Problem $1.$ A first$-$order temperature sensor has a static sensitivity of $0\mathrm{.}25F.$ The temperature sensor is initially in a $70F$ environment, and then is placed into a $212F$ water bath. After $4$ seconds, the sensor$\text{'}$s output is $50.$ What is the sensor$\text{'}$s time constant?

$($a$)0\mathrm{.}54$ s

$($b$)1\mathrm{.}62$ s

$($c$)1\mathrm{.}39$ s

$($d$)3\mathrm{.}733$

$($e$)0\mathrm{.}62$ s

$($f$)0\mathrm{.}72$ s

$($g$)$ None of the above

Problem $2.$ A first$-$order displacement sensor has time constant $0\mathrm{.}015$ s and static sensitivity $10\frac{V}{m}m.$ The input to the sensor is $0\mathrm{.}2sin(70t)mm.$ What is the amplitude of the sensor$\text{'}$s steady$-$state output $($in V$)?$

$($a$)0\mathrm{.}96$ V

$($b$)1\mathrm{.}85$ V

$($c$)2\mathrm{.}00$ V

$($d$)0\mathrm{.}69$ V

$($c$)1\mathrm{.}38$ V

$($I$)6\mathrm{.}90$ V

$($g$)$ None of the above

Problem $3.$ A first$-$order rotational speed sensor has time constant $0\mathrm{.}002$ s $.$ Of the following choices, which is the largest input frequency $($in Hz $)$ that can be accurately measured to within a dynamic error of $+-5\%?$

$($a$)10$ Hz

$($b$)20$ Hz

$($c$)30$ Hz

$($d$)40$ Hz

$($e$)50$ Hz

$($f$)60$ Hz

$($g$)70$ Hz

Problem $4.$ A second$-$order acceleration sensor has natural frequency $500$ rad s $,$ damping ratio $0\mathrm{.}4,$ and static sensitivity $100m\frac{V}{g}.$ The sensor$\text{'}$s steady$-$state output is $25\mathrm{.}0sin(300t)mV.$ What is the amplitude of the sensor$\text{'}$s input $($in g$\text{'}$s$)?$

$($a$)0\mathrm{.}05$ g

$($b$)0\mathrm{.}25$ g

$($c$)6\mathrm{.}25$ g

$($d$)0\mathrm{.}20$ g

$($c$)0\mathrm{.}01$ g

$($f$)1\mathrm{.}25$ g

$($g$)$ None of the above

Problem $5.$ A second$-$order sensor has a natural frequency $50ra\frac{d}{s}$ and damping ratio is $0\mathrm{.}7.$ Of the following choices, which the largest input frequency $($in Hz $)$ that can be accurately measured to within a dynamic error of $=10\%?($The plot shows

Ma versus $\frac{}{{}_m}$ for a damping ratio of $0\mathrm{.}7.)$

$($a$)10$ Hz

$($b$)25$ Hz

$($c$)75$ Hz

$($d$)100$ Hz

$($e$)250$ Hz

$($f$)500$ Hz

$($g$)$ None of the above