$4\mathrm{.}1$ A copper sphere initially at a uniform temperature T$0$ is immersed in a fluid. Electric heaters are placed in the fluid and controlled so that the temperature of the fluid follows a periodic variation given by $\backslash (\backslash$mathrm$\{$T$\}\backslash$infty$-\backslash$mathrm$\{$Tm$\}=\backslash$mathrm$\{$Asin$\}\backslash$omega $\backslash$tau $\backslash )$

where

Tm $=$ time$-$average mean fluid temperature

$\backslash ($ A$=\backslash )$ amplitude of temperature wave

$\backslash (\backslash$omega $\backslash )=$ frequency

Derive an expression for the temperature of the sphere as a function of time and the heat$-$transfer coefficient from the fluid to the sphere. Assume that the temperatures of the sphere and fluid are uniform at any instant so that the lumped$-$capacity method of analysis may be used.