#13 Relativity

Part 1: Time

Q1: The light traveling on the moving clock passes longer distance than the stationary clock.

Q2: The time required for the moving clock traveling back and forth is longer than the stationary clock.

Q3: On the moving clock, the distance and time is the same as the stationary.

Q4: If the velocity of the moving clock decrease, the difference in time also decreases.

Q5: t = 1.2x(2000/(3x10^8)) = 8x10^(-6), the simulation caters to the calculations.

Q6: 3 x 10^8 x 7.45 x 10^(-6) / 2000 = 1.12

Part 2: Length

Q1: Measuring in the frame, the time is the same for moving and stationary.

Q2: The time measured on earth is longer than the measurement in the light clock.

Q3: They will not be equal.

Q4:  1000 / 1.3 =769 m

#15 Laser

Q1: 3+183+20=206 for this specific moment. Other moments also shows conservation.

Q2: There is no preferred direction for which the photon goes.

Q3:  Although most of them emitted at the beginning, there can be a few atoms remaining in its exited states for a long time,so the time is not constant.

Q4: When the photon interact with the excited atom, the atom emits another photon that is in the same direction and same phase as the incoming photon.

Q5: 70 might be a good guess because it varies from 8:12 to 12:8 nearly all the time.

Q6: The odd direction photons are in spontaneous emission originally, but these photons sometimes interact with another exited photon thus causing a stimulated emission of photons.

  

#14 Spectra

Objective: 

The purpose of this experiment is to measure the wavelength of a few visible light and form a linear transformation of the derived wavelength, and use this to measure the spectra of hydrogen atom and an unknown to find out what the unknown is.

Procedures:

1.      Set up as instructed in the lab manual, put the light source as the white light.

2.      Record the position for each light seen through the diffraction grating. 

3.      Change the light source to hydrogen atoms and unknowns and perform same procedure.

Results:

       The acceptable visible light range is from 390 nm to 750 nm.

The wavelength is derived using formula:

L is 100 cm, and d is 1/500 mm.

For white light:

Color	Distance D (cm)	λ(nm)
Purple Begins	17.5	345
Purple -> Blue	22	430
Blue -> Green	24.5	478
Green -> Yellow	26	503
Yellow -> Red	28	539
Red	35	661
Red ends	40	743

           Using 345 and 743 to do the linear transformation, the equation is λ = 0.905λ'+78 in nm.

Then for the hydrogen atom:

Color	Distance D (cm)	λ(nm)
red	33.5	653
Yellow	28	566
Green	24	500
Purple	21	450

            

λ is derived from the early equation and linear transformation.

For the unknown, we derived following data:

Trial 1:
Color	Distance D (cm)	λ(nm)
Yellow	29	582
Green	27	550
Blue	21	450
Trial 2:
Color	Distance D (cm)	λ(nm)
Yellow	29.5	590
Green	28	566
Blue	21.5	458

            We suspect the unknown to be Mercury (Hg). The spectra for Mercury is:

          The spectra for Hg shows peak at 435, 545, and 580 nm, which is very close to the calculated data.

Discussion:

          The measurement is very inaccurate for the precise distance from diffraction grating to the light source is hard to measure, so does the displacement. The derived spectra for white light gives a fairly precise linear transformation to the system. The unknown is Hg as what professor told us, but the wacelength for blue light is comparably different from the theoretical one.

#12 light and matter waves

Objective: 

          The purpose of this experiment is by using Vpython to visualize the wave in the space.

          By creating a wave source at 0,0 producing a wavelength of 2mm, it yields as following:

     Contour as shown:

          Varing the wavelength to 4mm and 8mm, yields:

8mm 

4mm

          They show nearly the same pattern, and the peak is at the wave source.

          For double slits, the separation is 1.2cm, wave sources are set at (0,0.6) and (0.-0.6).

          Wavelength in 2mm, 4mm, and 8mm yields interference pattern as following:

2mm

4mm

8mm

          Thus again they perform a likely pattern. The intensity graph shown is:

          Shows it reaches peak near the central point, that is what shows in the graph.

          

#11 CD Diffraction

Objective: 

The purpose of this experiment is to measure the length of the gap between the grooves on the CD and DVD.

Procedures:

1.      Arrange the equipment as guided, shoot the laser to the CD/DVD.

2.      Record distance between two diffracted points on the white board behind the laser, make sure the laser is in the middle.

3.      Record wavelength of the laser and the distance from the board to CD/DVD.

Results:

       The wavelength of the laser is from 630 to 680 nm. Formula is given as:

       d*sin(theta)=d*y/L=n*lambda

       For CD:

        The distance from CD to screen is 47.5cm, distance between two point on the board is 49cm.Thus the angle is 26.32 degree.

    L = 47.5cm, theta = 26.32, y = 24.5cm, lambda from 630 to 680 nm.

    Derived value for d is from 1432 to 1545 nm, the proper distance from internet is 1500, which is in the range.

    For DVD:

    L = 20cm

    y = 19 cm
    Angle theta = 43.5
    Lambda unchanged
    d is from 915 to 988 nm. Proper distance is 740, which is not in the range.

       

Discussion:

    The measurement conducts well for CD. The range is mainly due to the uncertainty of the laser wavelength, but the theoretical value falls in the range. The experiment conducts not so well for the DVD, which does not fall in the range. Maybe DVD has different groove structure or different arrangements, which may cause error in this method of measurement.

#9 Microwave oven

Objective: 

By the sacrifice of brave marshmallows we shall be able to determine the frequency of the microwave by assuming a standing wave is producing in the oven. Furthermore, a rage of microwave shall be deduced in the dimensions of microwave oven. By heating up water we can acquire the power and then decided how many photons per seconds are oscillating in the microwave and what pressure do these photons exerts.

Data recorded:

      λ= 12 cm, dimension of the oven is 23 x 35 x 35 cm³, mass of water is 100 g, heating time is 30 s, T_i is 20 ℃, T_f is 57 ℃.

Calculation:

       The wave is electromagnetic wave, thus v = c.

       c =  λf , thus f = 3 x 10⁸ / 0.12 = 2.5 x 10⁹ Hz

       C_water = 4.186 J/(g℃)

Thus the energy is E = C_water x m_water x (T_f - T_i) = 15488.2 J

P = E / t = 516.3 W

Energy of photon is E_p = h f = 6.626 x 10^-34 x 2.5 x 10⁹ = 1.65 x 10^-24 J

Thus, total oscillating photon numbers are P / E_p = 3.13 x 10²⁶

Assume power exerts evenly on the inner surface, then

Area = 2 x (0.23 x 0.35 + 0.23 x 0.35 + 0.35 x 0.35) = 0.567 m²

Intensity is I = P / A = 516.3 / 0.567 = 910.6 W/m²

I = P_pressure x c, therefore P_pressure = 910.6 / 3 x 10⁸ = 3.04 x 10⁶ Pa

#8 Concave and convex mirrors

Objective: 

Give a qualitative observation of the image formed by both convex and concave mirrors.

Procedures:

1.      Describe the image formed by placing an object in front of a convex mirror.

2.      Move the object and describe the change of image.

3.      Repeat the process for concave mirror.

Results:

       Convex mirror:

       The image appears to be smaller and away from the center compared with object, it is not inverted. When moving towards the mirror the image grows larger and more like the original size.

 

       Diagram as shown:

d_o = 15 cm, h_o = 10 cm, d­_i = -7 cm, h­_i = 5 cm.

       Agrees with observation.

       Concave mirror:

       The image appears to be much larger than the object and more close to the center, but still not inverted. When moving towards the mirror the image becomes smaller while moving away from the mirror the image gain larger and larger and disappear then become inverted.

       Diagram as shown:

      d_o = 20 cm, h_o = 8 cm, d_i = 12 cm, h_i = -5 cm

       The diagram shows the situation when the object is further away from the mirror, which caters to the latter description.

Discussion:

       The theoretical diagram agrees with the observation. Experiment conduct successfully.

#6 Length of a pipe

Objective: 

The purpose of this experiment is to measure the length of a pipe using only the property of sound and standing wave.

Procedures:

1.      Record the frequency of a swing pipe when it reaches a harmonic, that is, a standing wave is created in the pipe.

2.      Record frequency when the swing pipe reaches next harmonic frequency.

3.      Using two frequencies to calculate the length of the swing pipe.

Results:

       Assume the length remains constant and the speed of sound remains to be 340 m/s, the first ω₁ = 3859 rad/s, ω₂ = 5068 rad/s. Thus, as f = ω / 2 π , f₁ = 614 , f₂ = 810. v = fλand v = 340 m/s, λ₁ =  0.56 m,λ₂ = 0.42.

       L = n (0.56) / 2 = (n + 1)(0.42)/2, so n is 3.

       Thus, the length L is 0.84 m

Discussion:

       The experiment conducts well. The tricky part is to understand the first harmonic sound is not actual first standing wave created in the pipe. The calculation is easy and error may occur in rounding the numbers.

#2 Fluid dynamics:

Objective: 

Using application problem to verify Bernoulli Equation.

Procedures:

1.      Set the equipment as instructed and record all values needed.

2.      Measure time for releasing certain amount of water.

3.      Calculate the theoretical value and compare with the actual measured data.

Bernoulli Equation:

Results:

Value obtained:

       Volume emptied: 450 mL  Height of the water surface: 7.6 cm

       Height of the hole: 2.5 cm       Radius of the hole: 2.6 mm

       We stick paper to the hole and draw the shape of the hole using pencil on the other side of the bucket. The radius of the hole is roughly determined to be the value above.

       T_theoretical  = V / (A·(2·g·h)^0.5)

= 450 ·10^-6 / (π·(2.6·10^-3)²·(2·9.8·(0.076-0.025))^0.5)

= 21.40 s

       Time measured:

      

# of run	1	2	3	4	5	6
Time / s	21.88	21.86	22.61	21.67	22.66	22.55
Average time / s:	22.205
Percent difference / %:	3.762
Theoretical hole radius / mm :	2.591

Discussion:

       The result is very close to theoretical value, proving the ideal equation works. However, by assuming the velocity of fluid is constant and water surface remains constant, the error can be created in the system. The actual data is with in 5 % of the theoretical one, the experiment conducts successfully.

#3 Mechanical wave:

Objective: 

The purpose of this experiment is to study the mechanical wave’s property, by measuring its wavelength and frequency; we can derive the speed of a wave.

Procedures:

1.      Create a steady wave. Measure the distance between two people who hold the spring.

2.      Determine the wavelength of the spring and record the period of the wave.

3.      Calculate the frequency. Set another steady wave with different distance.

4.      Repeat and record the data.

Results:

       We run three trails for each wavelength and count the time for ten periods.

Wavelength / cm	88	100	60
Average period / s	0.43	0.49	0.50
Average frequency / Hz	2.33	2.04	2.00
Product of wavelength and frequency	204.65	204.08	120.00

Discussion:

       Due to the fact that human generates this wave; the speed of the wave cannot be accurately controlled. The first two sets of data are good while the third has big difference with respect to the other two. Other factors may affect the results can be the insufficient precision in measuring both time and wavelength.

#5 Introduction to sound:

Objective: 

By using Labpro to capture the wave pattern of certain sound wave, we can further understand the property of sound.

Procedures:

1.      Say AAAAA steadily to the microphone while recording, and make sure the graph is in clear pattern.

2.      Let a different team member to say AAAAA again and record the pattern, compare with previous one.

3.      Collect data from a tuning fork by stinking it on a soft object. Compare with previous two.

4.      Measure the hair by using micrometer and compare with the calculated hair thickness.

Results:

       Values collected and calculated as shown:

	Collecting time / s	Patterns On the screen	Period / s	Frequ -ency / Hz	Wave -length / m	Amplitude		Theoretical Frequency of fork
First Person	0.03	3	0.01000	100	3.40	0.40		384 Hz
Second Person	0.03	8.5	0.00353	283	1.20	0.25		Speed of sound
Tuning Fork	0.03	11.5	0.00261	383	0.89	0.20		340 m/s

       The experimental frequency of the tuning fork is close to the theoretical one, thus the data is good. For different people saying AAAAA, the pattern dose not change while both the frequency and amplitude change. Also, by striking tuning fork lighter, the pattern and frequency of the sound wave produced does not change while the amplitude changed.
       Pattern for human voice of AAAA:

       
       Pattern for tuning fork:

       

Discussion:

       The experiment came with good data. By investigating the pattern, we found that certain sound of language may be carried in different loudness or frequency, but they will be in same pattern.