$($b$)$ Determine the torque and power necessary to rotate a vertical shaft $120$ mm diameter at $250$ rpm as shown in Fig. Q$1($b$),$ all dimensions are in mm$.$ The shaft passes through a sleeve of length $150$ mm which is stationary. The radial clearance between the shaft and sleeve is $0\mathrm{.}1$ mm and the space between them contains oil of dynamic viscosity $0\mathrm{.}2$ Pa s$.$ Assume a linear velocity distribution.

Fig. Q$1($b$)$

Hint: Using common notations $=2\frac{N}{60},T=Fr,W^{}=T$

$(10$ marks$)$