Let’s compute the kinetic energy.
Final Answer:
The kinetic energy of the blade assembly is approximately 369.5 kJ.
Would you like further details or have any questions?
Here are 5 questions to deepen your understanding:
- How does the moment of inertia depend on the mass distribution?
- What is the significance of the angular velocity in rotational motion?
- How would the kinetic energy change if the blades were twice as long?
- Can you derive the formula for the moment of inertia of a uniform rod?
- What factors affect the efficiency of wind turbines?
Tip: The angular velocity in radians per second is a key parameter in rotational motion, and it’s crucial to always convert rpm to this form when dealing with rotational energy.
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Math Problem Analysis
Mathematical Concepts
Rotational Kinetic Energy
Moment of Inertia
Angular Velocity
Formulas
KE = 0.5 * I * ω^2
I_blade = (1/3) * m * L^2
ω = 2π * (rpm / 60)
Theorems
Moment of Inertia for a Uniform Rod
Kinetic Energy of Rotational Motion
Suitable Grade Level
Grades 11-12 (Physics)
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