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:

KE=12Iω2KE = \frac{1}{2} I \omega^2

Where:

  • KEKE is the kinetic energy.
  • II is the moment of inertia.
  • ω\omega 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 II for a rod rotating about one end is:

Iblade=13mL2I_{\text{blade}} = \frac{1}{3} m L^2

Where:

  • mm is the mass of the blade (45 kg),
  • LL is the length of the blade (4.0 m).

So, for each blade:

Iblade=13×45kg×(4.0m)2=13×45×16=240kgm2I_{\text{blade}} = \frac{1}{3} \times 45 \, \text{kg} \times (4.0 \, \text{m})^2 = \frac{1}{3} \times 45 \times 16 = 240 \, \text{kg} \cdot \text{m}^2

Since there are 5 blades, the total moment of inertia for the blade assembly is:

Itotal=5×Iblade=5×240=1200kgm2I_{\text{total}} = 5 \times I_{\text{blade}} = 5 \times 240 = 1200 \, \text{kg} \cdot \text{m}^2

Step 2: Convert RPM to Angular Velocity

The angular velocity ω\omega is related to the rotational speed in rpm (revolutions per minute) by the formula:

ω=2π×rpm60\omega = 2\pi \times \frac{\text{rpm}}{60}

Given that the turbine spins at 237 rpm, we can calculate ω\omega:

ω=2π×23760=2π×3.9524.82rad/s\omega = 2\pi \times \frac{237}{60} = 2\pi \times 3.95 \approx 24.82 \, \text{rad/s}

Step 3: Calculate the Kinetic Energy

Now, we can use the kinetic energy formula:

KE=12×1200kgm2×(24.82rad/s)2KE = \frac{1}{2} \times 1200 \, \text{kg} \cdot \text{m}^2 \times (24.82 \, \text{rad/s})^2

Let’s compute the kinetic energy.

KE=12×1200×615.87369524J=369.5kJKE = \frac{1}{2} \times 1200 \times 615.87 \approx 369524 \, \text{J} = 369.5 \, \text{kJ}

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:

  1. How does the moment of inertia depend on the mass distribution?
  2. What is the significance of the angular velocity in rotational motion?
  3. How would the kinetic energy change if the blades were twice as long?
  4. Can you derive the formula for the moment of inertia of a uniform rod?
  5. 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)