A 1-m-tall barrel is closed on top except for a thin pipe extending 5 m up from the top. When the barrel is filled with water up to the base of the pipe ( 1 m deep ). The water pressure on the bottom of the barrel is 10 kPa. What is the pressure on the bottom when water is added to fill the pipe to it's top
To solve this problem, we can apply Pascal's principle, which states that the pressure applied to a fluid in a closed container is transmitted equally in all directions.
Given:
Height of the barrel = 1 m
Height of the pipe = 5 m
Water pressure on the bottom of the barrel when it is filled up to the base of the pipe (1 m deep) = 10 kPa
Step 1: Calculate the pressure due to the water column in the pipe.
Since the height of the water column in the pipe is 5 m and we know that the density of water is approximately 1000 kg/m³, we can use the equation P = ρgh, where P is the pressure, ρ is the density, g is the acceleration due to gravity, and h is the height of the water column.
Plugging in the values:
P = (1000 kg/m³) * (9.8 m/s²) * (5 m) = 49,000 Pa = 49 kPa
Step 2: Add the pressure at the bottom of the barrel when it is filled up to the base of the pipe to the pressure due to the water column in the pipe.
To find the total pressure at the bottom of the barrel when the water fills the pipe to its top, we simply add the two pressures calculated in steps 1 and 2:
Total pressure = 10 kPa + 49 kPa = 59 kPa
Therefore, the pressure on the bottom of the barrel when the water is added to fill the pipe to its top is 59 kPa.
To find the pressure on the bottom when the water is added to fill the pipe to its top, we can use the concept of hydrostatic pressure.
The hydrostatic pressure is given by the formula:
P = ρ * g * h
Where:
P is the pressure
ρ is the density of the fluid
g is the acceleration due to gravity
h is the height of the fluid column
In this case, the height of the water column is 5 m (the height of the pipe). We are given the pressure on the bottom when the water is filled up to the base of the pipe, which is 10 kPa.
To find the pressure at the bottom when the water is added to fill the pipe to its top, we need to consider the height of the water column from the base of the pipe to the bottom of the barrel. In this case, it is 1 m.
Now, let's calculate the pressure:
First, let's convert the pressure from kilopascals (kPa) to pascals (Pa):
10 kPa = 10,000 Pa
Next, we need to find the density of water. The density of water is approximately 1000 kg/m³.
Using the formula for hydrostatic pressure, we can calculate the pressure at the bottom when the water is added to fill the pipe to its top:
P = ρ * g * h
P = 1000 kg/m³ * 9.8 m/s² * (5 m + 1 m)
P = 1000 kg/m³ * 9.8 m/s² * 6 m
P = 58,800 Pa
Therefore, the pressure on the bottom when the water is added to fill the pipe to its top is 58,800 Pa.