Fluid Statics
The purpose of this lab was to practice and observe the force of buoyancy on submerged objects. This lab utilizes various equipment: force probe, string, overflow container, metal cylinders, and a meter stick.
The metal cylinder will be used as the object which weight will be measured on different circumstances and compared.
The mass was found to be 112 +/- 0.5 g (0.112 +/- 0.0005 kg), with weight of ~1.099 N.
When submerged, the weight (hang on string and measured with force probe, connected to LoggerPro) was 0.710 +/- 0.0005 N.
The force diagram looks something like the above figure ^
Since the entire object was at stand-still (floats in place), the sum of Forces can be said to be 0:
Gravitational Force = Buoyant Force + Tension Force
or
Buoyant Force = Gravitational Force - Tension Force
The buoyant Force value would be the Weight in air subtracted by the measured weight in air, which is ~1.099 N - 0.710 N = 0.389 N
The second method was to measure the overflow of water.
First, an empty beaker's mass was measured to be 0.217 +/- 0.0005 kg. This beaker was to be the overflow container later. Another beaker was set up and filled with water to the brim. The object was then submerged slowly, completely. Some water overflowed onto the overflow beaker/ container. The mass of such container was measured, and the measurement was 0.256 +/- 0.005 kg. The mass of water then was ~0.039 kg.
0.039 kg is roughly equivalent to ~0.383 N.
The last method was to measure the volume of displaced water.
The cylinder's height was 0.077 +/- 0.0005 m, while diameter 0.025 +/- 0.0005 m. Volume is Area * Height, or square of half of its diameter times pi time height (V = pi * (diameter/2)^2 * height)
Based on the measured dimensions, the volume was roughly found to be 3.78*10^-4 m^3.
Archimedes' principle states that
"When a body is completely or partially immersed in a fluid, the fluid exerts an upward force on the body equal to the weight of the fluid displaced by body."
Also, recall that mass is density * Volume. The force then is density * Volume * gravitational acceleration (Buoyant Force = density * gravitational acceleration * vol).
Combining all the data together, the Force was calculated to be roughly ~0.371 N
The values from the first, second, and third experiments, respectively, are: ~0.389 N, ~0.383 N, and ~0.371 N.
1. Error Analysis/ Uncertainties:
Error analysis/ uncertainties can be calculated by:
uncertainty =
((∂F/ ∂x * uX)^2 + (∂F / ∂y * uY)^2...)^(1/2)
(squareroot of the sum of partial derivatives times measured uncertainties squared)
First experiment:
Mass was measured to be 0.112 +/- 0.0005 kg. Weight would not gain "new calculated" uncertainties (we are assuming the acceleration of gravity to be constant 9.81), therefore weight in air is 1.099 N +/- 0.0005 N.
Measured weight was observed to be 0.710 +/- 0.0005 N.
The buoyant force, however, is the difference between both forces.
The new uncertainty is found to be squareroot of the sum of partial derivatives of Gravitational force in air minus in water.
Fg = force of gravity
Fw = force measured in water
W is weight function
∂W/∂Fg = 1,
1 * 0.0005 = 0.0005,
0.0005 ^2 = 2.5*10^(-7)
and ∂F/∂Fw = -1. The result will be the same. The sum will be 5.0 * 10^(-7). Squareroot of that is 7.07 * 10^(-4).
Therefore, the result from experiment 1 is 0.389 +/- 7.07*10^(-4) N.
Second experiment:
Mass of empty beaker was measured to be 0.217 +/- 0.0005 kg. Second mass (same beaker with water) 0.256 +/- 0.0005 kg. Uncertainty of their difference must be found using partial derivatives
Mw = mass of water
Me = mass of empty beaker
Mbw = mass of water+ beaker
W is weight as a function
Mw = Mbw - Me
∂W / ∂Me = -1
∂W / ∂Mw = 1
(the 1 and -1 value above will be multiplied by the same measurement uncertainties (0.0005) squared, which would give them both the same value of 2.5*10^(-7). The uncertainty in this experiment is the same as 1st.
Therefore, Weight is 0.382 +/- 7.07 * 10^(-4) N.
Third experiment.
Height was measured to be 0.077 +/- 0.0005 m
Diameter 0.025 +/- 0.0005 m; radius must be 0.013 +/- 0.0005 m.
Density of water is assumed to be constant of 1000 kg/ m^3.
Gravity is assumed to be constant 9.81 m/s^2
Weight is product of rho times pi times radius squared times height times gravity.
∂W / ∂r = rho * g * pi * h * 2r = 1000 * 9,81 * 3.14 * 0.077 * 2 * 0.013 = 61.7
∂W / ∂r * ur = 61.7 * 0.0005 =
(∂W / ∂r * ur)^2 = 9.51 * 10^(-4)
∂W / ∂h = rho * g * pi * r^2 * 1 = 5.21
∂W / ∂h * uh = 5.21 * 0.0005 = 2.60 * 10^(-03)
(∂W / ∂h * uh)^2 = 6.76*10^(-6)
((∂W / ∂r * ur)^2 + (∂W / ∂h * uh)^2)^(0.5) = 0.031
The result for last experiment is 0.371 +/- 0.031 N
2. Although third experiment yields the least error values, it is, however, the least accurate method, due to having to measure various dimensions (height, radius), which will accumulate more errors.
Both first and second method were equally as accurate. Finding Buoyant "weight" only involves simple arithmetic in experiment 1 and 2, resulting in least error. (Less measurement, less error accumulation).
3. Should cylinder touched the bottom, the buoyant force depends on the tension of the string. Touching the bottom (resting on bottom) will add a new Normal vector component pointing up; if the string is still taut, there would have been 3 vectors pointing upwards (tension, normal, and buoyant). This will result in lower value for buoyant force. Fb now, instead of Fg - Ft, becomes Fg - Ft - Fn; whereas (Fn = normal force)
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