Question
Show that motion of liquid in a U-tube is simple harmonic motion. Also find the time period of oscillation.
Solution
Consider a U-tube of uniform cross-section a.
Let a liquid of density ρ be poured into the tube such that length of the liquid in each limb is L, large as compared to the width of U-tube.
If liquid on the left side is depressed by a distance y, then the liquid will rise on the right side by y.
Therefore,
The height difference of liquid column in two limbs = 2y.
Restoring force exerted by weight of liquid column of height 2y is given by,
This force acts on the whole of liquid in U-tube.
The mass of liquid in U-tube is,
Therefore, acceleration of liquid in U-tube is,
Thus, acceleration is directly proportional to the displacement and directed towards the mean position. Hence, the motion is simple harmonic motion.
The time period of oscillation is given by,
Let a liquid of density ρ be poured into the tube such that length of the liquid in each limb is L, large as compared to the width of U-tube.
If liquid on the left side is depressed by a distance y, then the liquid will rise on the right side by y.
Therefore,
The height difference of liquid column in two limbs = 2y.
Restoring force exerted by weight of liquid column of height 2y is given by,
This force acts on the whole of liquid in U-tube.
The mass of liquid in U-tube is,
Therefore, acceleration of liquid in U-tube is,
Thus, acceleration is directly proportional to the displacement and directed towards the mean position. Hence, the motion is simple harmonic motion.
The time period of oscillation is given by,
