Problem $7_{?}$

Top View

The above figure shows the top and side views of a cylindrical drum of radius $R_i=0\mathrm{.}05[m]$ and height $H=0\mathrm{.}15[m]$ that is spinning at $=10[$radians $\frac{?}{s}.$ The drum is inside an open cylindrical can of radius $R_o=0\mathrm{.}06[m].$ The can is stationary and is concentric with the drum. The drum is surrounded by oil with viscosity $0\mathrm{.}107Pa-s.$ In the gap between the side of the drum and the side of the can, the tangential velocity of the oil is given by:

$v_{}(r)=\frac{V_{i}C}{({}^2-1)}[\frac{R_o}{r}-\frac{r}{R_o}],$ for $,R_{i}r{R}_o$

where $=\frac{R_o}{R_i},V_i$ is the oil velocity at the side wall of the drum, and $r$ is the radial distance from the axis of rotation.

a$)$ Estimate $V_i,$ the speed of the oil at the side wall of the drum.

b$)$ Determine the numerical value of the constant, $C.$

c$)$ Estimate the torque required to spin the drum at $=10[radian\frac{s}{s}].$ You may neglect the effect of the oil on the top and bottom surfaces of the drum.