$(30)$ A mechanical speed control works on the basis of centrifugal force, which is

related to angular velocity by the formula $F=mr{}^2,$ where $F$ is the force, $m$ is the mass

of the rotating weight, $r$ is the radius of rotation, and $$ is the angular velocity of the

system. The following specifications are given for the measurement system:

Mass $(m)$:$0-200g$ range, accuracy $+-4\%$ of reading

Radius $(r)$: $0-50mm$ range, accuracy $+-2\%$ of reading

Angular speed $()$ : $0-1000$ rad$/$s range, accuracy $+-3\%$ of reading

$($a$)(12)$ Calculate the systematic uncertainty in force measurement $($in N $)$ due to the

measurement devices for measured values of $160g,25mm,$ and $600ra\frac{d}{s}.$

$($b$)(13)$ In logging data with a high sampling rate, the calculated standard deviation

of large number of readings are $5g,0\mathrm{.}6mm,$ and $12ra\frac{d}{s},$ respectively, while the

mean values are $160g,25mm,$ and $600ra\frac{d}{s}.$ Determine the random uncertainty

of the force measurement $($in N$),$ at a $95\%$ confidence level.

$($c$)(5)$ What is the total uncertainty in the force measurement $($in N $)?$