Wind energy is gaining increased attention, generating an increased interest in windmill technology. Because windmill olades (vanes) rotate about a central axis, one of the most mportant physical properties of a windmill is its moment of inertia. Given is a picture of a typical windmill, where the geometry and center of mass of one of the vanes is illustrated. The mass of each vane is 411 kg. The distance from the center of mass of the vane to axis B is k₁ = 2.75 m. The distance from the center of mass of the vane to the center of the windmill mub is k₂ = 4.26 m. If the moment of inertia of a vane about axis A is 241 kg.m² and about axis B is 5860 kg-m², calculate the moment of inertia Itotal of the entire assembly about the axis that passes through the windmill's hub and is perpendicular to the screen. Ignore the hub and assume the vanes are flat.)
Wind energy is gaining increased attention, generating an increased interest in windmill technology. Because windmill olades (vanes) rotate about a central axis, one of the most mportant physical properties of a windmill is its moment of inertia. Given is a picture of a typical windmill, where the geometry and center of mass of one of the vanes is illustrated. The mass of each vane is 411 kg. The distance from the center of mass of the vane to axis B is k₁ = 2.75 m. The distance from the center of mass of the vane to the center of the windmill mub is k₂ = 4.26 m. If the moment of inertia of a vane about axis A is 241 kg.m² and about axis B is 5860 kg-m², calculate the moment of inertia Itotal of the entire assembly about the axis that passes through the windmill's hub and is perpendicular to the screen. Ignore the hub and assume the vanes are flat.)
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![Wind energy is gaining increased attention, generating an increased interest in windmill technology. Because windmill blades (vanes) rotate about a central axis, one of the most important physical properties of a windmill is its moment of inertia.
Given is a picture of a typical windmill, where the geometry and center of mass of one of the vanes is illustrated. The mass of each vane is 411 kg. The distance from the center of mass of the vane to axis B is \( k_1 = 2.75 \, \text{m} \). The distance from the center of mass of the vane to the center of the windmill hub is \( k_2 = 4.26 \, \text{m} \).
If the moment of inertia of a vane about axis A is \( 241 \, \text{kg} \cdot \text{m}^2 \) and about axis B is \( 5860 \, \text{kg} \cdot \text{m}^2 \), calculate the moment of inertia \( I_{\text{total}} \) of the entire assembly about the axis that passes through the windmill's hub and is perpendicular to the screen. (Ignore the hub and assume the vanes are flat.)
\[ I_{\text{total}} = \, \underline{\hspace{30mm}} \, \text{kg} \cdot \text{m}^2 \]
### Explanation of the Diagram
The diagram shows a windmill's vane structure:
- **Center of Mass:** The center of mass of a single vane is indicated by a dot.
- **Axes Explanation:**
- **Axis A** is oriented perpendicular to the vane at its tip.
- **Axis B** is parallel to axis A but passes through the center of mass of the vane.
- **Distances:**
- \( k_1 \) is the distance from the center of mass to axis B, marked as \( 2.75 \, \text{m} \).
- \( k_2 \) is the distance from the center of mass of the vane to the center of the hub, marked as \( 4.26 \, \text{m} \).
The challenge is to calculate the total moment of inertia about the central hub axis, considering the given details and assumptions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6a5c500a-af21-4189-b630-ff7bbb03bf84%2Ffe5a6d0f-2eda-4ca8-af46-2d59383a3e5a%2Fc9pojm_processed.png&w=3840&q=75)
Transcribed Image Text:Wind energy is gaining increased attention, generating an increased interest in windmill technology. Because windmill blades (vanes) rotate about a central axis, one of the most important physical properties of a windmill is its moment of inertia.
Given is a picture of a typical windmill, where the geometry and center of mass of one of the vanes is illustrated. The mass of each vane is 411 kg. The distance from the center of mass of the vane to axis B is \( k_1 = 2.75 \, \text{m} \). The distance from the center of mass of the vane to the center of the windmill hub is \( k_2 = 4.26 \, \text{m} \).
If the moment of inertia of a vane about axis A is \( 241 \, \text{kg} \cdot \text{m}^2 \) and about axis B is \( 5860 \, \text{kg} \cdot \text{m}^2 \), calculate the moment of inertia \( I_{\text{total}} \) of the entire assembly about the axis that passes through the windmill's hub and is perpendicular to the screen. (Ignore the hub and assume the vanes are flat.)
\[ I_{\text{total}} = \, \underline{\hspace{30mm}} \, \text{kg} \cdot \text{m}^2 \]
### Explanation of the Diagram
The diagram shows a windmill's vane structure:
- **Center of Mass:** The center of mass of a single vane is indicated by a dot.
- **Axes Explanation:**
- **Axis A** is oriented perpendicular to the vane at its tip.
- **Axis B** is parallel to axis A but passes through the center of mass of the vane.
- **Distances:**
- \( k_1 \) is the distance from the center of mass to axis B, marked as \( 2.75 \, \text{m} \).
- \( k_2 \) is the distance from the center of mass of the vane to the center of the hub, marked as \( 4.26 \, \text{m} \).
The challenge is to calculate the total moment of inertia about the central hub axis, considering the given details and assumptions.
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