A 1-m-tall barrel is closed on top except for a thin pipe extending 5.2m 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 its top?
To determine the pressure on the bottom of the barrel when the pipe is filled to the top, we need to consider the additional height of the water column within the pipe.
The pressure exerted by a fluid is directly proportional to its height and density. In this case, the difference in pressure will be due to the difference in height between the initial water level and the new water level inside the pipe.
First, let's calculate the height of the water column within the pipe. The pipe extends 5.2 m up from the top, and the barrel is initially filled with water up to 1 m deep, so the height of the water column within the pipe is:
5.2 m - 1 m = 4.2 m
Now, let's determine the pressure contribution from the water column within the pipe. The pressure at a certain depth in a fluid can be calculated using the formula:
Pressure = density x gravitational acceleration x depth
The density of water is approximately 1000 kg/m³, and the gravitational acceleration is 9.8 m/s². Plugging in these values and the height of the water column:
Pressure = 1000 kg/m³ x 9.8 m/s² x 4.2 m
Pressure = 41160 Pa
Finally, we need to convert the pressure from pascals (Pa) to kilopascals (kPa) by dividing by 1000:
Pressure = 41160 Pa / 1000 = 41.16 kPa
Therefore, the pressure on the bottom of the barrel when water is added to fill the pipe to its top would be 41.16 kPa.