Calculating Wheel Radius from Revolutions and Distance

This problem involves relating the distance a wheel travels to the number of revolutions it makes. The key idea is that the distance covered in one full revolution is equal to the circumference of the wheel.

Here's how we can solve this problem step-by-step:

1. Understand the Relationship:

• When a wheel completes one revolution, the distance it travels is equal to its circumference.
• The circumference \( C \) of a circle (or wheel) with radius \( r \) is given by the formula: \( C = 2 \pi r \).

2. Relate Total Distance to Revolutions:

• The total distance covered by the wheel is the number of revolutions multiplied by the distance covered in one revolution (the circumference).
• Total Distance \( D = \text{Number of Revolutions} \times C \)
• So, \( D = \text{Number of Revolutions} \times 2 \pi r \)

3. Convert Units:

• The distance is given in kilometers (km), but the options for the radius are in meters (m). We need to convert the distance to meters.
• 1 km = 1000 m
• Given distance = 60 km
• Distance in meters \( = 60 \times 1000 \) m \( = 60000 \) m

4. Substitute Values and Solve for Radius:

• We are given:
• Total Distance \( D = 60000 \) m
• Number of Revolutions \( N = 4000 \)
• We have the formula: \( D = N \times 2 \pi r \)
• Substitute the known values: \( 60000 = 4000 \times 2 \pi r \)
• Now, we need to isolate \( r \):
• \( 60000 = 8000 \pi r \)
• Divide both sides by \( 8000 \pi \):
• \( r = \frac{60000}{8000 \pi} \)
• Simplify the fraction:
• \( r = \frac{60}{8 \pi} \)
• \( r = \frac{15}{2 \pi} \)
• Now, calculate the numerical value using the value of \( \pi \approx 3.14159 \):
• \( r \approx \frac{15}{2 \times 3.14159} \)
• \( r \approx \frac{15}{6.28318} \)
• \( r \approx 2.3873 \) meters

5. Compare with Options:

Our calculated radius is approximately 2.3873 meters. Let's look at the given options:

1. 8 m
2. 8.25 m
3. 4.68 m
4. 2.39 m

The calculated value \( \approx 2.3873 \) m is closest to 2.39 m.

Final Answer Determination

Based on the calculation, the radius of the wheel is approximately 2.39 meters.

Revision Table: Key Concepts

Additional Information: Rolling Motion Basics

When a wheel rolls without slipping on a surface, the distance covered by the center of the wheel is exactly equal to the length of the arc traced on the circumference that has touched the ground. In one full revolution, the entire circumference touches the ground, hence the distance covered is equal to the circumference.

This principle is fundamental in understanding the mechanics of wheels and other rolling objects. It connects rotational motion (revolutions) to translational motion (distance covered).

It's important to always ensure units are consistent when performing calculations. In this case, converting kilometers to meters was a crucial step to get a radius in meters, as provided in the options.