The discussion centers on calculating the volume of liquid draining from a rotating vessel shaped like a paraboloid, described by the equation $$y=\frac{\omega^2x^2}{2g}$$. Participants clarify that the initial volume is given by $$\pi R^2h$$, and the volume of liquid that drains out can be expressed as $$V_{\text{out}}=\pi R^2 h - \frac{\pi\omega^2}{4g}\left(R^4-r^4\right)$$. The integration of the volume of thin cylindrical rings is crucial for determining the final volume after draining. The discussion emphasizes the importance of understanding the geometry and dynamics involved in the problem.
PREREQUISITESStudents and professionals in physics, engineering, and fluid dynamics, particularly those interested in the behavior of fluids in rotating systems and volume calculations in complex geometries.
If you just want an approximate answer, you should just send r to zero. The answer you gave contains a term proportional to ##r^4##. This term is negligible compared to the term you are missing due to not including the offset.Bling Fizikst said:It should be 'approximately' true i guess as ##r## is given to be small . Hence , the second order term ##r^2\approx 0## . So , i guess we can only get an approximate answer .