Measuring Thickness of Human Hair Using Young's Method

Purpose:

This experiment was a recreation of the famous Young’s double slit experiment originally carried out in 1800 by Thomas Young.  The slight variation on the experiment was that a hair was used as the divider of the light source.  This added an additional easement to the experiment that gave it a bit more depth.  When the light was projected on a whiteboard an interference pattern was viewed, confirmation of lights wave like properties.  This pattern was measured in an attempt to confirm the Young’s double slit equation.

Method and Results:

A hear was plucked from one of the members of the lab group and this hear was measured for its thickness (d).  Next the hair was taped to the hole of an index card in such a manner that divided the hole punched circle evenly.  This card was set in place and a laser of known pre measured wavelength was focused through the hole.  A white board was placed at varying distances to easily view the interference pattern.  The distance from the index card to the board was measured (R).  Next the interference pattern was measure from 4 interference fringes (y).  These values were then used to confirm the thickness of the hair measured with the micrometer.

Trial	R (m)	y (m)	M
1	2.00	0.0765	4
2	1.68	0.0630	4
3	3.62	0.1100	4

Wavelength:  670 nm +/- 20 nm

Measured Thickness of hair:  0.0245 mm+/- 0.0015 mm

Calculated thickness:

Trial 1: 0.0701 mm +/-  0.0030 mm

Trial 2: 0.0715 mm +/-  0.0030 mm

Trial 3: 0.0882 mm +/-  0.0030 mm

            Average: 0.0766 mm +/- 0.0090 mm

Conclusion:

The thickness measured with the Young’s double slit experiment gave precise results and two of the three trials were within the measured uncertainty.  These values seem much more reliable than the micrometer measurement as most groups were having trouble getting measurements from the equipment.  This coupled with the precision of the Young’s method leads me to believe the diameter of the hair is the average of the Young’s result.

Real Image through a Lens Lab

Purpose:

The purpose of this lab was to explore the actions of light as it moves through a lens.  A light source was used to project a real image through a magnifying glass and on to a white board to view how the light was refracted and altered.

Method and Results:

A magnifying glass was chosen and this glass was measured for its focal point.  The method for the measurement was to use the suns light to focus a light rays onto the ground.  Due to the magnifying glass being a convex lens it is know that the point at which the light is focused through it from a far off object is at the focal point.  Three measurements were taken from the focal point on the ground to the lens.  These measurements were taken for an average and to determine the uncertainty of the method.

f1= 14.10 cm
f2= 15.80 cm
f3= 15.00 cm
favg= 14.97 cm +/- 1.70 cm

The magnifying glass was set up distances from the light source of known multiples of the average focal length.  A white board was placed behind the lens and adjusted in tell the image was focused.  The distance from the lens to the white board was measured as was the height of the real image.

Relative to f	Object distance (cm)	Image distance (cm)	Object height (cm)	Image height (cm)	M	Type of image
5f	74.85	23.5	3	1.30	0.433	inverted/reversed
4f	59.88	24.5	3	1.75	0.5833	inverted/reversed
3f	44.91	28.7	3	1.95	0.65	inverted/reversed
2f	29.94	44.5	3	5.20	1.733	inverted/reversed
1.5f	22.46	332.5	3	53.60	17.867	inverted/reversed

A measurement was taken at distance 0.5f but the image could not be focused.  The object and image distance were graphed and an inverse relationship was found.

The inverse of both values was then taken and a linear relationship was discovered.

Conclusion:


The absolute value of the slope of the graph is 0.7738 which is a value that is off from the expected value of one as this is the value of n in air.  this discrepancy could be explained by the large uncertainty involved in our measurement of the focal point as the uncertainty of 1.70 cm is a large value.  More likely though the % error of 22.6% is due to the low number of points measured and the arbitrary nature in which the focused real image was obtained.  These cupeled with the fact that there was not an adequate light experiment setup lead to much uncertainty.  

Concave Convex Mirrors

Purpose:

The Purpose of this experiment was to explore the image formed by both a convex and concave mirror.  Concave and convex mirrors are both considered spherical mirrors and both behave differently than a traditional plane mirror.  One of the most apparent differences is in the distortion of the height and distance of the image.

Method and Results:

Objects were held in from of both a concave and convex mirror at separate times.  The object was moved in front of the mirror and inspected at different distances and orientations.  Next ray diagrams were constructed form each of the mirrors.

Convex Mirror

With the convex mirror the focal point is located on the inside of the mirror and this is on the opposite side of the reflected light rays.  This leads for the image to be on the inside of the mirror leading to a virtual image.  The closer the object is to the mirror the larger the image will be to the extent that if the image is close to the mirror the image will be larger than the image.

S= 6.20 cm               s= 1.95 cm
H= 3.10 cm              h= 0.75 cm

Concave Mirror

With a concave mirror the results are different.  The main result is that the image is on the same side as the outgoing rays making them a real image.  This image is inverted so long as the distance of the object is grater than the focal point which in this type of mirror is located on the same side as the object.  

S= 2.50 cm                                s= 0.75 cm
H= 11.40 cm                             h= 3.30 cm

Conclusion:

In conclusion the behavior of light rays in spherical mirrors is much different than in plane mirrors.  The image can be virtual or real.  The image can be erect or inverted.  The height of the object can vary drastically.  a majority of these differences occur due to the addition of the center of curvature which in a plane mirror goes to infinity and is no longer important leading to a simplified equation for light ray behavior.

Light Refraction Lab

Purpose:

The purpose of this experiment was to test the refraction of light through a class prism.  A light source was used with a circular protractor to measure the angle of refraction for the light source.

Method and Results:

A light source was set up with a focused beam of light.  A glass semi circle prism was set on a circular protractor and the light was focused into the prism at measured angles.  

First the light was incident to the strait diameter of the semicircle yielding these results.

theta i	theta r	sin(theta i)	sin(theta r)
144	60	0.587	0.866
148	55	0.53	0.819
150	50	0.5	0.766
152	45	0.47	0.707
156	40	0.407	0.643
158	35	0.375	0.574
161	30	0.326	0.5
163	25	0.292	0.423
167	20	0.225	0.342
172	15	0.139	0.259

Next the light was focused incident to the arc of the semi circle giving data table # 2.

theta i	theta r	sin(theta i)	sin(theta r)
124	30	0.829	0.5
135	25	0.707	0.423
149	20	0.515	0.342
152	15	0.469	0.259
167	10	0.225	0.174
174	5	0.105	0.087
180	0	0	0
190	-5	-0.174	-0.087
203	-10	-0.391	-0.174
211	-15	-0.515	-0.259

Conclusion:

When a graph of the two angles was taken, it was found to be linear, these results are not what was to be expected.  

A reason for both sets of theta vs. theta graphs incorrect equations is due to the fact that the results were not taken over a longer range of circular increments.  If the data was to be taken as every 10 degrees for 10 trials the results may have been less skewed.  The slope of the sin(O)1 vs sin(O)2 was taken to be the index of refraction which was 1.52 +/- .15. 

Harmonic Overtones Lab

Method and Results:

Using a Pasco Variable Wave Driver and a Pasco Student Function Generator standing waves were created of known frequency on a string (f).  

The other end of the string was attached to a pulley and weighed down with known mass (M).  This mass (M) was multiplied by gravity to determine the tension on the string (T).  The total length of the string was measured (L) and the amount of nodes, (n), created in the standing wave were recorded with respect to frequency.  

The mass of the string was recorded and divided by its length to determine the mass per unit length (m).  This information was used to determine the wavelength of the wave at certain frequency. 

frequency	muew	Tension	nodes	Velocity	Wavelength
16.0	1.88E-03	1.95902	2	37.5565	2.35
32.0	1.88E-03	1.95902	3	37.5565	1.17
46.0	1.88E-03	1.95902	4	37.5565	0.816
63.0	1.88E-03	1.95902	5	37.5565	0.596
76.0	1.88E-03	1.95902	6	37.5565	0.494
109.0	1.88E-03	1.95902	8	37.5565	0.345
frequency	muew	Tension	nodes	Velocity	Wavelength
27.0	1.88E-03	0.98196	2	22.8543	0.846
32.0	1.88E-03	0.98196	4	22.8543	0.714
54.0	1.88E-03	0.98196	6	22.8543	0.423
frequency	muew	Tension	nodes	Velocity	Wavelength
15.0	1.88E-03	3.92	2	45.663	3.04
29.4	1.88E-03	3.92	3	45.663	1.55
44.0	1.88E-03	3.92	4	45.663	1.04
54.4	1.88E-03	3.92	5	45.663	0.839
72.3	1.88E-03	3.92	6	45.663	0.632
100.3	1.88E-03	3.92	7	45.663	0.455

Conclusion: 

All data obtained was graphed and these graphs produced perfect linear equations when the inverse wavelength was taken and plotter with respect to the frequency. 

The slopes of each of these lines are the velocities of the waves on their respective strings.   When using a string with the same mass per length the frequency to obtain the same harmonic in different trials is directly proportional to the hanging mass used.

Standing Wave Lab

Purpose:

The purpose of this experiment was to investigate the relationship between the period of a standing wave (T) and its wavelength (L). 

Method and Results:

To achieve the desired data for this experiment a spring was secured and held at opposite ends so its frame was parallel with the ground.  The goal was to have both experimenters who held the opposite ends of the spring to apply an upward force in an attempt to create one standing wave. 

The length measured from one experimenter to the other was recorded as the wavelength (L).  Once the Standing wave was created a stopwatch was used to measure the elapsed time for ten complete cycles.  This time was then divided by ten too determine the period of one cycle (T). 

At least two trials were taken at this length.  The experiment was next repeated at different lengths.  Averages were taken at each length and the results we graphed.

L (m)                                    T (1/s)                                    Average T (1/s)

2.76 +/- .06                       0.855            +/- .045                        0.871            +/- 0.225           

                                          0.834            +/- .045                                   

                                          0.902            +/- .045

                                          0.858            +/- .045

                                          0.904            +/- .045

2.51 +/- .06                        0.777            +/- .045                        0.761  +/-  0.135

                                           0.755            +/- .045

                                           0.751            +/- .045

3.17 +/- .06                        1.014            +/- .045                        1.027  +/-  0.135

                                           1.030            +/- .045

                                           1.036            +/- .045

1.43 +/- .06                        0.436            +/- .045                        0.423  +/-  0.090

                                          0.409            +/- .045

Conclusion:

When graphed the slope of the line is equal to the velocity the wave travels through the medium, in this case the spring.  This value was determined to be 3.179 m/s +/- 0.285 m/s.

This result proves that the velocity of the propagation of the wave is expressed in the equation.

v= L/T

Fluid Dynamics (Flow of water out of bucket)

Objective:

The objective of this lab was to document how well the measured values of time agree with uncertainty, in order to properly prove the Bernoulli equation.

Method and Results:

A bucket that contained a drain hole on the bottom vertical wall was obtained and sealed with tape.  This Bucket was then filled with tap water up to a constant height.  Next a measurement was taken from the drain hole to the surface of the water (h). 

h= 0.110 m  +/-  0.0012 m

The diameter of the drain hole was given and then measured for accuracy.  That value was then divided by two to determine the radius of the drain hole (r). 

R*=  3.175e-3 m 

r = 3.675e-3  m +/-  1.5e-3 m

* given value

This value was used to determine the cross sectional area of the drain hole (A). 

A= (pi)r^2= 4.24e-5 m  +/- 3.00e-3 m

A Graduated cylinder was placed under the drain hole to measure the volume of water that was emptied from the bucket (V). 

V= 5.0e-4 m^3 +/- 1.5e-6 m^3

When the tape was removed from the drain hole a stopwatch was used to determine the amount of time (t) it took for the water to drain down the complete height (h). 

                  Time to empty (s)

1^st Run         20.30

2^nd Run         19.74

3^rd Run         19.87

4^th Run          20.30

5^th Run          20.89

6^th Run          20.21

The acceleration due to gravity (g) was taken as a constant 9.8 m/s^2.  These values were used to calculate the theoretical value of time (T).

T= 8.03 s  +/-  5.40 s

Conclusion and Analysis:

                                                               Time (s)

Avg                                                                20.22                                   

Range                                                      1.15

Standard deviation                                    0.404        

Error %                                                      60 %                                   

Measured Uncertainty                           5.40

Calculated uncertainty                           4.82

Although the uncertainty of the measured values is high, its predicted range of uncertainty is not adequate to explain the large gap between the predicted value of 8.03 s and the actual average of 20.22 s.  Reasons for error could be an underestimation of the actual intrinsic error involved with lab measurements.  One source of error that seems to stick out is the diameter of the drain hole, which was given as a ½ in which in turn was calculated to be 6.35e-3 m, this value was measured and found to be 3.68e-3, which is a large variance at that level.  Other reasons for the error may be that the height (h) of the water displaced was too large of a value.  With a smaller value the effects of the water draining may have been more accurately modeled by the equation for time derived from Bernoulli’s equation.  Interesting results though seem to be that there is only a slight variance between the measured uncertainty and the calculated uncertainty, which leads me to believe that there may have been a minor error in the recorded values of the calculated time.