Hydrostatics and Cylindrical Tank

In summary: The Attempt at a SolutionMy main doubt is whether i should integrate from 0 to 1 and add the weight of the water above or simply integrate from 49 to 50 and be done. Both yield significantly different results.Depends on what variable you integrating with respect to. Is it distance from the top or distance from the bottom?
  • #1
Shoelace Thm.
60
0

Homework Statement


I fill a cylindrical tank of length 50 ft and diameter 90 ft with water (to the brim). What is the force the bottom 1 ft band of water exerts on the lining?


Homework Equations





The Attempt at a Solution


My main doubt is whether i should integrate from 0 to 1 and add the weight of the water above or simply integrate from 49 to 50 and be done. Both yield significantly different results.
 
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  • #2
Depends on what variable you integrating with respect to. Is it distance from the top or distance from the bottom? Show your setup.
 
  • #3
I have one representation being [itex]\vec{F} = 62.4 * 2 * \pi*45 \int^1_0 x\,dx + \pi*45^2*49[/itex], where dx is a differential movement from the top of the 1 ft band to the base of the tank, and the other being [itex]\vec{F} = 62.4 * 2 * \pi*45 \int^{50}_{49} x\,dx[/itex], where dx represents the same. Both expressions take into account the fact that the force is greater because of there being water above, but nevertheless yield significantly different results.
 
  • #4
Shoelace Thm. said:
I have one representation being [itex]\vec{F} = 62.4 * 2 * \pi*45 \int^1_0 x\,dx + \pi*45^2*49[/itex], where dx is a differential movement from the top of the 1 ft band to the base of the tank, and the other being [itex]\vec{F} = 62.4 * 2 * \pi*45 \int^{50}_{49} x\,dx[/itex], where dx represents the same. Both expressions take into account the fact that the force is greater because of there being water above, but nevertheless yield significantly different results.

The pi*45^2*49 you are adding in the first case has nothing to do with the force exerted by the first 49 feet of water. What should it be? If you'd put units on things, you'd see it doesn't even have correct units.
 
  • #5
Would [itex]62.4lbs/ft^3*\pi*45^2ft^2*49ft[/itex] be correct?
 
  • #6
And if so the answers are still inconsistent.
 
  • #7
You want to add the force due to the pressure of the first 49 feet of water on the area around the side of the bottom foot of the pool. What would that be?
 
  • #8
I can't think of an expression because the 49 ft of water in question is not even in contact with the bottom 1 ft of the tank, in which case I can't find a reasonable way to incorporate the fact that the bottom 1 ft of water is "submerged" beneath 49 ft of water.
 
  • #9
Shoelace Thm. said:
I can't think of an expression because the 49 ft of water in question is not even in contact with the bottom 1 ft of the tank, in which case I can't find a reasonable way to incorporate the fact that the bottom 1 ft of water is "submerged" beneath 49 ft of water.

The pressure of the 49 ft of water adds to the pressure of the last foot of water. It doesn't have to be in direct contact. What's wrong with doing it with your 49 to 50 integral?
 

What is hydrostatics and how does it relate to cylindrical tanks?

Hydrostatics is the study of fluids at rest and their behavior under the influence of forces. Cylindrical tanks are often used to store liquids, and hydrostatics is important in understanding how the liquid exerts pressure on the tank's walls and base.

How is the pressure in a cylindrical tank calculated?

The pressure in a cylindrical tank is calculated using the equation P = ρgh, where P is the pressure, ρ is the density of the liquid, g is the acceleration due to gravity, and h is the height of the liquid in the tank. This equation is known as the hydrostatic equation.

What is the significance of the shape of a cylindrical tank in hydrostatics?

The shape of a cylindrical tank is important in hydrostatics because it determines how the pressure will be distributed on the tank's walls and base. The circular shape of a cylinder allows for even distribution of pressure, making it a stable and efficient design for storing liquids.

How does the height of the liquid in a cylindrical tank affect the pressure?

The pressure in a cylindrical tank increases as the height of the liquid increases. This is because the weight of the liquid increases with height, resulting in a higher hydrostatic pressure.

How is the stability of a cylindrical tank affected by the liquid inside?

The stability of a cylindrical tank is greatly affected by the liquid inside. If the liquid level is too high, it can create excessive pressure on the walls and cause the tank to collapse. On the other hand, if the liquid level is too low, the tank may become unstable and tip over. Properly managing the liquid level is crucial for maintaining the stability of a cylindrical tank.

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