Experiment 12

Exp 12
Diffraction

Purpose: To use diffraction of light to determine the length of CD grooves (width [a] of the openings) by measuring the length between the reflected rays using diffraction grating equations.

Materials: 
Diode laser
CD
Reflecting surface (white board)
Meter stick(s)
Stands (wooden blocks)
Diffraction film

Procedure:
Since λ of diode laser was not known, it has to be calculated first.
To calculate the wavelength of the laser, a diffraction film of known d (distance between slits) was used. In this case, it was 500 slits / mm. Reciprocating it,
d = 2*10^-3 mm.
The formula used was

d sin(θ) = m λ,

solving for λ,

λ = d sin (θ)/ m

The angle could be found experimentally. First, the laser was aimed perpendicular to the diffraction film; the ray would diffract and at least three rays would be shown on the reflecting surface. The angle was the arctangent of the distance between the diffraction film to the reflecting surface over the distance between the first light (1st maxima) to third light (3rd maxima).
Using formula above, θ was 40.32 degree. λ then, is  647.6 nm.

With λ known, the grooves can be calculated. The formula used is

a sin (θ) = m λ

a = m λ/ sin (θ

The CD was positioned so that when laser was aimed towards it, its ray would reflect to the surface orthogonal to it. The distance (d) from the CD to the surface was 68.5 + 0.1 cm The distance between the two lights observed was 28.8 + 0.1 cm. Using phytagorean principles, the angle was calculated to be 22.8 degrees.

Using the equation previously mentioned and letting m = 1 (first maxima), a was calculated to be 1671 nm.

Uncertainty and Error Analysis:

*The slits was assumed to be a constant number without error (it was not measured in the lab). 

ud = 0

*The angle, however, had uncertainty. The triangle was found using two length measurements. 

tan (θ) = (Length / Width);

θ = arctan (Length / Width);

uθ = ((L ln (W) 1/(1+L/W) uW)^2 + (1/W* (1/(1+L/W)) uL)^2)^(0.5)

uθ = 0.45 degree

λ(d, θ) = d sin (θ)/ m

dλ/dd = sin (θ) / m 

dλ/dθ = d cos(θ) / m

uλ = ((sin (θ) / m  * ud)^2 + (d cos(θ) / m * uθ)^2)^(0.5)

uλ = 3.43*10^-7 m = 343 nm

λ then, is 647.6 nm + 343 nm

The second calculation has angle with different uncertainty. Using the same formula:

uθ = ((L ln (W) 1/(1+L/W) uW)^2 + (1/W* (1/(1+L/W)) uL)^2)^(0.5)

uθ = 8.57 degrees

a(θ,λ)  =  λ / sin (θ)

da/ dλ = 1 / sin(θ)

da/ dθ = - λ cot(θ)/ sin(θ)

ua = ((1/ sin(θ) * uλ)^2 + (- λ cot(θ)/ sin(θ) * uθ)^2)^(0.5)

ua = 34070 nm

a = 1671 + 34070 nm

Normal CD grooves usually range between around 1500 nm. The calculated values, including the uncertainties, definitely falls within the range. 

Errors (comment on the large uncertainty value):

The large uncertainty value could be caused by several reasons, one of them being having to do the experiment twice. The wavelength of the laser was unknown initially and had to be determined by performing  an additional experiment. When performing this, some errors were recorded and accumulated throughout the lab. Those error later further were being accumulated and added up along with the second experiment's uncertainties, causing the values to substantially increased to what was calculated above. 

Further error analysis:

Although the calculated value of 1671 nm falls within about 10% of the expected value of 1500 nm, it was not exact due to several factors. One of them was the experimented CD had been used previously; it was a burned CD; a burned CD probably had a different slit density than a new CD. Secondly, the CD was from at least 2002, 10 years prior. CD-ROM technology had change the past decade. CDs today had an improved memory capacity than a decade ago; an increase in memory capacity implied that it had different slit density. 

Another room for error was when experiment was performed, it was not truly observed whether the ray was perfectly orthogonal to the surface or not. If it was not, then the calculated angle of 22.8 degrees would not be perfectly accurate, hence the calculated a value would be different. 

The inconsistency of the distance could not be found using the obtained data, because it would require two reflections on two different locations on the CD, which was not performed in this experiment, unfortunately.

Experiment 6

Exp 6
Speed of Sound

Purpose: To use fundamental frequency and spinning slinky to determine the speed of sound

Material:
Long, hollow, red plastic tube with one end closed
LoggerPro software
Measuring stick

Procedure:
Have the plastic tube spinning forming a circle. Use loggerpro software to measure the angular velocity, ω. Do the experiment several times to find several calculated values for velocity. 

v =λ f

(A.)

Frequency may be obtained by recalling that

ω = 2πf, giving 

f = ω/2π 

(B.)

While wavelength, λ, can be found by assuming that the wave pattern inside the tube would be harmonic wave:

(2n+1) λ/2 = L

 (C.)

whereas n is the number of harmonics and L length of pipe.

After λ and f were found, v, the speed of sound, can be calculated and compared.

Data and Uncertainty:

Length of pipe is 0.83 +/- 0.05 m

Two trials were conducted.

Trial 1:

ω = 3859 +/- 1 rad/s

f = 614 +/- 1 s^-1

Trial 2: 

ω = 5068 +/- 1 rad/s

f = 807 +/- 1 s^-1

The harmonics were found using the speed of sound, 340 m/s.

v = λf

v/f = λ

whereas from equation (C.), λ = 2L / (2n+1).

Both n's for trial 1 and 2 are 2+/- 0 and 3+/- 0, respectively (the uncertainty was 0 because if it were to be, for example, 2 +/- 0.05 carried from length, there is no such thing as 0.05 harmonics. Harmonics' value must be in integers).

n_1 = 2

n_2 = 3

From n, λ can be solved backwards,  using 

λ = 2L / (2n+1)

Trial 1:  λ = 0.56 +/- 0.05 m

Trial 2:  λ = 0.43 +/- 0.05 m

Putting the data together:

Trial 1: f = 614 +/- 1 s^-1, λ = 0.56 +/- 0.05 m

Velocity then, using equation (A.), is 343.84 +/- 1.05 m/s

Trial 2: f = 807 +/- 1 s^-1, λ = 0.43 +/- 0.05 m

v = 347.01 +/- 1.05 m/s

Whereas speed of sound at the sea level, according to https://www.google.com/, is 340.29 m/s

Although it does not fall within uncertainty values, the values were off by within less than 2.00% error, which can be considered insignificant. It was very close to the actual speed of sound.

Error Analysis:

Some of the possible rooms for error were the measurement of the length. The pipe had to be assumed initially to have constant diameter and did not deform when swung. Being a plastic pipe, some deformity must have occurred. This would alter the sound travel a little.

Second room for error was the rounding of harmonics, n. One value obtained was less than 3 (around 2.84, but it was rounded up to three). This altered the value by almost 10%.

END OF LAB 6.

Physics 4C complimentary guitar solo live performance

Experiment 3

Exp 3
Wavelength vs frequency



The purpose of this lab was to determine the relationship of waves' period, frequency, and wavelength.

Materials:
Simply,
     A spring
     A timer
     A measuring stick

Procedure:

Some thoughts...

A spring was oscillated consistently for 5 seconds. The length of the measured string was about 1 m long. The period would be found. Meanwhile, the spring's crests was observed to count how many waves throughout the entire oscillation. From those data, velocity could also be found using this relationship:

v = λf

Data, calculations, and errors:

Time, t, was 5.0 +/- 0.1 s throughout.

The quantity of how many times waves were seen to pass through was as seen on the side:

Trial 1: 13 waves +/- 0.25

Trial 2: 11 waves +/- 0.25

Trial 3: 11 waves +/- 0.25

The error was found using partial derivatives.

T (waves, t) = waves / t.

∂T/∂waves = 1/ut

∂T/∂t = - uwaves / ut^2

uT = squareroot of the sum of squares 

Period of each trial:

uT = 3.53. Since the data for all 3 trials are the same, the period's uncertainties, uT, are all the same, 3.53.

Period, T, is #waves / time (5s)

Trial 1 = 2.6 Hz +/- 3.53

Trial 2 = 2.2 Hz +/- 3.53

Trial 3 = 2.2 Hz +/- 3.53

Frequency is the inverse of Period

f = 1/T

f(T) = 1/T

∂f/ ∂T = - 1/T^2

the error for frequency is 

Trial 1 = 0.38 s +/- 0.08

Trial 2 = 0.45 s +/- 0.08

Trial 3 = 0.45 s +/- 0.08

Velocity is the product of wavelength. There were 2.5 waves observed throughout the three trials.

Since the length was 1 m, λ = 1/2.5 = 0.4 m +/- 0.35

(0.35 was obtained by summing all the errors together: 0.25 from waves uncertainty and 0.1 from length uncertainty)

Velocity for each trial is as follows:

v (λ, f) = λ* f

The sum of the errors, 0.35 + 0.08, is 0.43

Trial 1 = 0.15 m/s +/- 0.43

Trial 2 = 0.18 m/s +/- 0.43

Trial 3 = 0.18 m/s +/- 0.43

Error analysis:

The graph has a vertical line. This is because the experiment was done on the same time, 5 second. 

It should have been done through varying time, so the graph would look linear with real slope (slope shown here is infinity). 

All of the values look very similar / the same because again, due to the same time period. 

Another source of error would be when measuring the wavelength through observation. The entire system was in dynamic; the spring was constantly moving. It was hard to measure how many waves were observed at that moment. The 2.5 waves were simply an estimation. It would have been more accurate if a photograph of the spring was taken when oscillated and the number of crests could be counted to obtain a more accurate quantity of wavelength. 

Experiment 11

Exp 11
Human Hair Measurement

Purpose: To use the separation of light rays (interference) to rather accurately measure the thickness of human hair by measuring the distance between the reflected laser rays.



Materials/ tools:

Helium Neon Laser
Reflected Surface (whiteboard)
Taped, paper-punched-3x5 index card
Measuring Stick (LONG ruler)





                        




Procedures:

1. Obtain a strand of human hair. Tape it across the hole of the index card.





2. Set the card parallel to the reflective surface, 1 meter apart. Set up the HeNe laser and aim it ONTO the hair strand.



3. The reflective surface would show a diffraction pattern of light. Make sure that the laser ray is orthogonal to the surface. Measure the distance of the light diffraction.

The reflected light ray (center) has trailing, fading light diffraction immediately to its left and right

Data and Uncertainties:

Variables:

L = Distance between the surface to hair strand

y = Distance between light diffraction

λ = laser's wavelength

d = hair thickness

λ = d y / L

λ = 632.8 nm = 632.8 * 10^-9 m (Since it is the standard wavelength of HeNe laser, it is assumed accurate, and has no uncertainties)

Helium Neon laser is assumed to have a fixed wavelength... whoa!

Experiment 1:

L = 1.00 +/- 0.05 m

y = 0.25 +/- 0.1 cm  = 0.0025 +/- 0.001 m (large uncertainty due to difficulty measuring closely-spaced diffraction using large measuring stick)

d = λ L / y

d = 2.53 * 10^-4 m = 2.53 * 10^-1 mm = 253 μm

Experiment 2:

L = 1.02 +/- 0.05 m

y = 0.0052 +/- 0.001 m

d = 1.24 * 10^-4 m = 124 μm

Uncertainty by partial derivatives

d(y, L) = λ L / y

∂d/ ∂L = 1/ uy

∂d/ ∂y = uL * ln (y)

ud = ((∂d / ∂L * uL)^2 + (∂d / ∂y * uy)^2 ) ^(0.5)

ud = ((1/uy * uL)^2 + (uL * ln (uy))^2) ^(0.5)

Experiment 1:

ud = ((1/ 0.001 * 0.05)^2 + (0.05 * ln(0.0025) * 0.001)^2)^0.5

ud = (2500 + 0)^0.5 

ud = 50

Experiment 1 = 254 μm +/- 50 m = 254 +/- 50000000 μm

This uncertainty value makes no sense. 2nd method of uncertainty will be used by adding up all the uncertainties:

Uncertainty by adding up all the uncertainties:

Experiment 1 and 2: uL = 0.05 m, uy = 0.001m, uλ = 0

ud = Σ_uncertainties = 0.051

Experiment 1:

d = 2.53 * 10^-4 +/- 0.051 m = 253 +/- 51000 μm

Experiment 2:

d = 124 +/- 51000 μm

The average human hair thickness ranges from 40 to 250 μm, according to http://wiki.answers.com/Q/What_is_the_average_thickness_of_a_human_hair

Our experiment yields value of about 120 and 250 μm, which falls within the expected values (although the uncertainties are rather outrageously substantial).

Error analysis:

One of the possible cause of errors is the measuring of the diffraction. Since the diffraction was spread very closely to each other and the most intense ray (middle ray) outshone the other rays, it was highly possible that there was an additional ray hidden under the brightest ray. Miscounting one ray could throw off the data by magnitudes of ten. 

Another possible error is the position of the laser and the surface. During the experiment, it was merely being approximated that they were perpendicular to each other. If they were not truly perpendicular, then the distance, L, was not the true value. 

Another yet possible error is the condition of hair itself. It is possible that the person whom hair was being experimented on used a thickening hair product, causing the hair to be thicker than usual. One of the person was a female, so the chances of hair thickener being used would be higher. 

END

Exp 10

Exp 10
Lenses

Purpose:
To observe the behavior of light through converging (convex) lenses and how magnification and height is a function of the distance of the object from the light source; to also observe when an image is inverted and upright in front of converging lens.

Materials:
1. BIG rulers (meter stick)
2. Light source (light box with distinct shape on the opening)
3. Converging (convex) lens and lens holder
4. Flat, not diffused, surface

Procedures:

Since the lenses' focal length was unknown, it had to be predetermined before the lab could continue. This can be done by facing the lens against a light source infinitely far away (to get straight light rays) and measure the distance at which the converging light rays are strongest. 






 The height and horizontal distance from the meter stick was measured, forming a triangle. The hypotenuse would be the distance of focal point (f).

The triangle was a right, 6.0 * 7.0 triangle, with hypotenuse of 9.2 cm, the focal length.





Actual lab and questions:

The lens was put on the lens holder on a meter stick, against a light source. On the other end, a smooth surface was set up to show the shape of the image and measured.

Initially, the lens was put at distance four times focal length (f), which was 36.8 cm.
d_0 = 36.8 cm
The surface was adjusted to get the sharpest image. The distance between surface and light source was 44.2 cm. The distance between surface and the lens,
d_i = 7.5 cm



The height of filament (light box/ source) is
h_0 = 9.0 cm
whereas the height of the image
h_i = 3.1 cm
Magnification, M is h_i / h_0 = 0.33, a third the of original object.
The image is real.


When the lens is reversed, the measurements were still the same.
d_0 = 36.8 cm
d_i = 7.5 cm
h_0 = 9.0 cm
h_i = 3.1 cm
M = 0.33

The lens was moved back, towards the light source, so its distance was now 2f.
d_0 = 18.4 cm
d_i = 30.2 - 18.4 = 11.8 cm
h_0 = 9.0 cm
h_i = 6.0 cm
M = 0.66
The image was still the same as the above data when the lens was reversed.

The lens was moved further back, with distance 1.5 f
d_0 = 13.8 cm
d_i = 31.0 - 13.8 = 17.2 cm
h_0 = 9.0 cm
h_i = 12.0 cm
M = 1.33
Image was the  same reversed.

When half of the lens was covered, the image was still reflected with the same shape, but dimmer.


Lab part 2.
The lens was now set up at different distances.

h_0 = 9.0 cm

5f:
d_0 = 46.0 cm, d_i = 18.6 cm, h_i = 2.2 cm, M = 0.24
Image was inverted.
4f:
d_0 = 36.8 cm, d_i = 12.3 cm, h_i = 3.1 cm, M = 0.33
Image was inverted.
3f;
d_0 = 27.7 cm, d_i = 10.0 cm, h_i = 3.4 cm, M = 0.36
Image was inverted.
2f:
d_0 = 18.4 cm, d_i = 12.0 cm, h_i = 6.0 cm, M = 0.66
1.5f:
d_0 = 13.8 cm, d_i = 17.2 cm, h_i = 12.0 cm, M = 1.33

At 0.5f (4.6 cm) however, the image appeared too large to see. But when calculated/ predicted, it should have appeared upright.
Since it was not able to be observed directly, the image was only observable if seen through the lens directly. This type of image is virtual. The image, as predicted, was no longer inverted, but upright.

graph of d_i vs d_o

The graph was supposed to be analogous to y=1/x graph, though. 

Q. 8. The y intercept can not be found, because of the hyperbolic nature of the graph. The first graph does not look like to have a y intercept, as (10,27) was the uttermost left point (vertex).

The y value represents the negative inverse of object distance; since it is negative inverse, it represents the virtual distance.

Error and analysis:
The first possible error was during the calculation of the focal point; measuring the triangle's hypotenuse (distance) was done rather loosely; it was done on the hill and not a flat surface. The triangle was no longer perfectly orthogonal anymore.
The second one was observable by looking at the graph; there seemed to be a mistake when recording either the distance or height of the image at 4f. The graph showed a point at y= 36.8 cm (4f), whereas it should have been lower: distance less than the recorded image distance. The magnification then, is somehow off. The graph has two points that were misplaced; it should look like y=k/x, whereas the graph from experiment appeared hyperbolic instead.

1/S + 1/S' = 1/f, whereas S is object distance to lens and S' is the image distance from lens to reflective surface (surface where the reflection is the sharpest).
Solving for f, the following equation was obtained:

A. f = S(S') / (S' + S)
B. M is |S'|/ S, and S' = MS
C. S = S' / M
substituting that into equation A, f = (MS) / (M+1), and substituting C into equation A, f = S' / (M+1)
Focus depends only on either M and S or M' and s.

Conversely, S = S'f / (S' - f); object distance depends on image distance and focus over difference between image distance and focus.
Lastly, using M = |S'| / S, equation M = (S' - f)/ f was obtained. Using S, M= (S-f) / (f * S^2)
Magnification here is dependent on the sum of S and negative f over f times object distance squared. If S - f < 0, meaning object is placed where it is less than focus distance, it will give negative M. The image will be magnified differently. That explains when the object is placed close (S <= 0.5 f) as done in experiment to lens, it would appear upright and magnified and when  it is placed farther than half the focal length, it would appear smaller (0 < M < 1) and inverted.