JEE Main & Advanced Sample Paper JEE Main Sample Paper-43

  • question_answer
    An U tube of uniform bore of cross-sectional area A is set up vertically with open ends up. A liquid of mass M and density d is poured into it. The liquid column will oscillate with a period

    A)  \[2\pi \,\sqrt{\frac{M}{g}}\]              

    B)  \[2\pi \,\sqrt{\frac{MA}{dg}}\]

    C)  \[2\pi \,\sqrt{\frac{M}{Adg}}\]                      

    D)  \[2\pi \,\sqrt{\frac{M}{2Adg}}\]

    Correct Answer: D

    Solution :

     \[\Rightarrow \]where L is the length of the liquid is one of the limbs. However, if L is taken to be the length of the liquid column, then length of liquid in each limb is \[\frac{n'}{n}=\frac{v}{v-{{v}_{s}}}\]. \[\frac{\Delta n}{n}=\frac{{{v}_{s}}}{v-{{v}_{s}}}\]       \[\frac{2.5}{100}=\frac{{{v}_{s}}}{320-{{v}_{s}}}=\frac{1}{40}\] \[40{{v}_{s}}=320-{{v}_{s}}\]       \[{{v}_{s}}=\frac{320}{41}\simeq 8\,m/s\] \[V=\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{Q}{R} \right]+\frac{1}{4\pi {{\varepsilon }_{0}}}\left[ \frac{q}{r} \right]\]     \[=\frac{1}{{{\varepsilon }_{0}}}\left[ \frac{Q}{4\pi {{R}^{2}}}\times R+\frac{q}{4\pi {{r}^{2}}}\times r \right]\]

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