question_answer

A tube of length L is filled completely with an incompressible liquid of mass M and closed at both ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity. The force exerted by the liquid at the other end is:

A) \[\frac{ML{{\omega }^{2}}}{2}\]              

B) \[ML{{\omega }^{2}}\]       

C)      \[\frac{ML{{\omega }^{2}}}{4}\]  

D)      \[\frac{M{{L}^{2}}{{\omega }^{2}}}{2}\]        

Correct Answer: A

Solution :

\[dm=\frac{M}{L}dx\] \[dF=dm\,x{{\omega }^{2}}\] \[\int_{0}^{F}{dF}=\frac{M}{L}{{\omega }^{2}}\int_{0}^{L}{x\,dx}\] \[F=\frac{M}{L}{{\omega }^{2}}\frac{{{L}^{2}}}{2}=\frac{ML{{\omega }^{2}}}{2}\] Hence, the correction option is [a].