A graduated cylinder is inverted in a very large bucket of water. The depth of the water in the container is 6 m and the top of the bucket is open to the air in the room. The mouth (bottom) of the graduated cylinder is 4 m above the bottom of the container. Due to a pressure difference, the water in the graduated cylinder rises 3 m above the surface of the water in the container. calculate the pressure at the following locations:
A-Surface of water in bucket
B-bottom of bucket
C-at mouth of cylinder
D-in cylinder level with water in bucket
E-top of water in cylinder
To solve this problem, let's first establish the following information:
Given:
Depth of water in the container (h1): 6 m
Height of the mouth of the graduated cylinder above the bottom of the container (h2): 4 m
Additional height the water rises in the graduated cylinder (h3): 3 m
We can use the formula:
Pressure = Density * Gravity * Height
Step 1: Calculate the pressure at the surface of the water in the bucket (Location A).
Since the surface of the water in the bucket is open to the air in the room, the pressure is atmospheric pressure, which is approximately 101,325 Pa.
Step 2: Calculate the pressure at the bottom of the bucket (Location B).
The pressure at the bottom of the bucket can be calculated using the formula above.
First, we need to calculate the total height of the water column from the surface to the bottom of the bucket:
Total height (h1 + h2) = 6 m + 4 m = 10 m
Now, we can calculate the pressure at the bottom of the bucket:
Pressure B = Density of water * Gravity * Total height
= 1000 kg/m³ * 9.8 m/s² * 10 m
= 98,000 Pa
Step 3: Calculate the pressure at the mouth of the cylinder (Location C).
The pressure at the mouth of the cylinder is equal to the pressure at the bottom of the bucket since they are at the same height.
Pressure C = Pressure B = 98,000 Pa
Step 4: Calculate the pressure in the cylinder at the same level as the water in the bucket (Location D).
The pressure at this level is the sum of atmospheric pressure and the additional pressure due to the water column above it.
Pressure D = Atmospheric pressure + Pressure due to additional height
= 101,325 Pa + (Density of water * Gravity * Total height)
= 101,325 Pa + (1000 kg/m³ * 9.8 m/s² * 10 m)
= 201,325 Pa
Step 5: Calculate the pressure at the top of the water in the graduated cylinder (Location E).
The pressure at the top of the water in the graduated cylinder is equal to the pressure at the same level in the bucket (Location D).
Pressure E = Pressure D = 201,325 Pa
To summarize:
A - Surface of water in the bucket: Atmospheric pressure (approximately 101,325 Pa)
B - Bottom of the bucket: 98,000 Pa
C - Mouth of the cylinder: 98,000 Pa
D - In the cylinder level with water in the bucket: 201,325 Pa
E - Top of the water in the cylinder: 201,325 Pa
To calculate the pressure at each location, we can use the concept of pressure due to the height of a liquid column. The pressure at a certain depth is directly proportional to the height of the column of the liquid above that point.
Let's calculate the pressure at each location:
A) Surface of water in the bucket:
The pressure at the surface of water in the bucket is due to the atmosphere. Atmospheric pressure is commonly considered to be approximately 101,325 Pascals (Pa).
B) Bottom of the bucket:
The pressure at the bottom of the bucket is the sum of atmospheric pressure and the pressure due to the height of the water column above it. Since the depth of the water in the container is 6 m, the pressure at the bottom of the bucket would be the atmospheric pressure (101,325 Pa) plus the pressure due to the height of the water column (6 m).
C) At the mouth of the cylinder:
The pressure at the mouth of the cylinder is the same as the pressure at the bottom of the bucket, as they are at the same level.
D) In the cylinder level with water in the bucket:
The pressure at this location is also the same as the pressure at the bottom of the bucket and the mouth of the cylinder since they are all at the same level.
E) Top of the water in the cylinder:
The pressure at the top of the water in the cylinder is the sum of atmospheric pressure and the pressure due to the height of the water column above it. Since the water rises 3 m above the surface level, the pressure at the top of the water in the cylinder would be the atmospheric pressure (101,325 Pa) plus the pressure due to the height of the water column (3 m).
It's important to note that the pressure at these locations assumes ideal conditions, neglecting factors such as the density of the water and any pressure variations.