Experiment 9 --- Concave and Convex Mirror

 Introduction:
               The purpose of this experiment is to observe how the image will change with the changing distance between the object and the mirror (concave or convex), as to see how the image will change with different surface of the mirror being applied.
               In order to do this experiment, there are three different kinds of mirror would be used for completing this experiment, the plane mirror, the concave mirror, and the convex mirror. For this experiment, the plane mirror would be used to be the control mirror, as well as the comparison for the other two mirrors. It is because the image refracted by the plate mirror would be the same size as the object. Besides, the image would be virtual and erect, with the same distance as the distance between the mirror and the object. To be short, the object distance would equal to the image distance. Therefore, with all properties provided, the image produced by the plane mirror could be the control image.
               After knowing the size of image produced by the plat mirror, now the object would be put in front of the convex mirror.As the object was put in front of the convex mirror, the image appeared was smaller than the object. However, no matter what the object distance was, the image was always erect and smaller. By comparing the image appeared in the convex mirror and the image appeared in the plane mirror, the image appeared in the convex mirror seemed to be further than the plane mirror. When the object was moved away from the mirror, the image distance also increased proportionally with the object distance, but the image was always smaller and erect.
               As the object was put in front of the concave mirror, the image appeared was larger than the object when the object distance was very small. However, as the object was gradually moved away, the image suddenly turned inversely and smaller. After passing this point, the image started to become smaller proportionally with the increasing object distance. Now it showed us that the point, where the image was turned upside down, was known as the focal point. Therefore, when the image appeared in the concave mirror (inside the focal point) was compared to the image appeared on the plane mirror, the mirror appeared on the formal one was closer in the mirror than the later one. The image appeared on the concave mirror when the object was inside the focal point was always larger and erect. However, the image appeared on the concave mirror became further than the image appeared on the plane mirror when the object was moved outside and away from the focal point. Besides, the image appeared on the concave mirror when the object was outside of the focal point was always smaller and inverted.

Plane mirror, the image has the same size as the object 

concave mirror, the image was inverted when the object was away from the mirror

concave mirror

convex mirror, the appeared image was erect

Diagrams, data, and the calculations

Convex Mirror:
Ray diagram---

h0	hi	d0	di
2.2 cm ± 0.05cm	0.7 cm ± 0.05cm	5.6 cm ± 0.05cm	-2.0 cm ± 0.05cm

Calculate magnification:

M = y’/y = -s’/s

M=0.7/0.22=0.318

  =-(-2.0/5.6)=0.357

Average M= (0.318+0.357)/2=0.3375

Concave Mirror:

Ray diagram---

h0	hi	d0	di
2.3 cm ± 0.15cm	-0.7 cm ± 0.05cm	11.9 cm ± 0.45cm	3.4 cm ± 0.2cm

Calculate magnification:

M = y’/y = -s’/s

M=-0.7/0.23=-0.304

  =-(3.4/11.9)=-0.286

Average M= (-0.304-0.286)/2=-0.295

Uncertainty:

Conclusion:

               For the convex mirror,since the M obtained from the –s’/s is positive number, the appeared image would be erect, and since the image was formed behind the mirror, which was the opposite direction of the outgoing light, the image was virtual.

               For the concave mirror,since the M obtained from the –s’/s is negative number, the appeared image would be inverted, and since the image was formed in front of the mirror, which was the same direction of the outgoing light, the image was real. However, this diagram only showed the image appeared when the object was outside of the focal point, if the object was put on the center point, the image would not be formed, and this was the instantaneous moment when the image changed from erect to inverted. If the object was put inside the focal point, the m would be positive number, but the image is going to appear behind the mirror, which is the opposite side of the outgoing ray. Therefore, the image would be virtual and the image would be erect.

               The magnification constant calculate from y’/y  should  equal to -s’/s, however, from the above calculations, the magnification constants calculated from each one did not equal to each other. With the uncertainty, the value could not be matched to each other, so the average of the magnification constant would be taken in this case.The inaccuracy for both the convex and the concave diagram could be due to the inaccuracy of the its diagram. 

               As a result, it showed us that when the object was put in the different distance from the mirror, the image appeared would also be different. The changing factor would produce the different distance of the image, erect or the inverted image was formed, the virtual or the real image was formed, and also the position the image was formed.

Lab Quiz - Electromagnetic Standing Wave

Introduction:

                For this experiment, the frequency of the microwave, the possible dimensions for the microwave, the total energy content of the cavity, the numbers of the photons per second oscillated in the microwave, and the pressure exerted by the photons on the side of the microwave have to be analyzed or calculated. In order to get the needed data to answer all the questions, first, a plate of marshmallows had to be micro waved so that the standing wave could be produced. When the marshmallows were heated, some spots rose up and formed some sequent hills. After the marshmallows stopped being heated, some dents then would be formed, and they were known to be the antinodes of the standing waves. The distance between two antinodes had to be measured so that the wavelength of the After heating the marshmallows, a cup of water, which was 100 grams, was also micro waved for 30 seconds. Then the temperature of the heated water had to be in order to obtain the energy. Finally, with these two objected being micro waved, the dimensions of the microwave were measured with ruler. With all the measured and recorded data, the questions mentioned above had to be calculated or deduced.

 The Marshmallows were ready to be micro waved

The pattern for the heated marshmallows (the dents were where the anti-nodes w)

Then the 100 grams water were also micro waved

Data table:

Approximate pattern of the marshmallows
Distance from two antinodes	12±0.5 (cm)
Increased temperature	20±0.5 Celsius  ~  57±0.5 Celsius
Dimensions of the microwave	Width	35±0.5(cm)
	Length	35±0.5 (cm)
	Height	23±0.5(cm)

Calculation:

frequency = velocity/wavelength = 3x10^8/0.24 = 1.25x10^9 (Hz)

Dimensions:

since the standing wave is three dimensional, distance from one pop to another pop would just be the half of the wavelength.

          Length:1.5(#of waves)*24=36(cm)

          Width:1(#of waves)*24=24(cm)

           Deduced height: 20 (cm)  

Total energy content of the cavity:

          Equations: Q=mc(change of Temp)

                              =(0.1kg)*(4.184J/kg)*(57 Celsius - 20 Celsius)=15480.8(J)

Energy for one photon

          Equation: E=(hc)/λ

                           =6.626*10^(-34)*(3*10^8)/0.24=8.2825*10^(-25) (J/per photon)

Numbers of photon oscillate in the microwave per second:

          Equation:Q/E/time=15480.8/8.2825*10^(-25)/30=6.23*10^26 (photons/s)

Power:

          Equations:Power=Energy/time

                                    =15480.8/30=516.03(W)

Pressure on each side of the microwave:

          Equation: pressure=power/(area*c)

               top/bottom sides:516.03/(0.35*0.35*3*10^8)=2.14*10^(-5) (Pa)

                                 sides:516.03/(0.35*0.23*3*10^8)=4.21*10^(-5) (Pa)
          

Conclusion:

As a result, since the marshmallows were not put on the entire inside surface of the microwave, the deduced length and the width were not as the ratios of the measured data. With the concept of conserved energy, the equation, Specific Heat Capacity could be applied to obtain the total energy content of the cavity. With this obtained energy, the numbers of photon oscillated per second could also be calculated. The microwave created waves when it is heating, since the microwave is a closed box, the wave would then be bounced back to create a standing wave. Standing wave is also a type of periodical wave in which then the frequency of the wave is constant so that the wavelength could also be calculated. 

Introduction:

For the first part in this experiment, the purpose was to find the length of the pipe by applying the concept of speed sound. To be more specific, the length of the pipe had to be found without using the ruler to measure it. The tools which were used in this experiment were the pipe for creating the sound, the microphone for detecting the sound, and the Logger Pro to record the data. In order to find the lengh of the pipe, the pipe had to be spun in a slow speed (but the sound had to loud enough to be heard). Then the sound would be detected by the microphone and be sent to Logger Pro to get recorded. Since the length of the pipe is always constant and in order to calculate it, the pipe had to be spun with a higher speed, thus created a different frequency. This sound would then become the next scale, which also reffered to the next level of the tone, or it could also be called as the first overtone. The reason that the sound was to be believed as the next scale was because if one sclae was skipped(which is the third level of the tone), then the sound would not be heard as a clear sound but a mixed different level of the sounds.

Data and Analysis:

Recorded Data:

Pipe’s spinning speed	ω (rad/s)
Slow	3859±0.9736
Fast	5068±1.016

 

After obtaining the angular frequency (since the pipe was spun so it had a centripital acceleration), the equation v=λf, then the equation was rearranged to be λ=f/v, where f=2πω, and the v is the speed of sound, as well as 343(m/s). Since two different frequencies were obtained, so then two different wavelengths would also be calculated, one of the wavelengh would represent the first note (n/2), and the second would represent the next level of the tone(n+1)/2, which was the second note. (n/2)*λ=(n+1)/2*λ since the lenght of the pipe is constant, and n could be calculated with this equation. After obtaining n, substitue the value into either of the equations above, then the lengh of pipe could eventually obtained.

Calculated Data:

Pipe’s spinning speed	Frequency (Hz)	λ (m)
Slow	614.18	0.558
Fast	806.60	0.425

Length of the pipe 

Pipe’s spinning speed	Lengthmeasured(m)	Lengthexperimental(m)	Percent error (%)
Slow	0.800	0.837	4.63
Fast		0.850	6.25

 Uncertainty:
Length=First note:(n/2)*λ=(n/2)*v/k=(n/2)*v/(ω/2π)=(n/2)*2πv/ω
            or=Second note:((n+1)/2)*λ=((n+1)/2)*v/k=((n+1)/2)*v/(ω/2π)=((n+1)/2)*2πv/ω
U_L = √[(∂L/∂*uω)^2]= First note: √[n*(2π/ω^2)*uω]^2
                                    =Second note: √[(n+1)*(2π/ω^2)*uω]^2

Pipe’s spinning speed	Lengthuncertainty(m)
Slow	5.70x10^-6
Fast	3.91x10^-6

   Conclusion:
As a result, even with the uncertainty, the percent error could not be reduced. However, on the other side, the length of pipe obtained from both trials which were conducted with different speed had percent error less than 10%. This showed us that the length of the pipe could be calculated by measuring its angular frequency with the application of the sound's speed, hence obtaining the expected results without actually measuring it. If the factor which produced the error in this experiment, the error could be contributed by the temperature, or the flexibility of the pipe. With higher temperature, the speed of the sound might have a higer value. As for the flexibility of the pipe, the pipe could be stretched more when being spun due to the centriputal force. Therefore, if all the factors of errorness could be taken into consideration and got rid of, the results for the lengh of the pipe would definetely be closer to the measured value.
 

Experiment 5 : Introduction to Sound

In this lab, different frequencies were compared by observing the graph for each frequency. For all the different graph of frequencies, the frequencies were created by different voices made by human and also the tuning fork. First, in order to analyze the frequency, one person needed to say "AAA" into the voice detector, after that the other person with different voice would also say "AAA" to the voice detector. From these two obtained graph, different pitch of the voice could be analyzed by the graphs produced by different frequency of the voice. After obtaining voice produced by the human, the tuning fork was used to create the sound so that different pattern of the frequency could be obtained. With all the obtained graphs, numbers of waves, the period of those waves, the frequency of those waves, and the amplitude of thoes waves should all be calculated or analyzed. The numbers of the wave would be determined by counting the numbers of the repeated pattern on the graph. Then the time taken by one wave could also be obtained from the graph. After obtaining the time, the equeation 1/T would be applied to calculate the frequency of the wave. The amplitude would also be calculated with all the obtained data.

Figure 1. Sound produced by one person talked to the voice detector (lower frequency)

 Figure 2. Sound produced by the other person talked to the voice detector (higher frequency)

Figure 3. Sound produced by tuning fork striking on a soft object

From the above graphs, we notice that the sound graph is a periodic wave. it shows that the graphs go to a higher position than go down gradually with several curves, then the pattern would repeat with the sound goes on. From the graphs above, figure one shows that the sound produced about three waves in 0.027(s), and the figure 2 shows that the sound made by the second person produced about six waves in 0.03(s). From here, we notice that the second person had higher frequency, hence higher velocity. 
Since three waves were produced in 0.027 (s) in figure 1, frequency could be calculated by applying 1/T, so the frequency obtained for the figure one would be 111.11 (Hz).
To answer question f, which the speed of sound was assumed to be 340m/s, what would the wavelengh be? Here the equarion v=fλ would be applied to calculate the λ, the equation was rearranged and appeared to be λ=v/f, in whice velocity of sound, 340 m/s and frequency obtained earlier, 111(Hz) were substituted in the equation. The final obtained answer for wavelengh then would be 3.06 (m).


For figure 2, five waves were produced in about 0.025 seconds, which means the time taken for a wave to go was 0.005 seconds. Therefore, the frequecny could also be obtained by applying the equarion used for figure one, the frequency in this case would be 200(Hz), so the wavelengh in this case would be 1.7 (m).



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	Time (s)	Frequency(Hz)	Wavelength(m)
Figure 1	0.009±0.0005	111±0.5	3.06±0.005
Figure 2	0.005±0.0005	200±0.5	1.70±0.005

By obseving figure 3, we notice that the curves, or the waves went regularly and constantly. The result turned out as expected that the tuning fork prouced the sound with constant frequency.
If we use the same tuning fork to collect data for a sound that is not as loud, then the amplitude of the wave should be smaller. The answer was given because higher volume has larger amplitude and lower volume has smaller amplitude. 

Experiment 4- Standing Waves

The prupose of this experiment was to compare the final velcity of the wave when the wavelength and the mass of the string were made to be constant but the frequency and the tension of the spring were changed.


This experiment constisted two cases. For both cases, the mass and the length of the string were set to be constant. However, for the first case the tension was given the mass of0.2(kg), then the tension was changed to have the mass of 1/4 of the first case in the second case. Therefore, the tension of the string would be different in these two cases. In order to get the tension of the string, the equation, T=mg (T=tension(N),m=mass(kg),g=gravitational acceleration(ms²)) was applied. After that, the frequency had to be adjusted in order to obtain the desired loops of the wave (such as one loop, two loops, three loops,...etc). The same process would also be applied to the second case when the tension of the spring was changed.
After all the data were obtained, two equations for calculation the wave speed were both be applied for the results comparison. Theses two equations are :

v=fxλ       v= √(T/μ)

However, in order to find the wavelengh of the wave, the equation λ_n= 2L/n will be used to calculate the wavelengh. Here, L equals to the measured lengh of the string and n equals the numbers of the produced loops.
After all the results are obtained, the ratio of the wave speeds from two different equations need to be compared, and the ratio of the frequency for different hormonic will also be compared between case 1 and case 2.

The set up of this experiment

The function generator for adjusting the frequency

the wave driver which allows the spring to create loops

Weight was hung on the one end of the spring 

loops were created

Data:
chart1:

		Mass of the string			1.41 (g)	Length of the string			110 (cm)
Case one
# of loops	μ (m/lengh)		Tension (N)	Frequency(KHz)		λ [2L/n] (m)	v=f*λ(m/s)	v=√ (T/μ)(m/s)
1	0.00128 ±0.000005		1.96 ±0.005	0.015 ±0.0005		2.20 ±0.005	33.0 ±0.05	39.1 ±0.05
2	0.00128 ±0.000005		1.96 ±0.005	0.037 ±0.0005		1.10 ±0.005	40.7 ±0.05	39.1 ±0.05
3	0.00128 ±0.000005		1.96 ±0.005	0.056 ±0.0005		0.733 ±0.0005	41.1 ±0.05	39.1 ±0.05
4	0.00128 ±0.000005		1.96 ±0.005	0.076 ±0.0005		0.550 ±0.0005	41.8 ±0.05	39.1 ±0.05
5	0.00128 ±0.000005		1.96 ±0.005	0.095 ±0.0005		0.440 ±0.0005	41.8 ±0.05	39.1 ±0.05
6	0.00128 ±0.000005		1.96 ±0.005	0.114 ±0.0005		0.367 ±0.0005	41.8 ±0.05	39.1 ±0.05
7	0.00128 ±0.000005		1.96 ±0.005	0.133 ±0.0005		0.314 ±0.0005	41.8 ±0.05	39.1 ±0.05
Average wave speed (m/s)							40.28 ±0.05	39.1 ±0.05
Case two
1	0.00128 ±0.000005		0.49 ±0.005	0.009 ±0.0005		2.20 ±0.005	19.8 ±0.05	19.6 ±0.05
2	0.00128 ±0.000005		0.49 ±0.005	0.019 ±0.0005		1.10 ±0.005	20.9 ±0.05	19.6 ±0.05
3	0.00128 ±0.000005		0.49 ±0.005	0.028 ±0.0005		0.733 ±0.0005	20.5 ±0.05	19.6 ±0.05
4	0.00128 ±0.000005		0.49 ±0.005	0.038 ±0.0005		0.550 ±0.0005	20.9 ±0.05	19.6 ±0.05
5	0.00128 ±0.000005		0.49 ±0.005	0.048 ±0.0005		0.440 ±0.0005	21.1 ±0.05	19.6 ±0.05
6	0.00128 ±0.000005		0.49 ±0.005	0.058 ±0.0005		0.367 ±0.0005	21.3 ±0.05	19.6 ±0.05
7	0.00128 ±0.000005		0.49 ±0.005	0.070 ±0.0005		0.314 ±0.0005	22.0 ±0.05	19.6 ±0.05
Average wave speed (m/s)							21.0 ±0.05	19.6 ±0.05

The ratio between f200g and f50g
#of loops	1	2	3	4	5	6	7	average
f200g /f50g	1.7	1.9	2.0	2.0	2.0	2.0	1.9	1.9

Graphs:

 

figure 1: case one 

figure 2:　case two

Uncertainty:
Derivation 

v=f*λ=f*2L/n , where f and  L were measured:

v=√(T/μ)=√T/(mass/lengh), where mass and length were measured:

Results of uncertainty:

chart 2:

# of loops	Case1  v=f*λ(m/s)	Case 1 v=√ (T/μ)(m/s)	Case2  v=f*λ(m/s)	Case2 v=√ (T/μ)(m/s)
1	33.0 ±0.015977	19.8 ±0.194	39.1 ±0.009054	19.6 ±0.0972
2	40.7 ±0.037793	20.9 ±0.194	39.1 0.021425	19.6 ±0.0972
3	41.1 ±0.056388	20.5 ±0.194	39.1 0.02935	19.6 ±0.0972
4	41.8 ±0.076286	20.9 ±0.194	39.1 0.039006	19.6 ±0.0972
5	41.8 ±0.095159	21.1 ±0.194	39.1 0.0488	19.6 ±0.0972
6	41.8 ±0.122197	21.3 ±0.194	39.1 0.058664	19.6 ±0.0972
7	41.8 ±0.137033	22.0 ±0.194	39.1 0.070859	19.6 ±0.0972

Percent Error (%)

Case one:
40.28±0.05=40.23~40.33

21.0±0.05=20.95~21.05

Case two:

39.1±0.05=39.05~39.15

19.6±0.05=19.55~19.65

Case one:
v=f*λ
(42.979-40.28)/42.979x100%=6.28%

 v=√ (T/μ)(m/s)
(27.079-21.0)/27.079x100%=22.45%

Case two:
v=f*λ
(42.979-39.1)/42.979x100%=9.04%

 v=√ (T/μ)(m/s)
(27.079-19.6)/27.079x100%=27.62%

Results:

velocity obtained by the slope:

200g:   42.979(m/s)

50g:   27.079(m/s)

Conclusion:

First the first case of this experiment will be analyzed. Reasonably, the slope of the frequency v.s. 1/λ should be equal to the wave speed. Two equations were applied to calculate the wave speed. The results obtained by applying the equation v=fxλ has smaller error compared to the results obtained from applying equation v= √(T/μ). Since both of the equations have the variable, the length of the wave, so the factor which caused this error difference would be the inaccurate measure of the spring's mass. When the mass was measured, the obtained mass was the whole spring. However, only partial spring was used to created loops. Therefore, the mass of the used spring contributed to the huge error difference. On the other side, the results obtained by equation v=fxλ in both cases have error within 10%. Therefore, the experiment showed that the wave speed is determined by the mass, length, and also most importantly, the frequency.