A 1-m-tall barrel is closed on top except for a thin pipe extending 4.8m 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? (Neglect the pressure due to the atmosphere in the following calculations.)
To find the pressure on the bottom of the barrel when the pipe is filled to its top, you need to consider the additional height of water in the pipe.
Here's how you can calculate it:
1. Start by determining the pressure at the base of the pipe when it is empty.
Since the water is filled up to a depth of 1 m, the pressure at the base of the pipe is simply the pressure due to the height of the water column above it.
Using the formula for pressure, P = ρgh, where:
P = pressure,
ρ = density of water,
g = acceleration due to gravity,
and h = height of the water column,
Plug in the values:
ρ = 1000 kg/m³ (density of water),
g = 9.8 m/s² (acceleration due to gravity),
h = 1 m (height of the water column).
P = (1000 kg/m³) x (9.8 m/s²) x (1 m) = 9800 N/m² = 9800 Pa = 9.8 kPa.
So, when the barrel is filled with water up to the base of the pipe, the pressure on the bottom is 9.8 kPa.
2. Next, calculate the pressure when the pipe is filled to its top.
Now, you need to account for the additional height of water in the pipe, which is (4.8 m - 1 m = 3.8 m).
Using the same formula, P = ρgh, with the same values for ρ and g, but with h = 3.8 m (height of the water column above the base of the pipe),
P = (1000 kg/m³) x (9.8 m/s²) x (3.8 m) = 37240 N/m² = 37240 Pa = 37.24 kPa.
Therefore, when the water is added to fill the pipe to its top, the pressure on the bottom of the barrel is 37.24 kPa.