Experiment 4: Standing Waves

Experiment 4: Standing Waves

Introduction:
The purpose of this experiment is determine a relationship between wavelength and frequency in order to understanding the standing waves driven by an external force.
We are going to measure and compare different wavelength of waves on a single string by changing the frequency of the wave.The weight and length of the string to be 3.08 g and 140 cm; also the hanging mass is 199.8g. We had two cases for this lab, by changing the amount of tension we give to the string we should be able to change the wavelength and thus the frequency of the string.  We repeated this measurement multiple times to get an idea of the uncertainties involved.

Calculations & Graphs:

Case 1
Osciallation Frequency (Hz)	16.5	32.5	45.6	56.7	74.6	87.4
Nodes	2	3	4	5	6	7
Wave Length (m)	1.4	0.67	0.45	0.37	0.26	0.21
1/λ	0.71	1.49	2.22	2.70	3.85	4.76
Case 2
Osciallation Frequency	21.8	29.5
Nodes	4	5
Wave Length	0.46	28

Both cases' frequencies followed the patter of f = vn/2L where v is the wave velocity, n is the number of antinode, and L is the length of the string. As we expected, when the wave lengths are the same, the lower velocity requires lover frequency inorder to reach certain resonance frequency. On the other hand, the higher the velocity requires higher frequency to reach the resonance frequency.

By looking at the graph, it is clear that there is a linear relationship between frequency and wavelength. As the wavelength is increased, the frequency correspondingly decreases. The ratio of velocities between case 1 is about 1.48, which is close to radical 2 as we expected. However, we cannot get enough data for case 2 since the tension on the string is too small.

Conclusion:

The uncertainties relating to the wavelength is due to our ability to correctly measure the spring using a large 2-meter stick, and to correctly hold that measured length during that phase of the experiment.  Finally, the uncertainty related to the frequency is naturally derived from the uncertainties related to the time.

Experiment 2: Fluid Dynamics

Experiment 2: Fluid Dynamics

Introduction:
The purpose of this experiment is to apply Bernoulli equation to determine the diameter of the hole on the bottom of the bucket.

First, we began the experiment by filling a large bucket with water and then allowed water to flow out of a small hole drilled near the bottom of the bucket. We took 6 trials and record the time took for a 16 ounces of water to exit from the bottom of the bucket. All measurements are in seconds and have an uncertainty of +/- 0.1s.

	1st run	2nd run	3rd run	4th run	5th run	6th run
Time to empty(tactual)	29.1 +/- 0.1s	28.9 +/- 0.1s	28.8 +/- 0.1s	29.2 +/- 0.1s	29.3 +/- 0.1s	28.9 +/- 0.1s
Volume of water before (inch)	3 1/6	3 1/8	3  1/16	3	3	3
Volume fo water after (inch)	2 1/2	2 1/2	2  7/16	2  7/16	2  7/16	2  7/16
Volume emptied (V): 16 ounces = 1.67 × 102 ft3
Area of drain hole (A): πr2 = 4.144 × 10 -4 ft 2
Acceleration due to gravity (g) : 32 ft/s2
Height of water (h): 3 inches = 0.25 ft
ttheoretical: V / A√2gh = (1.601 ×10-2)/ (4.144 × 10 -4)(√2(32)(5.58)) = 25.5 s
% error	1st run	2nd run	3rd run	4th run	5th run	6th run
	14.90%	13.30%	12.94%	14.51%	14.90%	13.33%
taverage : 29.03 s
A theoretical: V / t√2gh = (1.601 ×10-2)/ (29.03)(√2(32)(5.58)) = 0.000292 ft2
Calculated Diameter: 0.588 cm
Given Diameter: 0.700 cm
% error: 16 %

Conclusion:
Compare the experimental value and the theoretical value, we have 16 % error. The 16 % error are due to several uncertainties and unaccounted human errors. First one is during the experiment, when we use stopwatch to catch the time until the beaker reach 16 ounces, there have some time delay. Second, the hole was made with a drill and never properly finished, this would create variations in stream diameter due to roughness around the hole.

Experiment 1: Fluid Statics

Introduction:

The purpose of this experiment is to calculate the buoyant force to the metal cylinder by three different methods. By using three different methods to determine for the buoyant force, in order to  compare which of the three methods is the most accurate.

Part A: Underwater Weighting Method

 When the cylindrical mass is completely submerged in the water, there are three forces acting upon it. The force of gravity, the tensional force by the string and the buoyant force by the water.

T+B-W=0, or B=W-T.

             The weight of the metal by hanging it in the air and using the force probe to read the data: 1.053±0.05N. Then we completely submerged the solid in water and used the force probe to read the data: 0.670±0.05N. Finally, using the above calculation, the buoyant force is 1.053N-0.670N= 0.383±0.05N.

Part B. The Displaced Fluid Method

For this method, we uses the weight of a displaced fluid to calculate the buoyant force. The weight of a displaced fluid is equal to the buoyant force of that fluid, or W=B.

The mass of the dry beaker is : 0.1090kg and the mass of the beaker + water: 0.230kg. Then, we slowly lowered the cylindrical mass into a beaker full of water.  When the water has completely stopped dripping. Then we calculated the mass of the runoff water by subtraction: 0.230kg-0.1090kg = 0.0391kg. Plugging this mass into the equation:  B=0.0391kg*9.81m/s^2 = 0.383±0.05N.

Part C. Volume of Object Method

The third method we use the volume of the metal cylinder to find the volume of displace water. The formula is W=B, or W=p*V*g.

The volume of a cylinder is V=pi*r^2*h. We used calipers to measure the diameter, d=0.025m and the height, h=0.076m. We then used the formula to calculate the volume, V=(pi)(0.0125)^2(0.076)= 3.729*10^-5 m^3. Using the density of water as 1000kg/m^3, we calculated the buoyant force: B=(1000kg/m^3)(3.729x10^-5 m^3)(9.81m/s^2)= 0.366±0.05N..

Conclusion:

Among all three of the methods, two of the results are the same. We have got the buoyant force to be 0.383N±0.05N for both part A and part B. For part C, the buoyant force is 0.366±0.05N.

The first method is flawed in its inherit method of collecting data values for weight. The force probe readings are not quite stablize, therefore, the results given do have lot of uncertainty.

           The second method which uses the mass of the displaced fluid also is quite inaccurate. We weren't able to completely catch all of the runoff water, and thus some of the mass could have been discarded. This would yield less weight calculated and ultimately less buoyancy.

The third method is the most accurate but it also have some uncertainty. The critical measurement in this method is the measurements for the volume, the diameter and the height. The humans reading the calipers would each be subject to uncertainty. Since the density of water of 1000kg/m^3 which is at STP. However, the classroom temperature was much higher than 0°C, and this could have influenced the calculation. By compare to other two methods, the third method is the most accurate.

In Part A, if the metal cylinder touching the bottom of water container, the buoyant force will be higher then currect value. because if the cylinder touch the bottom, there is another force support the cylinder up which have the same direction as the buoyant force, so the value of the buoyant force will be smaller than the value that we have got in the experiment.