Relativity of Length Activity

Question 1: Yes. The movement changes the observed distance.
Question 2: Longer due to time dilation.
Question 3: Yes. It must in order to withhold the Lorentz factor relationship.
Question 4: It would be 769.23 meters while moving.

Relativity of Time Activity

Question 1: The distance is greater for the moving frame than for the stationary one.
Question 2: The moving frame will experience a longer time interval.
Question 3: No, the rider would experience the same result as the stationary frame because it is the equivalent of a stationary frame of reference.
Question 4: There would be less of a difference in time experienced if the velocity was decreased because the distance the light traveled in the moving frame of reference would decrease.
Question 5: It would be the time measured by the observer riding the light clock multiplied by the Lorentz factor. in this case it is 8x10^-6 seconds for a time of 6.67x10^-6 and a Lorentz factor of 1.2.
Question 6: The Lorentz factor would have to be 1.1169

Experiment 7: Reflection and Refraction

Introduction: The purpose of this experiment is to examine some simple examples of reflection and refraction and relate them to Snell's law

Procedure: We set up a light box with a slit of light pouring out. We then directed that slit at the flat side of an acrylic semicircle. We set the semicircle on a protractor and rotate the entire set up to measure the angle of refracted light coming out the other side. We did this for 10 trials, taking data down starting at 10 degrees and ending on 70 degrees.

 We then attempted to do the same thing on the rounded side of the acrylic semicircle but were forced to stop at 44 degrees, only taking down 8 data sets.

Data:

Flat side of acrylic:

Theta 1 (degrees)	Theta 2 (degrees)
10	7
15	12
20	14
30	20
35	23
40	25
45	30
50	32
60	36
70	41

This is Theta 1 vs. Theta 2

Sin(Theta 1) vs. Sin(Theta 2)

Round side of acrylic:

Theta 1 (degrees)	Theta 2 (degrees)
0	0
5	7
10	15
15	24
20	35
30	53
40	75
44	90

Theta 1 vs. Theta 2

Sin(Theta 1) vs. Sin(Theta 2)

Analysis: We were not able to complete the 10 trials on the round side of the acrylic because we witnessed total internal reflection at 44 degrees. This is the last angel we could muster before total internal reflection.

The equation of the line from the flat side of the acrylic can be rewritten as sin(theta 1)/sin(theta 2)=1.48. Rearranging Snell's law, sin(theta 1)/sin(theta 2)=n two/n one. Therefore, 1.48 equals n of air/n of acrylic, given n one is air in this case. For the rounded side of the acrylic, the equation of the line can be rewritten as sin(theta 1)/sin(theta 2)=0.667. Using the rearranged version of Snell's law, 0.667 equals n of air/n of acrylic, given that n one is acrylic in this case because the light is being refracted after it has already entered the acrylic. Indeed, the second slope is the inverse of the first one.

Experiment 5: Introduction to Sound

Introduction: The purpose of this lab is to record different sound waves and analyze the differences in their wavelengths, amplitudes, and frequencies.

Procedure: Have a person say aaaaaa into a microphone and collect the sound wave data.

We then had a different person say aaaaaaaaa into the microphone and recorded the data

We then struck a tuning fork hard and then soft and recorded both results in a microphone

Data:

First person's voice

Second person's voice

Tuning fork struck hard

Tuning fork struck soft

Analysis: 

1-a) The wave is periodic because of its definite repeating pattern

1-b) We determined four waves by only counting the completed ones

1-c) The sound was only captured for .03 seconds so about a blink of an eye

1-d) The period is 7.5 x 10^-3 seconds. This was decided upon by counting the number of waves and dividing the total time of the sample by that number

1-e) Frequency is 1/period = 133 Hz

1-f) velocity=wavelength x frequency so wavelength=velocity/frequency which is 2.55 meters; about the length of a lab desk

1-g) The amplitude can not be determined because there is nothing to reference it against, but there was an arbitrary pressure of about 2.7.

1-h) The only thing that would change is the amount of waves on the graph. Everything else would stay relatively unchanged.

2) Period=1.67 x 10^-3, frequency=600 Hz, nearly 4 times greater than the first, wavelength=0.567 m, nearly 4 times shorter than the first, amplitude is roughly the same.

3) Period=3.75 x 10^-3, frequency=2.67 Hz, wavelength=1.275 m. The wave is much smoother and uniform than a human voice.

4) Nothing changed but the amplitude and the smooth nature of the wave. We hit the fork on a softer object to make it softer.

Experiment 4: Standing Waves

Introduction: The purpose of this experiment is to observe the relationship between wavelength, frequency, and wave velocity at resonant conditions on a standing wave on string.

Procedure: We attached an oscillator to a frequency generator.

We then attached a length of string to a frequency generator via a clamp.

We put the other end of the string over a pulley and attache a 200g mass to it.

We changed the frequency of the frequency generator until we got 2 nodes and measured the distance between the nodes.

We did this for 3 nodes and 4 nodes up until 12 nodes and recorded the lengths between each.

We then replaced the 200g mass for 1/4 of it or 50g and repeated the entire process, only recording 7 different node distances rather than 11.

Data:

Case 1: 200g mass

Number of nodes	Frequency (Hz)	Distance d between nodes (m)	λ=2d (m)	1/λ	n=2L/λ (L=1.97)
2	22	1.97	3.94	0.253807106598985	1
3	44	0.98	1.96	0.510204081632653	2.01
4	59	0.652	1.304	0.766871165644172	3.02
5	77	0.44	0.88	1.13636363636364	4.48
6	97	0.393	0.786	1.27226463104326	5.01
7	120	0.33	0.66	1.51515151515152	5.97
8	130	0.295	0.59	1.69491525423729	6.68
9	170	0.245	0.49	2.04081632653061	8.04
10	190	0.215	0.43	2.32558139534884	9.16
11	200	0.185	0.37	2.7027027027027	10.6
12	215	0.175	0.35	2.85714285714286	11.3

  

 Case 2: 50g mass

Number of nodes	Frequency (Hz)	Distance d between nodes (m)	λ=2d (m)	1/λ	n=2L/λ (L=1.887)
2	10	1.887	3.774	0.264970853206147	1
4	29	0.662	1.324	0.755287009063444	2.85
5	34	0.481	0.962	1.03950103950104	3.92
6	50	0.395	0.79	1.26582278481013	4.78
7	59	0.33	0.66	1.51515151515152	5.72
8	70	0.277	0.554	1.80505415162455	6.81
14	130	0.115	0.23	4.34782608695652	16.4

Mass of String: 0.99g +/– 0.01g

Experimental mass of 200g mass: 199.9g +/– 0.01g

Experimental mass of 50g mass: 49.89g +/– 0.01g

String Length (oscillator to mass): 2.66m +/– 0.1m

Mass per unit length: 3.72 x 10^-4 kg/m

Analysis: Here is a graph of the inverse of wavelength versus frequency. The slope should represent the wave velocity.

Using the equation velocity=sqrt(Tension/mass per unit length), we get velocity=72.62 meters per second. That yields a percentage error of 4.91%. There exist uncertainty in both the equation value and the graph value, however I assumed the equation value to be the true value here. There is a small uncertainty that comes from the the measurement of the length and mass of the string as well as the fact that the string and pulley are not ideal. This is very small and that is why the equation can be assumed more accurate than the graph data. The graph data relies on the distance between nodes which was more difficult to measure. For one, we could not bring the meter stick too close to the string while it was vibrating. Also as the nodes became more numerous, it became difficult to pinpoint the point of stillness on the vibrating string. These measurements were less accurate and will most likely account for this error. The final contribution is the uncertainty of the frequency generator, given the lack of precise markings on the knob.

Here is the same graph for case 2

The ratio case 1/case 2 for velocity from the graphs is 2.64. The ratio of velocity from the equation is 2.00. The ratios are not equal, but similar. The difference here is probably due to the inability to measure the distance between nodes accurately as well as the uncertainty of the knob on the frequency generator. The values are close so it can be assumed that the measurements were close to the true measurements.

The resonant frequency times n does not equal the set frequency for any value in case 1, but they are relatively close. This can be caused by any of three things: the wave velocity used is inaccurate, the value for n is incorrect, or the frequency recorded from the knob is wrong. It is probably a mixture of all but given how the velocity is derived, it is one of  most influential factors.  The other is the value for n. It is certainly incorrect because n is always one number less than the number of nodes.

The frequency for case 1 is just about double that of case 2 for the same harmonic. This makes sense because we used 4 times the tension in case 1 compared to case 2, and the formula for velocity includes tension under a radical.

Experiment 3: Introduction to Waves

Introduction: The purpose of this lab is to examine the relationship between frequency and wavelength

Procedure: We decided to generate our wave at a length of 2,4,6,8, and 10 meters. We then measured the proper length of wave.

We then shook the string, attempting to generate something similar to simple harmonic motion. We recorded the time it took to complete 20 oscillations.

Data: We found period by dividing the time it took to complete 20 oscillations by 20. We then found frequency by dividing 1 by the period

Analysis: Frequency seems to be inversely proportional to wavelength as one can see from this graph

Experiment 2: Fluid Dynamics

Introduction: The purpose of this lab is to confirm Bernoulli's equation, specifically the special case of large volumes: Flow Rate = V÷t = A*sqrt(2gh)

Procedure: We took a large bucket with a hole drilled in it and measured the diameter of the hole. We then taped off the hole and filled the bucket with 14cm of water.

 We removed the tape and one of us placed our finger over the hole. Another readied a 500ml beaker.

 The person holding the whole withdrew and started a stopwatch. We timed how long it took to fill the beaker to 300ml.

 When the water level in the beaker reached 300ml, we stopped the time and plugged the hole again.

We then refilled the water level in the bucket to 14cm and repeated five more times.

Data: Hole diameter: 5.1mm, Average time: 13.86 s

Trial	Time
1	14.91s
2	13.86 s
3	13.20 s
4	13.55 s
5	14.26 s
6	13.39 s

 

Analysis: Using t = V÷(A*sqrt(2gh)) we calculated t theoretical to be 8.87 s. Here is the percentage error between actual and theoretical values of t for all trials 

Trial	Percentage Difference
1	68.1%
2	56.3%
3	48.8%
4	52.8%
5	60.8%
6	51.0%

 

Using is the above equation solved for A and the average experimental time, I calculated the theoretical diameter of the hole in the bucket. Using the equation (experimental diameter–theoretical diameter)÷Theoretical diameter x 100%, I found a 25% error. There is a steady error of about 55% between theoretical time and experimental time. The imprecise nature of this data can be explained by our inability to stop the watch at exactly 300mL. We also had uncertainty of about plus or minus a centimeter when measuring the 14cm water level. The inaccurate nature of this data can be explained by the diameter of the hole drilled in the bucket. We could not get a consistent measure on the hole because it was not round. It was irregular and conical shaped. The inaccuracy of our data is much greater than our lack of precision so it can be assumed that our consistent mistake can be attributed to the incorrect measurement of the hole.