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?

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:

\[
KE = \frac{1}{2} I \omega^2
\]

Where:
- \( KE \) is the kinetic energy.
- \( I \) is the moment of inertia.
- \( \omega \) 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 \( I \) for a rod rotating about one end is:

\[
I_{\text{blade}} = \frac{1}{3} m L^2
\]

Where:
- \( m \) is the mass of the blade (45 kg),
- \( L \) is the length of the blade (4.0 m).

So, for each blade:

\[
I_{\text{blade}} = \frac{1}{3} \times 45 \, \text{kg} \times (4.0 \, \text{m})^2 = \frac{1}{3} \times 45 \times 16 = 240 \, \text{kg} \cdot \text{m}^2
\]

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

\[
I_{\text{total}} = 5 \times I_{\text{blade}} = 5 \times 240 = 1200 \, \text{kg} \cdot \text{m}^2
\]

### Step 2: Convert RPM to Angular Velocity

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

\[
\omega = 2\pi \times \frac{\text{rpm}}{60}
\]

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

\[
\omega = 2\pi \times \frac{237}{60} = 2\pi \times 3.95 \approx 24.82 \, \text{rad/s}
\]

### Step 3: Calculate the Kinetic Energy

Now, we can use the kinetic energy formula:

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

Let’s compute the kinetic energy.

\[
KE = \frac{1}{2} \times 1200 \times 615.87 \approx 369524 \, \text{J} = 369.5 \, \text{kJ}
\]

### 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.

Learn how to calculate the kinetic energy of a wind turbine's blade assembly. The problem involves 5 blades, each 4.0 m long and 45 kg in mass, rotating at 237 rpm. Key concepts include rotational kinetic energy, moment of inertia, and angular velocity. Suitable for high school physics students.

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