Objective
The objective of this experiment is to verify a special form of the Bernoulli equation for a pressurized fluid flowing out of a container.
Lab Equipment
- Bucket with small hole drilled in the bottom
- 5 gallons of tap water
- Beaker with volume marks
- Ruler
- Stopwatch
Lab Procedure (Data Collection)
For this experiment we measure the amount of time it takes for a given amount of water to drain from the bottom of a large container of water as well as the actual diameter of the circular hole drilled into the container and compare the two using the special case Bernoulli relationship V = At(2gh)^(0.5).
1) Obtain a bucket with a small hole drilled near the base. Stop the hole with duct tape and fill the bucket with water to a certain height. Measure the distance from the hole to the surface of the water. This measurement will act as the height of water in the Bernoulli equation.
2) Place a beaker near the base of the bucket so that when the duct tape is removed water will flow out of the hole and into the beaker.
3) Measure the amount of time it takes to fill the beaker to 500 ml. Repeat this process for 6 different trials.
Data
Initial height of bucket = 0.1 m (error = 0.005 m)
Final height of bucket = 0.084 m (error = 0.005 cm)
Time Trials (error = 0.5 s)
1st Run: 26.27 s
2nd Run: 25.61 s
3rd Run: 24.51 s
4th Run: 26.21 s
5th Run: 25.77 s
6th Run: 25.78 s
avg time = 25.69
Volume emptied = 0.0005 m^3 (error = 0.000025 m^3)
Radius of drain hole = 0.0025 m (error = 0.00005 m)
Area of drain hole = 1.96 * 10^(-5) m^2 (error = 5 * 10^(-6) m^2)
(error for area of drain hole is increased due to irregularities in the shape of the hole)
Acceleration due to gravity = 9.81 m/s^2 (error = 0.001 m/s^2)
Calculations
Theoretical time to empty = V/(A(2gh)^(0.5)) = 20.9 s (error = 7.2 s)
Error Analysis
error between theoretical time and actual time = ([theoretical time - actual time])/theoretical time
e1 = 25.6 %
e2 = 22.5 %
e3= 17.3 %
e4 = 25.4 %
e5 = 23.3 %
e6 = 23.3 %
All experimental time values do agree with the uncertainty of the theoretical time value. (time is between 13.7s and 28.1s)
Theoretical area of drain hole = V/(t(2gh)^(0.5)) = 4.6 * 10^(-6) m^2
Theoretical radius = 0.0012 m
error = (0.0025 - 0.0012)/0.0012 = 52 %
Analysis
Our lab had an unusually large amount of error in predicting the time taken to empty a volume of water from a bucket. Small amounts of error can be attributed to errors in measurement as well as the fact that the water level changed significantly as the water drained from the bucket. However, I believe that most of the error originated in a flaw in the hole in the bucket. we measured the radius of the hole from the outside but upon further inspection realized that the actual hole was nearly half as small as the outer diameter led us to believe. A smaller hole would theoretically cause the water to drain more slowly and would result in longer draining times. In order to reduce these errors we would have to find a bucket with a less irregularly shaped hole and perhaps create a model for draining time that took into account the changing water level in the larger bucket.
2) Place a beaker near the base of the bucket so that when the duct tape is removed water will flow out of the hole and into the beaker.
3) Measure the amount of time it takes to fill the beaker to 500 ml. Repeat this process for 6 different trials.
Data
Initial height of bucket = 0.1 m (error = 0.005 m)
Final height of bucket = 0.084 m (error = 0.005 cm)
Time Trials (error = 0.5 s)
1st Run: 26.27 s
2nd Run: 25.61 s
3rd Run: 24.51 s
4th Run: 26.21 s
5th Run: 25.77 s
6th Run: 25.78 s
avg time = 25.69
Volume emptied = 0.0005 m^3 (error = 0.000025 m^3)
Radius of drain hole = 0.0025 m (error = 0.00005 m)
Area of drain hole = 1.96 * 10^(-5) m^2 (error = 5 * 10^(-6) m^2)
(error for area of drain hole is increased due to irregularities in the shape of the hole)
Acceleration due to gravity = 9.81 m/s^2 (error = 0.001 m/s^2)
Calculations
Theoretical time to empty = V/(A(2gh)^(0.5)) = 20.9 s (error = 7.2 s)
Error Analysis
error between theoretical time and actual time = ([theoretical time - actual time])/theoretical time
e1 = 25.6 %
e2 = 22.5 %
e3= 17.3 %
e4 = 25.4 %
e5 = 23.3 %
e6 = 23.3 %
All experimental time values do agree with the uncertainty of the theoretical time value. (time is between 13.7s and 28.1s)
Theoretical area of drain hole = V/(t(2gh)^(0.5)) = 4.6 * 10^(-6) m^2
Theoretical radius = 0.0012 m
error = (0.0025 - 0.0012)/0.0012 = 52 %
Analysis
Our lab had an unusually large amount of error in predicting the time taken to empty a volume of water from a bucket. Small amounts of error can be attributed to errors in measurement as well as the fact that the water level changed significantly as the water drained from the bucket. However, I believe that most of the error originated in a flaw in the hole in the bucket. we measured the radius of the hole from the outside but upon further inspection realized that the actual hole was nearly half as small as the outer diameter led us to believe. A smaller hole would theoretically cause the water to drain more slowly and would result in longer draining times. In order to reduce these errors we would have to find a bucket with a less irregularly shaped hole and perhaps create a model for draining time that took into account the changing water level in the larger bucket.
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