A load cell like the one used in Lab $9$ is intended to measure force. However, it does not provide force values directly. Instead it provides a voltage that is proportional to the force applied on the load cell. In order to calibrate it$,$ we conduct an experiment in which we apply known forces and record the voltages. These measurements are summarized in the table below.

After fitting a straight line to the $4$ points on the table above, we obtain the following relationship between Force and Voltage: $\backslash ($ F$=3\mathrm{.}97\backslash$cdot V$+0\mathrm{.}08\backslash ),$ where F is force in N and V is voltage in mV $.$ The force values have an uncertainty of $1$ N $.$ The voltage values were measured with a DAQ board presenting a sensitivity and accuracy of $\backslash (0\mathrm{.}07\backslash \%\backslash )$ and $\backslash (0\mathrm{.}05\backslash \%\backslash )$ of the rate output $($RO$),$ respectively. The RO for such DAQ board is $200$ mV $.$

a$)$ Find the calibration uncertainty for a $\backslash (95\backslash \%\backslash )$ probability.

b$)$ We now use the load cell to measure an unknown force produced by a pneumatic cylinder. We record a voltage of $9\mathrm{.}8$ mV from the load cell with a multimeter that has a resolution of $0\mathrm{.}1$ mV $.$ Find the force produced by the pneumatic cylinder and provide its uncertainty $(\backslash (95\backslash \%\backslash )).$

a$)\backslash ($ u$\_\{$c e l$\}=1\mathrm{.}416\backslash$mathrm$\{$~N$\}\backslash )$

b$)\backslash ($ F$=38\mathrm{.}98\backslash$pm $1\mathrm{.}429\backslash$mathrm$\{$~N$\}\backslash )$