Two circular wheels of the same radius $r$ have their central hubs at a distance of $a$ from one another. The minimum length of a fan belt which will pass around both the wheels is

### Concept & Length of Belts on Pulleys When a belt wraps around two circular wheels of the same radius, the total length of the belt consists of two straight segments connecting the wheels and two curved segments wrapping around the outside of the wheels. $$ \text{Circumference of circle} = 2\pi r $$ ### Step-by-Step Solution 1. The straight segments of the belt are parallel to the line connecting the hubs. Since the wheels have the same radius, the length of each straight segment exactly equals the distance between the hubs, $a$. 2. Total length of straight segments = $a + a = 2a$. 3. The belt wraps around exactly half of each wheel (180 degrees) because the wheels are of identical size. 4. Length of the curved segment on one wheel = $\frac{1}{2} \times 2\pi r = \pi r$. 5. Total length of curved segments (for two wheels) = $\pi r + \pi r = 2\pi r$. 6. Total minimum length of the belt = Straight length + Curved length = $2a + 2\pi r = 2(a + \pi r)$. ### Exam Strategy & Shortcut For any two pulleys of equal radius $r$ separated by distance $d$, the belt length is always $2d + 2\pi r$. You can think of the two semicircular wraps as forming one complete circle of circumference $2\pi r$, and the straight paths contributing $2d$. ### Common Pitfall Forgetting that there are *two* straight sections and *two* semi-circular wraps, leading to answers like $a + \pi r$ or $2a + \pi r$. ### Final Answer Therefore, the correct answer is **$2(a + \pi r)$**.