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QJEEM22Ph48b6: JEE Main 2022 Physics - Class 11 JEEM22Ph48b6 Solution: A tube of length is filled completely with an incompressible…

A tube of length 50 cm50 \mathrm{~cm} is filled completely with an incompressible liquid of mass 250 g250 \mathrm{~g} and closed at both ends. The tube is then rotated in horizontal plane about one of its ends with a uniform angular velocity xF rad s−1x \sqrt{F} \,\mathrm{rad} \,\mathrm{s}^{-1} . If F\mathrm{F} be the force exerted by the liquid at the other end then the value of xx will be __________.

Step-by-Step Verified Solution

JEE Main 2022 (Online) 29th July Evening Shift Physics - Properties of Matter Question 107 English Explanation
F=∫(dm)ω2x\mathrm{F}=\int(\mathrm{dm}) \omega^2 \mathrm{x}

=∫0L(mLdx)ω2x=\int_0^{\mathrm{L}}\left(\frac{\mathrm{m}}{\mathrm{L}} \mathrm{dx}\right) \omega^2 \mathrm{x}

=mLω2L22=\frac{\mathrm{m}}{\mathrm{L}} \omega^2 \frac{\mathrm{L}^2}{2}

=mω2 L2=\frac{\mathrm{m} \omega^2 \mathrm{~L}}{2}

ω=2 mLF\omega=\sqrt{\frac{2}{\mathrm{~mL}}} \sqrt{\mathrm{F}}

=20.25×0.5F=\sqrt{\frac{2}{0.25 \times 0.5}} \sqrt{\mathrm{F}}

=16F=\sqrt{16} \sqrt{\mathrm{F}}

=4F=4 \sqrt{\mathrm{F}}

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