#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.

#10 Measuing a human hair:

Objective: 

By applying the diffraction of light, measuring small thickness is feasible. The object is to measure the thickness of human hair and use micrometer to verify the measurement.

Procedures:

1.      Make a hole in a card and tape a single human hair across the hole and keep the hair taut.

2.      Stabilize the card so that the card is parallel to the whiteboard.

3.      Obtain the wavelength of the laser, and then point the laser perpendicular to the whiteboard while shinning through the hair. Record the distance from the card to the wall and the distance between nearest bright point to the central diffraction.

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

Results:

Value obtained:

Distance from the card to the board (L): 110 cm      Wavelength: 632.8 nm

Distance from the first bright point to the center (y): 0.16 cm

Thus, thickness is:

D = 2.18 x 10^-4 m

Measured value from micrometer:

Percent difference:
Because of time and equipment limitation and missing lab partner, we were unable to measure the hair thickness by micrometer, thus I decided to search on the Internet about the usual human hair thickness.

Discussion:

       The error may results from imprecise measurement from the meter stick. As searched on the Internet, the human hair usually lands in 5~8 x 10^-5 m, thus the experiment are not conducted so well.

#4 Standing wave:

Objective: 

The objective of this lab is to gain knowledge and understanding of standing waves driven by an external force. Resonant conditions for standing waves on a string will be investigated.

Procedures:

1.      Measure and record length and mass of string.

2.      Tie the string to two clamps. Tie about 200 g on the end of pulley.

3.      Attach the string to the wave driver, set up the function generator as instructed.

4.      Adjust the frequency to reach the fundamental mode, record frequency and number of nodes. The length participating the oscillation must be recorded as well. Repeat and record information under different node numbers.

Results:

       The recorded values as shown:

Frequency / Hz	12.45	24.35	40.35	49.35	65.35	87.35	102.35
Node to node / cm	168.5	88.5	57.5	42.5	29.5	26.5	22.5
Number of nodes	2	3	4	5	6	7	8
Wave length / cm	337	177	115	85	59	53	45
Wave speed /ms-1	41.9565	43.0995	46.4025	41.9475	38.5565	46.2955	46.0575

       Average wave speed is 43.4736 m/s, and its standard deviation is 2.9486.

       The string is 195.3 cm in length and 2.36 g in mass, thus the density is 1.224 x 10^-3 kg/m. By a tension of nearly 0.2x9.8 = 1.96 N, the theoretical speed is calculated as 40.0163 m/s

       Percent difference is (43.4736 - 40.0163) / 40.0163 x 100 = 8.639 %.

       The uncertainty in measuring the mass is 0.01 g and length is 0.2 cm, thus the uncertainty in density is 0.265 x 10^-3 kg/m. Therefore the error in speed should be with in (45.2083-43.4736) / 43.4736 x 100 = 3.990%, and 8.639 % exceed that value.

      

Discussion:

       Due to the technical difficulty and time limitation we were unable to perform the latter part of the lab. From the data of the first lab, the data is considered a little over expected error but as it is still within ten percent, the experiment is not a total failure.
       To my lovely lab partner: Where did you guys get the data of part II...

#9 Lenses:

Objective: 

       This experiment is designed to investigate some characteristics of converging lens.

Procedures:

1.      Determine the focal length of the lens. The focal length is 20 cm

2.       Place the lens in the holder, and place the filament on one side of the lens. The distance between the lens and filament is set in different values as 5, 4, 3, 2, 1.5 focal length.

3.       Place the whiteboard at the other side and find the clear image if applicable. Record the distance to the lens.

4.       Reverse the lens to see if the direction would affect the lens’s optical property.

Results:

Recorded values as shown:

Object Distance /cm	Image Distance /cm	Image Height /cm	Magnification	Type of image		Object height
100	24	2.5	0.284	Real and inverted		8.8 cm
80	25	3.5	0.398	Real and inverted		Focal length
60	28	4.5	0.511	Real and inverted		20 cm
40	37	8.5	0.966	Real and inverted
30	50	14.5	1.647	Real and inverted
10	\	\	\	Virtual and same

The image distance with respect to 10 cm object distance is immeasurable, by observation, the image is in the same side as the object and the height is about half of the object and 5 cm to the lens.

Discussion:

       By reversing the lens, the image did not change, so reserving the lens does not affect the optical property of the lens. By covering half of the lens, the image became much dimmer, but still kept the same image height and type; thus, covering the lens does not affect the type of image.

       The virtual image formed by 10 cm object lens is shown as:

       Which is verified by observation.

       The measured distance satisfy the lens as 1/d_o + 1/d_i =1/f

      

Object distance/cm	100	80	60	40	30		1/f
1/d0 + 1/di	0.052	0.053	0.052	0.052	0.053		0.050
Percent difference	3.333	5.000	4.762	4.054	6.667

The error may caused by human error in finding the clearest image and the focal length of lens which may not be 20 cm as marked.