$(30$ points$)$ see Figure $2-$An electric motor that powers a gyroscope in an autonomous vehicle

navigation system is drawing more current than expected during operation. This is draining the

battery faster, thereby reducing the range of the vehicle. You suspect that the guidance algorithm

is not accounting for the viscous effects of the motor bearing. Model the bearing as a thin layer of

engine oil that lubricates the shaft of the motor.

a$.$ If the shaft is spinning at $20,000rpm,$ what is the angular velocity $\frac{m}{s}$ of the shaft relative

to the bearing?

b$.$ Starting with the $2$D CoM and CoMom equations in Cartesian coordinates, derive an

equation for the fully developed velocity profile of the oil, i$.$e$.u=u(y).$ Assume no pressure

is driving the flow.

c$.$ What is the torque $N^{**}(cm)$ exerted on the shaft associated with the viscous interaction with

the oil? The shaft is $3$ cm long, and use values for oil at $350$ K $.$

d$.$ What is the power $[$W$]$ required to rotate the shaft against this viscosity induced torque?

e$.$ If the battery stores $12$ kJ of energy and the time of flight is $90$ sec $,$ what fraction of battery

use will be devoted to the viscous interaction of the shaft with the oil?