Lab 7: Lab Quiz (3/22/2012) s

Question: 

A microwave oven is in the back of the class room. We have placed some marshmallows in the microwave to make some measurements of the standing wave. Determine the frequency of this microwave. From this deduce a range of possible dimensions for microwaves including the smallest possible microwave. In this lab we also microwaved a cup of water. What is the total energy content of the captivity? How many photons per second are oscillating in the microwave? What pressure do these photons exert on the side of the microwave?

Data:

• Distance between peak energy: 12 ± 1cm
• Estimate mass of water: 100 ± 1g
• Change in water temperature: 37 ± 1 ^oC 
• Dimension of Microwave in W*L*H:  (36 ± 1cm) *(36 ± 1cm)*(23 ± 1cm)

Method:

 

Figure 1: Microwave and marshmallow used in this lab.

Analysis:

• The measured distance between peak energy also represents the distance between two adjacent anti-nodes. Thus  λ can be calculate by multiplying 2 to the distance.
• λ = d * 2
•  Since the microwave must contain the entire wavelength, the dimension of the microwave must be a positive integer multiple of the wavelength. 
•  width =  m * λ,  length =  n * λ ,    whereas n and m is positive integer
• the smallest dimension would be when n and m equal one
• The energy in captivity is in direct relationship with the change of temperature
• E = mcΔT
• The numbers of proton can be find by dividing the total energy in captivity to the energy a proton holds.
• Energy of a proton in microwave: E = hf
• c = f λ,     f =  λ / c 
• Pressure on the side is directly proportional to Poynting vector and speed of light
• P = S / c (assuming the microwave is black body)
• S ≈ I,      in which I = intensity = power/area
• power = E / t  

 

Figure 2: Calculation for wavelength, frequency, energy and number of photons.

Table 1: Wavelength and frequency

λ (m)	f (Hz)
0.24 ± 0.01	1.25*109

Table 2: Possible Dimension

Width (m)[ j Є Z+]	Length (m) [ k Є Z+]
0.24 * j	0.24 * k

Table 3: Energy and power.

Total Energy (J)	Power (W)	Energy per proton (J/proton)
15481 ± 200	516 ± 16	(8.29 ± 0.04)*10-25

Table 2: Photon and pressure.

Number of photons (photon)	Pressure (Pa)	Photon per Second (photon/s)
(1.87 ± 0.06)*1028	(1.33 ± 0.048)*10-5	(6.23 ± 0.02)*1026

Conclusion:

In this lab we have discussed the behavior of electromagnetic standing wave through analyzing the physical properties of the microwave. We found that the dimension of the microwave have to include the wavelength of the microwave spectrum. Thus the dimensions of a microwave will always be an integer multiple of the wavelength.

Lab 6: Lecture Lab (3/16/2012) s

Purpose:

In this lab we will measure the angular velocity of a spinning pipe in two distinct harmonic to deduce the length of the pipe.


Method:

We will determine the modes of a harmonic through solving the unknown n  by equating the length of the pipe represent by L = n(λ/2) and L = (n+1)*(λ/2). The wavelength will be measure through the angular velocity of the pipe using logger pro. Once n is determined, we can substitute the mode into the length equation to calculate L.


Data and Analysis:

 

Figure 1: Calculation of the length of the pipe. 

ƒ = ω/ 2π

λ = v / ƒ 

μ ƒ  = μω/ 2π

μλ = (v / ƒ_max) - (v / ƒ_min)

Table 1: Measurements from spinning pipe

Harmonic	ω(rad/s)	ƒ(Hz)	λ(m)
Low	3859 ± 0.51	614.49± 0.08	0.5598 ± 0.000145
High	5068 ± 1.10	807.01 ± 0.18	0.425 ± 0.0000845

Figure 2: Calculation of the length of the pipe. 

L = (n₁ / 2) *λ₁

L = ((n₁+1) / 2) *λ₂

n₁ = 3.1 ≈ 3 

(Harmonics are whole numbers)

Substitute n

and

μL =  (μλ / 2) * 3

L = 0.8397 ± 0.0002175 m

Conclusion:

In this lab we have determined the length of the pipe by measuring the angular velocity of two distinct harmonics. The length of the pipe is inversely related to the frequency of the harmonics. Thus by equating the two standing wave equation in terms of length we were able to determine the n of the harmonics and deduce the length of the pipe. We have calculated our uncertainty by setting an upper and lower bound on the calculations. The range between the bound will serve as the uncertainty. We believe that the error in the can be entirely attribute to the precision of the lab equipment.

Lab 5: Introduction to Sound (5) s

Purpose:

In this lab we will examine the periodic nature of sound waves by analyzing sound of human voice and tuning forks.



Method:

We will be using logger pro and an attach microphone to obtain a graph of sound pressure vs. time to analyze the nature of sound waves.

Figure 1: Microphone recording sound wave from tuning fork.

Data and Analysis:

Figure 2: Sound Pressure vs. Time (human) 

• Sound wave is periodic as evidence by the sinusoidal behavior of the graph.
• There are 7 waves in the data collected in Figure 2, we designate the gap between two crest to be one wave.
• The time frame in Figure 2 is comparable to a flash of eye.
• T = 0.0038 ± 0.0001 s, the period is define by the time it takes to complete one cycle.
• ƒ = 262 ± 7.59 Hz,  ƒ  = 1/ T  
• λ = 1.30 ± 0.039 m,  λ = v / f, the length is comparable to a meter stick.
• Amplitude A = 2.718 ± 0.2 W, we took the average of the two different peak to determine the amplitude of the wave. 

Figure 3: Sound Pressure vs. Time (human, t = 0.3)

• All characteristic of Figure 3 will be similar to Figure 2 except the time frame will be ten times as long.

Figure 4: Sound Pressure vs. Time (human) 

• There are 3 waves in Figure 4.
• ƒ = 129 ± 5.81 Hz
• T = 0.01132 ± 0.002 s 
• λ = 2.63 ± 0.057 m
• A = 0.14 ± 0.02 m
• The sound wave in Figure 4 has a lower frequency and smaller amplitude than that of Figure 2.

Figure 5: Sound Pressure vs. Time (tuning fork) 

• The graph made by tuning fork is much smoother due its ability to produce sound waves in a single set of frequency. 

Figure 6: Sound Pressure vs. Time (tuning fork)

• The loudness of a frequency is determined by the amplitude of the wave, thus Figure 6 has a lower amplitude than Figure 5.

Conclusion:

In this lab we have analyze the property of the sound wave graphically. We find that sound waves are generally periodic as shown in the repetitive pattern. And the intensity and loudness of sound is directly proportional to the amplitude of the wave. We also discover that single frequency sound wave act as a sinusoidal wave. This proves that human voice does not produce one frequency but multiple while talking.  

Lab 4: Standing Waves (4) x

Purpose:

In this lab we will examine the properties and characteristics of a standing waves driven by external force.


Method:

Trials	Node	d Between node	F	W
1	2	204+/- 0.01 cm	10.6+/-0.01 Hz	408+/- 0.01 cm
2	3	102+/- 0.01 cm	19.2 +/-0.01 Hz	204+/- 0.01 cm
3	4	73+/- 0.01 cm	29.66 +/-0.01 Hz	146+/- 0.01 cm
4	5	52+/- 0.01 cm	42.59 +/-0.01 Hz	106+/- 0.01 cm
5	6	43+/- 0.01 cm	51.5 +/-0.01 Hz	84+/- 0.01 cm
6	7	37+/- 0.01 cm	60.8+/-0.01 Hz	74+/- 0.01 cm
7	8	30+/- 0.01 cm	71.9 +/-0.01 Hz	60+/- 0.01 cm
8	9	25+/- 0.01 cm	83.04 +/-0.01 Hz	50+/- 0.01 cm

Wave Speed

Case 2

Trial	Node	d between node	F	W
1	2	93.25+/- 0.01 cm	11.9+/-0.01 Hz	186.5+/- 0.01 cm
2	3	62.2+/- 0.01 cm	16.25 +/-0.01 Hz	124.4+/- 0.01 cm
3	4	46.6+/- 0.01 cm	21.77+/-0.01 Hz	93.2+/- 0.01 cm
4	5	37.3+/- 0.01 cm	27.97 +/-0.01 Hz	74.6+/- 0.01 cm
5	6	31+/- 0.01 cm	33.77 +/-0.01 Hz	62+/- 0.01 cm
6	7	26.6+/- 0.01 cm	39.83+/-0.01 Hz	53.2+/- 0.01 cm
7	8	23.3+/- 0.01 cm	45.73 +/-0.01 Hz	46.6+/- 0.01 cm
8	9	20.7+/- 0.01 cm	51.23 +/-0.01 Hz	41.4+/- 0.01 cm

Conclusion:

Due to the linear relationship between frequency and wavelength, we have validate the frequency wavelength equation. The velocity of the oscillating wave corresponds to the slope of the trendlines. The two graph have similar velocity thus we conclude that the equation is valid.