Problem $2$: Assume that for the problem above the surrounding temperature is changing periodically as $T_s(t)=cos(2t).$

Find an expression for the temperature of the thermometer as a function of time.

giving

$T(t)=m{e}^{-mt}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(t+T_0)e^{mt}dt=m{e}^{-mt}[(t+T_0)(\frac{1}{m}{e}^{mt})-{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(\frac{1}{m}{e}^{mt})dt]$

$=m{e}^{-mt}[\frac{1}{m}(t+T_0)e^{mt}-\frac{}{m}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{mt}dt]=m{e}^{-mt}[\frac{1}{m}(t+T_0)e^{mt}-\frac{}{m^2}{e}^{mt}+C]$

~~$t+T_0-\frac{}{m}+Cm{e}^{-mt}$

Problem $1$: Assume that a thermometer with a spherical test section of radius $r=0\mathrm{.}3cm$ is initially at a temperature

$T_i=50C,$ the density of the thermometer medium is $=1000k\frac{g}{m^3},$ its specific heat is $c=3000\frac{J}{k}gC,$ the heat transfer

coefficient $h=1500\frac{W}{m^2}C,$ the initial temperature of the surroundings $T_0=100C,$ and the rate of change of the sur$-$

roundings temperature $=5\frac{C}{s}.$ How does the temperature of the thermometer compares to that of the surrounding

medium at long times? Plot the ratio of the thermometer's temperature to the surrounding temperature as a function of

time.

Problem $2$: Assume that for the problem above the surrounding temperature is changing periodically as $T_s(t)=cos(2t).$

Find an expression for the temperature of the thermometer as a function of time.