This problem involves calculating the radius of a wheel rolling on a larger concentric circle, given information about another wheel rolling on a smaller concentric circle.

Understanding Concentric Circles and Rolling Wheels

We are given two concentric circles with radii:

• Smaller circle radius, $ R_1 = 100 \text{ m} $
• Larger circle radius, $ R_2 = 110 \text{ m} $

A wheel (Wheel 1) with radius $ r_{w1} = 30 \text{ cm} $ rolls on the smaller circle ($R_1$). Another wheel (Wheel 2) with an unknown radius $ r_{w2} $ rolls on the larger circle ($R_2$).

The key information is that after the circles complete one revolution, the number of times each wheel rolled on its axis is equal. Let $N$ be this number of rolls.

Calculating Wheel Revolutions

When a wheel rolls without slipping along a path, the distance it covers is equal to the number of its revolutions multiplied by its circumference. If the path is the circumference of a circle, the distance covered is the circumference of that circle.

For Wheel 1 rolling on the smaller circle (radius $R_1$):

• Circumference of the path (smaller circle): $ C_1 = 2 \pi R_1 $
• Circumference of Wheel 1: $ c_{w1} = 2 \pi r_{w1} $
• Number of times Wheel 1 rolled: $ N_1 = \frac{C_1}{c_{w1}} = \frac{2 \pi R_1}{2 \pi r_{w1}} = \frac{R_1}{r_{w1}} $

For Wheel 2 rolling on the larger circle (radius $R_2$):

• Circumference of the path (larger circle): $ C_2 = 2 \pi R_2 $
• Circumference of Wheel 2: $ c_{w2} = 2 \pi r_{w2} $
• Number of times Wheel 2 rolled: $ N_2 = \frac{C_2}{c_{w2}} = \frac{2 \pi R_2}{2 \pi r_{w2}} = \frac{R_2}{r_{w2}} $

Solving for the Unknown Radius

The problem states that the number of rolls is equal for both wheels:

$ N_1 = N_2 $

Substituting the formulas:

$ \frac{R_1}{r_{w1}} = \frac{R_2}{r_{w2}} $

We need to find $ r_{w2} $. Rearranging the equation:

$ r_{w2} = r_{w1} \times \frac{R_2}{R_1} $

Before substituting, ensure units are consistent. Convert the circle radii from meters (m) to centimeters (cm):

• $ R_1 = 100 \text{ m} = 100 \times 100 \text{ cm} = 10000 \text{ cm} $
• $ R_2 = 110 \text{ m} = 110 \times 100 \text{ cm} = 11000 \text{ cm} $
• $ r_{w1} = 30 \text{ cm} $

Now substitute these values into the equation for $ r_{w2} $:

$ r_{w2} = 30 \text{ cm} \times \frac{11000 \text{ cm}}{10000 \text{ cm}} $

$ r_{w2} = 30 \text{ cm} \times 1.1 $

$ r_{w2} = 33 \text{ cm} $

Therefore, the radius of the other wheel is 33 cm.