An open$-$loop positioning system uses a stepper motor connected to a ball

screw with a $4$:$1$ gear reduction $($four turns of the motor for each turn of the ball

screw$).$ The stepper motor has a step angle of $7\mathrm{.}2\backslash$deg $.$ The ball screw pitch is $5$

mm$.$ Mechanical inaccuracies can be described by a normal distribution whose

standard deviation $=0\mathrm{.}005$ mm$.$ The range of the worktable axis is $500$ mm$.$

What is the minimum number of bits that the binary register must have so that

the mechanical drive system becomes the limiting component on control

resolution?