Q.

A cylindrical shaft of radius R rotating with angular speed ω is supported on a horizontal surface such that there exists a thin film of viscous fluid of thickness h. Coefficient of viscosity of liquid is η. The power supplied to the shaft must be proportional to the nth power of angular speed of the shaft. Then n is

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Detailed Solution

External toque = viscous torque in equilibriumViscous farce X radius due to an elemental ring viscous torque.viscous Force =Fv=ηAdvdx=ηArωdx Power, P=FvV=ηArωdxrω=ηAr2ω2dxP∝ω2
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A cylindrical shaft of radius R rotating with angular speed ω is supported on a horizontal surface such that there exists a thin film of viscous fluid of thickness h. Coefficient of viscosity of liquid is η. The power supplied to the shaft must be proportional to the nth power of angular speed of the shaft. Then n is