There are two concentric cylinders of radii `R_(1)` and `R_(2)` as shown. The narrow space between the two cylinders is filled with liquid of viscosity `mu`. The inner cylinder is held stationary by means of a torsional spring of stiffness G. the outer cylinder is rotated with constant angular speed `omega`. The torque acting on the inner cylinder due to shearing action of thhe liquid:

Options

A. `(pimuomegaR^(4))/(2h)+(2pimuomegaHR_(1)^(2)R_(2))/(R_(2)-R_(1))`
B. `(pimuomegaHR_(1)^(2)R_(2))/(R_(2)-R_(1))`
C. `(pimuomegaHR_(1)^(2)R_(2))/(R_(2)-R_(1))+(muomegaR^(4))/(3h)`
D. `(pimuomegaR^(4))/(2h)`

Answer

A

Explanation

The torque due to liquid in annular space is
`(2piR_(1)H)xxmu((omegaR_(2))/(R_(2)-R_(1)))xxR_(1)`
where velocity gradient `=(omegaR_(2))/(R_(2)-R_(1))`
Also torque due to liquid in between space in botto of thickness 'h'
`underset(0)overset(R_(1))int(x)(2pixdx).(omegax)/(h)=(pimuomegaR^(4))/(2h)`