Wheels of diameters 7 cm and 14 cm start rolling simultaneously from $X$ and $Y$, which are 1980 cm apart, towards each other in opposite directions. Both of them make the same number of revolutions per second. If both of them meet after 10 seconds, the speed of the smaller wheel is

### Concept & Logic When two objects move towards each other, the total distance covered is the sum of the individual distances traveled. $$\text{Circumference } (C) = \pi d$$ $$\text{Relative Speed} = \text{Speed}_1 + \text{Speed}_2$$ ### Step-by-Step Solution * **Given:** * Diameter of smaller wheel ($d_1$) = $7\text{ cm}$ * Diameter of larger wheel ($d_2$) = $14\text{ cm}$ * Distance between $X$ and $Y$ = $1980\text{ cm}$ * Time to meet ($t$) = $10\text{ seconds}$ * **Circumferences:** $$C_1 = \pi d_1 = \frac{22}{7} \times 7 = 22\text{ cm}$$ $$C_2 = \pi d_2 = \frac{22}{7} \times 14 = 44\text{ cm}$$ * **Combined distance in 1 revolution:** $$C_1 + C_2 = 22 + 44 = 66\text{ cm}$$ * **Let $n$ be the number of revolutions per second made by each wheel:** $$\text{Total distance covered in 10 seconds} = 10 \times n \times (C_1 + C_2)$$ $$1980 = 10 \times n \times 66$$ $$1980 = 660n \implies n = \frac{1980}{660} = 3\text{ rev/sec}$$ * **Speed of the smaller wheel:** $$\text{Speed}_1 = n \times C_1 = 3 \times 22 = 66\text{ cm/sec}$$ ### Exam Strategy & Shortcut Relative speed $= \frac{1980}{10} = 198\text{ cm/sec}$. Since diameters are in the ratio $7 : 14 = 1 : 2$, the distances and speeds are divided in the ratio $1 : 2$. $$\text{Speed of smaller wheel} = \frac{1}{1 + 2} \times 198 = \frac{1}{3} \times 198 = 66\text{ cm/sec}$$ ### Common Pitfall Dividing 1980 directly by the single circumference instead of the sum of both circumferences. ### Final Answer Therefore, the correct answer is **66 cm/sec**.