Question

What is the calculation for determining the closest and farthest distances from a person to a spinning wheel, given the person's height, distance from the wheel's center and the wheel's diameter?

Answer 1

To determine the closest and farthest distances from a person to a spinning wheel, we need to consider a few key measurements:

1. **Height of the person (H)**: The vertical distance from the ground to the person's head.
2. **Distance from the center of the wheel to the person (D)**: This is the horizontal distance from the center of the spinning wheel to the person standing or sitting at a certain position.
3. **Diameter of the wheel (d)**: This gives us the radius of the wheel (R), which is \( R = \frac{d}{2} \).

### Closest Distance

The closest distance between the person (at height H) and the edge of the wheel occurs when the person is closest to the wheel. The closest point on the wheel is at the edge closest to the person, which is \( D - R \). If the person's height is H, we are essentially looking at a straight horizontal line since the problem does not involve any vertical components affecting the wheel's edge.

Therefore, the closest distance (\( D_{closest} \)) can be determined as follows:
\[
D_{closest} = D - R
\]

However, if the person is standing level with the wheel's center, the vertical height does not affect this calculation directly, and thus it simplifies to:
\[
D_{closest} = D - \frac{d}{2}
\]

### Farthest Distance

The farthest distance from the person to the spinning wheel is when the distance is maximized, which occurs when they are at the point directly opposite the closest edge of the wheel from their position. This distance can be calculated as:
\[
D_{farthest} = D + R
\]

Which simplifies to:
\[
D_{farthest} = D + \frac{d}{2}
\]

### Summary Formulas

1. Closest distance:
\[
D_{closest} = D - \frac{d}{2}
\]

2. Farthest distance:
\[
D_{farthest} = D + \frac{d}{2}
\]

### Important Consideration

Make sure to account for whether the person is at the same level (height) as the center of the wheel. If the height component is significant (i.e., if the height impacts how close or far a person is from the wheel due to the position), you might need a more nuanced analysis. However, in most simplified scenarios, especially when the height of the person is not significantly above or below the center of the wheel, these calculations should suffice.