One end of a $U - tube$ containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the $U - tube$ executes simple harmonic motion

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Solution

As suction-pump is removed from one end mercury try to restore its equilibrium position [mean position] with restoring force.

Restoring force $F_{R}$ = Weight of fluid column

$F_{R} = - m g$

$= - ρ A h g$

$F_{R} = - (ρ A h g)$

$F_{R} \propto - h$

This equation indicating that
mercury in $U - tube$ column executing simple harmonic
motion

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One end of a U−tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U−tube executes simple harmonic motion

One end of a $U - tube$ containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the $U - tube$ executes simple harmonic motion