Two identical wooden barrels are fitted with long pipes extending out their tops. The pipe on the first barrell is 1 foot in diameter, an the pipe on the second barrell is only 1/2 inch in diameter. When the larger ptpe is fill with water to a height of 20 feet, the barrell bursts. To burst the second barrell. will water have to be added to an height less than, equal too, or greater than 20 feet? Explain.
Thank you.
same height
pressure = density * g * height
Same height (20 feet). The pressure in the barrel depends only upon the height of the column of water above.
To determine whether water needs to be added to a height less than, equal to, or greater than 20 feet in order to burst the second barrel with the smaller pipe, we need to consider the principles of fluid pressure.
The pressure exerted by a fluid at a certain depth is directly proportional to the height of the fluid above that point. This principle is known as Pascal's law.
In this scenario, since the barrels are identical, we can assume that the same amount of water will be added to each barrel.
When the larger pipe is filled to a height of 20 feet and bursts, it means that the pressure at the bottom of the barrel due to the weight of the water is sufficient to exceed the structural strength of the barrel.
To determine whether the second barrel will burst, we need to compare the pressure exerted by the water in the large barrel (with a 1-foot diameter pipe) at a height of 20 feet to the pressure exerted by the water in the small barrel (with a 1/2 inch diameter pipe) at a certain height.
Using Pascal's law, we can calculate the pressure at the base of the large barrel:
Pressure = Density × Gravity × Height
Assuming the density of water is constant, we can ignore it in this comparison. Therefore, the pressure at the base of the large barrel is directly proportional to the height of the water column (20 feet).
For the small barrel to burst, its pressure at the base needs to exceed the pressure exerted by the water in the large barrel. However, since the height of the small barrel is the same as the large barrel, the pressure at the base of the small barrel will be lower due to the smaller diameter of the pipe.
This means that, in order to burst the second barrel, water will need to be added to a height greater than 20 feet. The smaller pipe diameter restricts the flow of water and thereby decreases the pressure at the base of the second barrel. The reduced pressure at the base means that a greater height is required to exert enough pressure to burst the barrel compared to the larger one with the larger pipe diameter.
Therefore, water will need to be added to a height greater than 20 feet to burst the second barrel with the smaller pipe.
To determine if the second barrel will burst, we need to understand the concept of hydrostatic pressure. Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity acting on it.
In this scenario, we have two identical barrels but with different diameters for their pipes. The larger pipe has a diameter of 1 foot, while the smaller pipe has a diameter of 1/2 inch.
The height to which the larger pipe is filled is given as 20 feet. For the larger barrel to burst, the hydrostatic pressure at the base of the barrel must exceed the strength of the barrel itself.
To determine if the smaller barrel will burst, we need to compare the hydrostatic pressure at the base of the barrels. The hydrostatic pressure is directly proportional to the height and the density of the fluid. Since both barrels are filled with water, the density remains constant.
First, let's calculate the hydrostatic pressure exerted by the water in the larger barrel. We use the formula: P = ρgh, where P represents pressure, ρ represents density, g represents gravity, and h represents height.
For the larger barrel, the diameter of the pipe is 1 foot, so the radius is 0.5 feet. The area of the pipe's base is then calculated as A = πr^2 = π(0.5)^2 = π(0.25) = 0.7854 square feet.
Plugging in the values, we have:
P = ρgh = (62.43 lb/ft^3)(32.17 ft/s^2)(20 ft) = 39965.52 lb/ft^2
Now let's calculate the hydrostatic pressure exerted by the water in the smaller barrel. The diameter of the pipe is 1/2 inch, which is equivalent to a radius of 1/4 inch or 0.021 feet. The area of the pipe's base is then calculated as A = πr^2 = π(0.021)^2 = π(0.000441) = 0.00138 square feet.
Plugging in the values, we have:
P = ρgh = (62.43 lb/ft^3)(32.17 ft/s^2)(h)
To determine the height required for the smaller barrel to burst, we set the pressure equal to the hydrostatic pressure of the larger barrel and solve for h:
39965.52 lb/ft^2 = (62.43 lb/ft^3)(32.17 ft/s^2)(h)
h = (39965.52 lb/ft^2) / [(62.43 lb/ft^3) x (32.17 ft/s^2)]
By simplifying the equation, we find:
h = 20.4 ft
Therefore, to burst the second barrel, water must be added to a height greater than 20 feet, specifically 20.4 feet. Adding water to a height greater than 20.4 feet will exert a hydrostatic pressure on the base of the barrel that exceeds its strength, causing it to burst.