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Model Paper 6 — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsModel Paper 65 Marks
Question
A tube of length $L$ is filled completely with an incompressible liquid of mass $M$ and closed at both the ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity $\omega$. Determine the force exerted by the liquid at the other end.
✓

Answer

Consider a small element of the liquid of length dx at a distance x from one end. 
Image
Mass of the small element $=\frac{M}{L} d x$
Centripetal force associated with the element
$
d F=\left(\frac{M}{L} d x\right) x \omega^2\left[\because F=m r \omega^2\right]
$
Force exerted by the liquid = Total centripetal force at the other end
$
\begin{aligned}
& F=\int d F=\int_0^L \frac{M}{L} \omega^2 x d x=\frac{M}{L} \omega^2\left[\frac{x^2}{2}\right]_0^L \\
& =\frac{M}{L} \omega^2 \frac{L^2}{2}=\frac{1}{2} M \omega^2 L
\end{aligned}
$

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