A$1)$ a$)$ While performing an "inspection route" in an industrial facility, you observe that the housing of a journal bearing running at $800$ rpm is "very hot", at $\backslash (200^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ Though this may be an acceptable temperature for the bearing itself, you decide to undertake a deeper look. You shut off the system and allow it to cool to ambient $(\backslash (\backslash$left$\mathrm{.}20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash$right$)\backslash ).$ You then turn the system back on and find the housing rises to $\backslash (200^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ again in $5$ minutes. Assuming the primary system mass is represented by the housing $($plain carbon steel$)$ and is $0\mathrm{.}5$ kg $,$ and that there are no "heat losses" until the steady $\backslash (200^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ is reached, what is the "friction torque" of the system? What is the annual cost of the bearing operating like this if the shaft is driven by an electric motor where electricity costs $\backslash$$$0\mathrm{.}165/$kWh$?$

b$)$ It is determined that the bronze $($SAE $660)$ bushing of the journal bearing needs to be replaced. Take the inside diameter of the steel housing to be $60\mathrm{.}0$ mm $.$ The bronze bushing is expected to have a $0\mathrm{.}05$ mm interference fit $($based on diameter$)$ on installation, as well as a minimum overall $($based on diamerer$)$ clearance of $0\mathrm{.}1$ mm during installation. Validate $($with suitable equations$)$ that cooling into liquid nitrogen satisfies the required assembly clearance$($s$).$