This problem will let you understand better whether a composite can be viewed as

homogeneous or inhomogeneous depending on different scales. In Fig I, we have the

following composite filled with randomly oriented discontinuous fibers. We view the

material at two different locations A and $B$ with three different scales, which are

represented by solid $(A)$ or dashed $(B)$ circles. Case $1$ : The circle has a radius of $1$ mm $.$

Case $2$: The circle has a radius of $2$ mm $.$ Case $3$ : The circle has a radius of $5`$

each$)$

$(1)$ In all cases, first count the number of fibers inside the circle and then calculate the

density of fibers. $($Hint: The density of fibers in a circle here is approximated as the ratio

of the number of fibers to the area of the circle.$)$

$(2)$ Calculate the difference of fiber density between A and $B,$ for all three scales. How

does the value of the difference change when increasing the scale$/$radius$?$

Fig. I for solving problem $(1)$ and $(2)$

$(3)$ In general, please analyze whether the material below in Fig. II can be viewed as

homogeneous or inhomogeneous at scale A$,$ B and C$.$

Fig. Il used for solving question $(3)$