Question

A sieve is made of a square mesh of wires. Each wire has diameter d, and the holes in the mesh are squares whose side length is w. A spherical particle of radiusr is dropped on the mesh. What is the probability that it passes through? What is the probability that it fails to pass through if it is dropped n times? (Calculations such as these are relevant to the theory of sieving for analyzing the size distribution of particulate matter.

Answer 1

Answer:

(Prfail)=(1−Prpass)=(1−(−2)2(+)2)

Step-by-step explanation:

If is the width of a mesh hole, then (12++12)=(+) is the distance from one wire midpoint to the midpoint of the wire on the opposite side of the hole. If is the number of holes, then the total area of the mesh is

=(+)2

In order to pass through the hole without touching a wire, the center of the spherical particle needs to be positioned within a smaller square that has the side (−2). The total area that allows pass-through is therefore

=(−2)2

For a single drop, the probability of passing through is

r==(−2)2(+)2

And the probability of failing(Prfail) = 1 - Prpass

Assuming each drop is an independent event, the probability of failing drops is

(Prfail)=(1−Prpass)=(1−(−2)2(+)2)