There are two concentric circles . The radii of the two circles are 100 m and 110 m respectively. A wheel of radius 30 cm on the smaller circle and another wheel on the larger circle. After they have completed one revolution, it is found that the two wheels rolled equal number of times on their respective axes. What is the radius of the other wheel?

The problem involves two concentric circles with radii $R_1 = 100$ m and $R_2 = 110$ m. A wheel of radius $r_1 = 30$ cm rolls on the smaller circle, and another wheel of radius $r_2$ rolls on the larger circle. When the wheels complete one full revolution around the center of the concentric circles, they cover distances equal to the circumferences of their respective paths: $2\pi R_1$ and $2\pi R_2$ [1]. The number of rotations $N$ a wheel makes on its own axis is the total distance traveled divided by its own circumference $2\pi r$. Given $N_1 = N_2$, we have $(2\pi R_1) / (2\pi r_1) = (2\pi R_2) / (2\pi r_2)$, which simplifies to $R_1 / r_1 = R_2 / r_2$. Substituting the values: $100 / 30 = 110 / r_2$. Solving for $r_2$ gives $r_2 = (110 \times 30) / 100 = 33$ cm.

Sources

1. [1] Science-Class VII . NCERT(Revised ed 2025) > Chapter 12: Earth, Moon, and the Sun > Repeat Activity 12.2 but replace the torch with an electric lamp. Then place the globe at diff erent positions on a circle around the lamp while maintaining the tilt of the globe. > p. 186