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  4. A U tube of uniform bore of cross-sectional area A has been set up vertically with open ends facing up. Now m gm of a liquid of density d is poured into it. The column of liquid in this tube will oscillate with a period T such that

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Q. A $U$ tube of uniform bore of cross-sectional area $A$ has been set up vertically with open ends facing up. Now $m gm$ of a liquid of density $d$ is poured into it. The column of liquid in this tube will oscillate with a period $T$ such that

Oscillations

Solution:

If the level of liquid is depressed by $y cm$ on one side, then the level of liquid in column $P$ is $2 y cm$ higher than $B$ as shown.
image
The weight of extra liquid on the side $P=2 A y d g$.
This becomes the restoring force on mass $M$.
$\therefore $ Restoring acceleration $=\frac{-2 A y d g}{M}$
This relation satisfies the condition of SHM i.e., $a \propto-y .$
Hence time period
$ T =2 \pi \sqrt{\frac{\text { Displacement }}{\mid \text { Acceleration } \mid}} $
$=2 \pi \sqrt{\frac{\frac{y}{2 A y d g}}{M}} $
$\Rightarrow T=2 \pi \sqrt{\frac{M}{2 A d g}} $