One end of a U-tube containing mercury is connected to a suction pump and the other end to the 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.

As an experienced tutor registered on UrbanPro, I'd be glad to explain this concept to you. Let's delve into the physics behind this scenario.

When a U-tube containing mercury is connected to a suction pump and the other end to the atmosphere, a small pressure difference is maintained between the two columns. This pressure difference causes the mercury levels in the two arms of the U-tube to be different, with one side being higher than the other.

Now, when the suction pump is removed, the pressure inside the U-tube equalizes with the atmospheric pressure. As a result, the mercury in the higher arm of the U-tube begins to fall while the mercury in the lower arm rises until both levels stabilize at the same height.

However, due to the inertia of the mercury, it doesn't stop immediately but overshoots the equilibrium position, creating a restoring force that brings it back towards the equilibrium position. This process repeats, causing the column of mercury to oscillate back and forth around the equilibrium position.

This oscillatory motion of the column of mercury in the U-tube is essentially simple harmonic motion (SHM). SHM occurs when a restoring force is proportional to the displacement from the equilibrium position and acts in the opposite direction to the displacement.

In the case of the U-tube, the restoring force is provided by gravity pulling the mercury back towards the equilibrium position. The displacement from the equilibrium position is directly proportional to the pressure difference between the two arms of the U-tube. Thus, the motion of the column of mercury in the U-tube can be described as simple harmonic motion.

Understanding this phenomenon not only helps in grasping the concept of SHM but also provides insights into fluid dynamics and pressure systems. If you have any further questions or need clarification on any point, feel free to ask!