Standing Waves Activity

Objective

Determine the velocity of a wave in a spring by observing the properties of standing waves created in the spring, specifically the wavelength and frequency of oscillation.

Lab Procedure

1) Measure a length of spring and have one student hold the spring on either end.

2) Have one of the students holding the spring oscillate his end vertically at a frequency which creates a standing wave with 2 nodes and 1 anti-node.

3) Have a third student use a stopwatch to measure the amount of time taken for 10 oscillations in order to determine the frequency.

4) Repeat these steps for 3 different lengths of spring

Data 

Trial 1

Length of spring = 1.2 m, wavelength = 2.4 m
time for 10 oscillations = 4.16 s, frequency = 2.4 hz

Trial 2

Length of spring = 1.4 m, wavelength = 2.8 m
time for 10 oscillations = 4.73 s, frequency = 2.11 hz

Trial 3

Length of spring = 1.6 m, wavelength = 3.2 m
time for 10 oscillations = 6.09 s, frequency = 1.64 hz

Analysis

using the data from the trials we found the velocity of a transverse wave in our spring to be between 5.2 and 5.9 m/s. One problem that we encountered when doing this activity was that when we changed the length of the spring we also adjusted the tension in order to make the amplitude less. However, by adjusting the tension in the spring we unknowingly changed the wave velocity and compromised the experiment. Also, the wavelengths used were too close together to graphically determine the relationship between wavelength and frequency. We could reduce these errors by using a single length of spring and trying to produce different standing waves. This would reduce the tension problems and would widen the range of wavelengths that we could observe in the experiment.

Fluid Dynamics Lab

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.

Fluid Statics Lab

Lab Objective

Experimentally verify the theoretical effects of the bouyant force on an object submerged in water.

Lab Equipment

-Lab Pro set up with Force Probe

-String

-Overflow can

-Beaker

-Metal cyinder with hooks

-Meter stick

-Vernier Calliper

Lab Procedure (data collection)

1) use a force probe to measure the weight of a metal cylinder suspended in the air.

2) suspend the metal cylinder in the overflow can and measure the new reading on the force probe as well as the amount of water that spills out of the overflow can

3) measure the dimensions of the metal cylinder

Collected Data

Force Probe reading- 1.099N in air, 0.710N in water (error = 0.001N)

Mass of water spilled from overflow can- 0.0387kg (error = 0.005kg)

Mass of metal cylinder- 0.112kg (error = 0.001kg)

Diameter of metal cylinder- 0.025m (error = 0.01m)

Height of metal cylinder- 0.077m (error = 0.01m)

Calculations and Analysis

In order to find the experimental value of the bouyant force we find the difference between the two force probe readings.

Fb = Fa - Fw

Our measured value for the bouyant force lies between 0.387N and 0.391N

Next we compare this value to the weight of displaced water, assuming g = 9.81 m/s^2

Fb = W = g(Mw)

The measured weight of displaced water lies between 0.374N and 0.384N

These two values are close but the intervals do not overlap. This was most likely caused by water sticking to the overflow can as it spilled out. This would result in the measured weight of displaced water to be less than the actual bouyant force.

We also compared these bouyant force values to the theoretical weight of displaced water found by measuring the volume of the metal cylinder and multiplying by the density of water and acceleration due to gravity, assuming rho = 1000kg/m^3, g = 9.81 m/s^2

Fb = W = V(rho)(g) = pi(d/2)^2(h)(rho)(g)

This value for the bouyant force lies between 0.337N and 0.406N

This interval agrees with both previous intervals but is much wider due to the propagation of errors in measuring the dimensions of the cylinder.

Summary

1) Compare the 3 values for the bouyant force
The 3 intervals we obtained for the bouyant force (3.87-3.91, 3.74-3.84, 3.37-4.06) seem to agree with eachother fairly well. The intervals were constructed by calculating the lowest and highest possible values of the bouyant force given the initial measurements and errors. The first and second intervals did not overlap, but this was probably due to water residue from the overflow attaching to the overflow can.

2) Which method do you think was most accurate and why?
I think that the most accurate method for obtaining the bouyant force was by using the force probe because it measured the force more directly than the other methods and there was little room for experimental errors when simply using a force probe to measure a force. Also, the interval constructed from the force probe measurements was significantly smaller than the other two intervals.

3) If the cylinder had been touching the bottom of the water container, how would that have changed your value for the bouyant force?
If the cylinder had been touching the bottom our measurement for the bouyat force would have been much higher than the actual value. This would have been due to the normal force between the cylinder and container which would act upward on the cylinder and cause the measuerements of the force probe to be significantly lower.