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The angular velocity , omega of a rotating body or shaft can be measuredby attaching an open cylinder of liquid , as shown in Fig and measuring the change in the fluid level , H- h_(0) , caused by the rotation of the fluid . Determine the relationship between this change in fluid level and the angular velocity . Assume no fluid is spilling out of the vessel. |
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Answer» Solution :The HEIGHT h , of the free surface above the tank bottom can be determined as follows `h - (omega^(2) r^(2))/(2 g) + h_(0)`  The INITIAL volume of fluid in the tank , `V_(i) = pi R^(2) H` The volume of the fluid with the rotating tank can be found by considering a differential ELEMENT of cylindrical shape as shown in Fig . This cylindrical shell is TAKEN at some arbitary radius , r and thickness dr . Thus , its volume is `dV = 2 pi r h dr` The total volume is, therefore , `V. = 2 pi int_(0)^(R) r ((omega^(2) r^(2))/(2 g) + h_(0)) dr = (pi omega^(2) R^(4))/( 4 g) + pi R^(2) h_(0)` Since the volume of the fluid in the tank must remain constant (given that none spills over the top) , it follows that`pi R^(2) H = (pi omega^(2) R^(4))/(4 g) = pi R^(2) h_(0)` or `H - h_(0) = (omega^(2) R^(2))/(4 g)` |
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