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To solve this problem, let's consider the concept of rotational inertia (also known as the moment of inertia) about the points \(a\), \(b\), and \(c\) on the metal plate. The moment of inertia depends on both the mass distribution relative to the point and the distance of the mass from that point. Here’s the analysis:

### Observations:
1. **Uniform Plate**: The metal plate was originally square, and 25% of it was removed, which likely shifted the center of mass.
2. **Points**: Points \(a\), \(b\), and \(c\) are located at different parts of the plate:
   - **Point \(a\)** is at the top left.
   - **Point \(b\)** is near the center.
   - **Point \(c\)** is at the bottom right, near where the corner was removed.

### Concept of Rotational Inertia:
The rotational inertia about a given point depends on how mass is distributed relative to that point:
- **Point closer to the center of mass** will generally have **lower rotational inertia**.
- **Point further from the mass distribution** will have **higher rotational inertia**.

### Ranking Analysis:
1. **Point \(a\)**: Since it’s on the far edge of the remaining plate, there’s more mass distributed at a distance from it. Therefore, this point likely has the **greatest rotational inertia**.
   
2. **Point \(b\)**: Located near the center of the remaining part of the plate, close to the new center of mass. This point should have the **smallest rotational inertia** due to the proximity of mass.

3. **Point \(c\)**: Located near the bottom right, where mass has been removed. However, it still has more mass distributed at a distance compared to \(b\), making its rotational inertia **greater than \(b\) but less than \(a\)**.

### Final Ranking:
Based on the distribution:
- **Point \(a\)**: Greatest rotational inertia.
- **Point \(c\)**: Intermediate rotational inertia.
- **Point \(b\)**: Smallest rotational inertia.

So the ranking from greatest to least is:
- **Point \(a\)** (Greatest)
- **Point \(c\)** (Intermediate)
- **Point \(b\)** (Least)

Would you like further details on any part of the analysis?

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Here are 5 related questions to deepen understanding:
1. How does the shape of an object affect its rotational inertia about different points?
2. What would happen to the rankings if the shape of the removed section was different?
3. How does the location of the center of mass influence rotational inertia?
4. What is the mathematical formula for calculating rotational inertia for a given shape and axis?
5. How would adding mass near one of the points change the rankings?

**Tip**: Remember, rotational inertia is not just about mass, but also how far that mass is from the axis of rotation.

Rank points A, B, and C on a uniform metal plate with a removed section, based on rotational inertia about a perpendicular axis through each point.

Learn how to rank points A, B, and C on a uniform metal plate with a section removed, based on rotational inertia. This problem explores concepts such as rotational inertia, center of mass, and mass distribution. Ideal for students studying physics at the undergraduate level.

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