Toricelli's Law Lab Report
Autor: Maryam • February 13, 2018 • 3,043 Words (13 Pages) • 1,915 Views
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There are three types of pressure and they are: Total Pressure, Static Pressure, and Dynamic Pressure. All three pressures can be represented in this equation:
Ptotal = Pstatic + Pdynamic
Total pressure is the sum of static and dynamic pressure. Static pressure represents when a fluid is at rest. Dynamic pressure represents the kinetic energy of the moving fluid.
Bernoulli’s equation is represented as:[pic 5]
The variables P1, v1, and h1, represent the pressure, speed, and the height of the fluid at point 1. In addition, P2, v2, and h2, represent the pressure, speed, and the height of the fluid at point 2. The ‘p’ in 1/2ρv^2 represents the density of the fluid.
Deriving Torricelli’s Equation from Bernoulli's Equation:
- P1 + 1/2pv1² + pgh1 = P2 + 1/2pv2² + pgh2
- P1 + 1/2p(0)² + pgh1 = P2 + 1/2pv2² + pg(0)
- P1 + pgh1 = P2 + 1/2pv2²
- Patm + pgh1 = Patm + 1/2pv2²
- Gh = 1/2pv2²
- V2² = 2gh
- V2 = √2gh
Steps of Derivation Explained
- Write Bernoulli's equation down.
- Assume that the velocity is 0 at location 2 at the top of the fluid, meaning there is no kinetic energy present. Also assume that there is no gravitational potential energy at location 1 since it is at a height of 0.
- Write the remaining components cancelling the kinetic energy from the left side and the gravitational potential energy on the right side.
- The pressure at both locations is equal to the atmospheric pressure since at both points, the fluid is exposed to the bare atmosphere.
- Cancel out the atmospheric pressures at both sides as well as the density.
- Simplify to get V2² by itself with everything else on the other side of the equation.
- Square root both sides of the equation with the resulting final equation which is Torricelli’s equation.
Theoretical Result
In this experiment, we will be calculating for the flow rate and the height since calculating for velocity will produce inaccurate results according to our means of performing the experiment. We will still apply the flow rate to velocity conversion factor in order to further validate our data and theory in the end. This is further explained in the Analysis section of the report. Furthermore, the relationship between the flow rate and the height should also theoretically produce a square root relationship because both flow rate and velocity describes the motion of a fluid but in different units, and they are directly correlated to each other. The relationship is interrelated. The flow rate of a fluid is the volume of the fluid that goes through a surface per unit time. On the other hand, the velocity is how far the fluid travels per unit time and can be found dividing the flow rate by the cross sectional area of the bottle.
Hypothesis
If water leaks out the hole at the bottom of the bottle, it will leak at a rate that is proportional to the square root relationship of the depth of the water as if the water were to be in free fall from that height of h.
Observations and Analysis
Circumference of Two Litre Bottle
Volume of Water Below Hole
0.34 m
426 mL
Volume Lost over 5 seconds (±1 mL)
Volume of Water (mL)
Trial 1
Trial 2
Trial 3
Average Volume Loss (mL)
Error in Measuring Volume of Water (mL)
600
47 mL
49 mL
46 mL
47.0
6
700
61 mL
60 mL
63 mL
61.5
7
800
70 mL
71 mL
69 mL
70.0
8
900
73 mL
73 mL
73 mL
73.0
9
1000
76 mL
77 mL
75 mL
76.0
10
1100
80 mL
82 mL
80 mL
80.5
11
1200
82 mL
84 mL
83 mL
83.0
12
1300
86 mL
85 mL
87 mL
86.0
13
1400
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