A U-tube is filled with a single homogeneous liquid. The liquid is temporarily depressed in one side by a piston. The piston is removed and the level of the liquid in each side oscillates. We have to show that the period of oscillation is where L is the total length of the liquid in the tube.

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A U-tube is filled with a single homogeneous liquid. At equilibrium liquid levels on both sides of the U-tube will be equal. When the level of the liquid say in the left-side of the U-tube has been depressed by x units it will be lifted up by x units in the other side of the U-tube.

The liquid column will experience a downward force , where A is the cross-sectional area of the U-tube, because of gravity or equivalently due to the pressure difference. Mass of the liquid column is , as L is the total length of the liquid column. Motion of the liquid column can be considered in terms of the motion of its centre of mass (cm). Equation of motion of the cm of the liquid column is

It is an oscillatory motion with angular frequency

Period of oscillation T is related to as

Therefore, the period of oscillation of the liquid column is