Conceptual Pressure Question: Fluid Pressure in a U-Tube

In summary, a U-tube filled with water has two arms that are capped and cylindrical in shape. The right arm has twice the radius of the left arm and both caps have negligible mass and are watertight. A one-inch depth of sand is poured onto each cap. After the caps have moved to reestablish equilibrium, the right cap will be lower than the left cap due to Pascal's Principle and the larger area on the right side.
  • #1
mantillab
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0

Homework Statement


A U-tube is filled with water, and the two arms are capped. The tube is cylindrical, and the right arm has twice the radius of the left arm. The caps have negligible mass, are watertight, and can freely slide up and down the tube. A one-inch depth of sand is poured onto the cap on each arm. After the caps have moved (if necessary) to reestablish equilibrium, is the right cap higher, lower, or the same height as the left cap?

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Homework Equations



p = p_0 + pgd

The Attempt at a Solution


I'm slightly confused by this problem because my text doesn't really discuss closed liquid systems.

It seems that if the left cap is wider, yet they both have a 1" depth of sand poured in, that there would be more sand (more mass) on the left cap. Would that result in a greater surface pressure (p_0)?

I know that if the water is in hydrostatic equilibrium and if the container is open, then the pressure under both caps is the same and they would rise to the same height.

But since the u-tube is capped/closed would the left cap (larger diameter) sink lower than the right cap in order to reestablish equilibrium?
 
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  • #2
Apply Pascal's Principle, the pressure in a closed system is constant. Assuming that the right side has a larger area than that of the left side, what can be said about the forces of the right side to the left side.
 
  • #3


I would approach this problem by first considering the basic principles of fluid pressure and hydrostatic equilibrium. In a closed system like a U-tube, the pressure at any given point must be equal in all directions. This means that the pressure at the bottom of each cap must be the same, regardless of the size or shape of the cap.

Next, I would consider the weight of the sand on each cap. Since the caps are watertight and can freely slide up and down the tube, the sand will exert a downward force on each cap. However, the larger cap will have a larger surface area, which means that the force per unit area (pressure) will be less compared to the smaller cap. This means that the left cap will experience a greater downward force due to the weight of the sand.

To reestablish equilibrium, the left cap will need to sink lower than the right cap in order to balance out the greater downward force from the sand. This means that the right cap will be higher than the left cap.

In summary, the right cap will be higher than the left cap in this closed system due to the difference in surface area and the weight of the sand. This is in accordance with the principle of hydrostatic equilibrium and the equation for fluid pressure.
 

What is fluid pressure?

Fluid pressure is the force per unit area that a fluid exerts on the walls of its container or any object immersed in it. It is caused by the constant collisions of fluid particles with the walls or surface of an object.

How is fluid pressure measured?

Fluid pressure is measured in units of force per unit area, such as pounds per square inch (psi) or pascals (Pa). It can be measured using devices such as pressure gauges or manometers.

How does fluid pressure change in a U-tube?

In a U-tube, fluid pressure is the same at any given horizontal level, regardless of the shape or width of the tube. However, the pressure decreases as you move upwards due to the weight of the fluid above pushing down, and increases as you move downwards due to the weight of the fluid below pushing upwards.

What factors affect fluid pressure in a U-tube?

The main factors that affect fluid pressure in a U-tube are the density and height of the fluid columns. As the density of the fluid increases, the pressure also increases. Similarly, the higher the height of the fluid column, the greater the pressure at the bottom of the tube.

What is the relationship between fluid pressure and depth?

The relationship between fluid pressure and depth is directly proportional. This means that as depth increases, so does the fluid pressure. This is due to the weight of the fluid above pushing down and creating a greater pressure at lower depths.

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