A carpet lying on ground has length `L`, width a and a small thickness `d`. It is rolled over a light cylindrical pipe of radius `r = (L)/(100 sqrt(pi))` and kept on a level ground. Increase in gravitational potential energy of the carpet is `DeltaU_(1)` (compared to its initial position when it was lying flat). In another experiment the carpet was folded to give it a shape of a cuboid (see figure) having width `b`. When this is placed on level ground its gravitational potential energy is `Delta U_(2)` higher than its initial position (flat on ground). It is given that `d = 10^(-4) L`. Find `b` for which `DeltaU_(1) = Delta U_(2)`. [Take `sqrt((pi)/(2)) = 1.25`]

Answer

`(L)/(260)`