Relativity of Time and Length

Introduction

                The purpose of this experiment is to understand how the time interval between the occurrence of two events and the length of an object in space depends on the inertial frame of reference. According to the relativity of simultaneity, an observer moving with a constant velocity, comparable to the speed of light, relative to the rest frame sees the clock of the rest frame to run slower. This is demonstrated by the relationship between the time interval of the moving and the stationary light clock,

∆t = γ∆t₀,                                             (1)

where ∆t is the time interval of the moving light clock, γ is the Lorentz factor, and ∆t₀ is the time interval of the stationary light clock. Not only the time interval but also the observed length differs depending of the frame of reference. An observer moving with a constant velocity relative to the rest frame sees the length of an object to be contracted. This idea is also described by a mathematical relationship,

l= γl₀,                                                     (2)

where l represents the length seen by moving observer, and l₀ is the length in the rest frame.

                In this experiment, these concepts were explored by using the animations of relativity of time and length on <Modern Physics>. The questions given in this website were used as a tool to analyze these concepts.

Images

Image 1: Image depicting the relativity of time at γ = 1.20

Image 2: Image depicting the relativity of time at γ = 1.12

 Image 3: Image depicting the length contraction at γ = 1.41

 Image 4: Image depicting the length contraction at γ = 1.30

Conclusion

Image 1 and 2 explain the concept of relativity of simultaneity. As shown in image 1, the distance travelled by the moving light clock is larger than that by the stationary light clock. Since the moving light clock is travelling with a constant velocity, the light pulse leaving the green object and coming back to this object happen at different positions in space. Therefore, the path of the light pulse is hypotenuses in the moving light clock. Since its path is hypotenuses, it takes longer time for the moving light clock to do the round-trip than the stationary light clock. If I am riding on the light clock, the time interval of the round-trip of the light pulse does not change because I am moving with the same speed of the light clock. In my frame of reference, the light clock is, in this case, always stationary. Thus, the light pulse leaving the green object and reaching this object will occur at the same point; hence, the time interval, which is 6.67μs, remains unchanged. If the velocity of the light clock decreases, the difference between the stationary and the moving time intervals will decrease since the moving light clock travels less distance. Therefore, the hypotenuses or the distance travelled by the light pulse on the moving light clock will approach that distance by the light pulse on stationary light clock. This is confirmed by image 1 and 2, in which the time interval of the moving light clock in image 1 is 8 μs and that in image 2 is 7.47 μs. Assuming the Lorentz factor relative to the earth is 1.2, the time interval records on earth will be 8μs. This result is confirmed by image 1. Besides, if the time interval of the moving light clock is 7.45μs, the Lorentz factor will become 1.12 as shown in image 2. These differences in time values reveal that the observer moving relative to a rest frame observe the clock of the rest frame to run slower.

Image 3 and 4 explain the length contraction in space. If I am riding on the left end of the light clock, the time interval for a pulse of light to travel to the right end and to reflect to the left end of the light clock will not change whether the light clock is moving or stationary relative to the earth. This is because I am moving with the light clock, so in my frame of reference, the light clock is stationary. Therefore, the time interval will always be 6.67 μs as shown in image 3. However, if I am to measure the time interval on the earth, the light clock is moving with a certain speed in my frame of reference. Then the light pulse has to travel longer than the length of the object; hence, it will take longer time interval for the light pulse to reflect back to the left end of the light clock. This is numerically confirmed by image 3, in which the time interval of the stationary light clock is 6.67 μs and that of the moving light clock is 8 μs. The length of the moving clock shown in image 3 and 4 are smaller than the stationary clock due to length contraction. If the lengths are the same, the round-trip time interval as measured on the earth will not be equal to t₀/γ. This is because the light clock is moving with a certain speed, so the time interval will become even greater since it has to travel a greater distance. In mathematical way, in equation 2, γ has to become 1 if the lengths are the same. This will contradict the situation since the light clock is moving with a speed comparable to c; thus, γ cannot equal to 1. If the light clock is 1000m long when measured at rest and the Lorentz factor is 1.3, the earth-bound observer will measure the length to be 749 m as confirmed in image 4. 

Experiment 12: CD Diffraction

Introduction

                The purpose of this experiment is to test a DVD to know whether the distance between the grooves causes distortion to the music. The groove spacing will be computed by employing the reflection-grating concept. The standard groove spacing for a DVD is 740 nm*. The experiment was conducted using a compact disc, a meter stick, a laser pointer, stands, and wood blocks. A compact disc was set up in front of the laser pointer so the laser beam would fall on the disc surface perpendicularly. The disc was adjusted so that the zero order maximum would strike the hold of the laser pointer. The first order maxima were found by putting two wood blocks, aligned with each other on each side of the laser pointer. The perpendicular distance between the screen (wood blocks) and the disc was recorded. The distance between two bright fringes on each side of the laser was also measured. The angle from the zero order maximum to the first order maximum was computed. The distance between the grooves was also calculated, and this value was compared with the standard value.

Figure 1: A compact disk was set up to remain perpendicular to the laser beam

 Figure 2: Laser was adjusted so that the zero order maximum stroke the hole of the laser

 

Data and Analysis

Table 1: Measured and computed data required for the grooves distance calculation

λ (nm)	Distance between the CD and the screen (cm)	Distance between 2 adjacent bright fringes (cm)	Angle between perpendicular line to the CD & the first bright fringe (˚)
632.8	11.70 ± 1.00	19.65 ± 1.00	59.23 ± 0.04

Table 2: Comparison between the experimental and the standard distance between the grooves

Experimental width (nm)	Standard width (nm)	% error (%)	% uncertainty (%)
736.5 ± 19.2	740	0.48	2.6

Uncertainty calculation

Conclusion

                According to table 2, the percent error between the experimental distance between the grooves and the standard values were 0.48% with the experimental distance of 736.5nm. In addition, the standard value was within the uncertainty of experimental error since the percent error was a lot smaller than the percent uncertainty. The error that could possibly contribute to this experimental result was that scratches on the disk and the uncertainties in measurement of the distances.

In order to improve the capacity of the DVD, the DVD can be added more layers. However, these layers have to have different pits so that the incoming waves will not interfere destructively when they reflect from each layer.

*Groove spacing on a DVD

Experiment 11: Measuring a human hair

Introduction

                The purpose of this experiment is to measure a human hair using the concepts of light interference. This experiment was conducted using a human hair, a 3x5 notecard, a whiteboard, and a microscope. A human hair was taped across a hole in 3x5 notecard, and let it parallel to the whiteboard at a known distance. A laser was used to point toward the white board through the hair. The distance between the first 2 fringes was recorded. The diameter of the human hair was measured experimentally using a micrometer, and this result was compared with the actual diameter of the hair. The theoretical result was computed using

d=λL/y

where d= diameter of the hair

λ= wavelength of the laser

L= Distance between the laser and the whiteboard

y= distance between 2 adjacent diffraction pattern.

Figure 1: A laser beam perpendicular to the axis passing through the hair

Figure 2: Diffraction pattern formed by the laser beam due to light interference

 Figure 3: Measuring the diameter of the hair using a micrometer

Data and Analysis

Table 1

Wavelength,λ (nm)	632.8
Distance,L (cm)	109 ± 1
Distance between 2 fringes,y (cm)	0.8 ± 0.05
Experimental diameter of the hair (μm)	90.0 ± 3
Actual Diameter of the hair (μm)	86.2 ± 1.25
% error (%)	4.41

Conclusion

As shown in table 1, the experimental diameter of the hair was measured to be 90 μm while the actual diameter was 86.2 μm. Therefore, the percent error was very small, and these values were within the uncertainties of each other since the smallest experimental diameter was 87 μm and the largest actual diameter was 87.45 μm.       

Based on the observation of this experiment, light interfered constructively and destructively depending on the distance travelled by each of the secondary light source. At the middle of the light source where both secondary light sources travelled equal distance, the light was the brightest. However there was no light when their distance travelled was a half wavelength difference. Therefore, there was a dark space between 2 bright fringes as shown in figure 2. Since the distance between the lens and the white board was a lot larger than the diameter of the hair, the equation mentioned could be used to compute the diameter of the hair. Otherwise, this equation could not be applied in this case. In addition, the laser beam had to be perpendicular and pass through the hole equally to form the accurate diffraction patter. Despite these disadvantages, the light interference method could accurately measure the diameter of the hair or the width the slits, but the micrometer had more uncertainties in measurement. Yet using the micrometer could be done faster to obtain an approximately accurate result, and there was no need to consider about the distance between the micrometer and the hair.                 

Experiment 10: Lenses

Introduction

The purpose of this experiment was to observe how the image changes based on the object distance. In this experiment, a meter stick, a converging lens, and a socket lamp with filament, a lens holder, and a cardboard. First, the focal length of the lens was determined by measuring the distance between the lens and the focus of the sun rays. Then the object distances varying from 1.5f to 5f were computed using the focal length. These computed distances were used to position the lens distance from the filament. The set up of the apparatus was shown in figure 2. The height of the filament was also measured. After turning the power on to let the light shine through the filament, the cardboard distance from the lens was adjusted till a sharp image was obtained. This image distance and its height were also measured. After recording all these data, the magnification of the lens was computed, and two graphs were constructed to observe the relationship between the image and object distance, and the focal length.  

 Figure 1: Measuring the focal length of the converging lens

Figure 2: Experimental set-up

Figure 3: Image forming through the converging lens

Data and Analysis

Focal length (f) = 9.5 ± 0.25 cm

Table 1: Recorded data of the object and image distances and heights

Object distance as a multiple of f(cm)	Object distance(cm)	Image distance(cm)	Object height(cm)	Image height(cm)	Type of image
5f	47.5 ± 1.25	16.50 ± 0.25	8.80 ± 0.25	3.30 ± 0.25	Inverted, real
4f	38.0 ± 1.00	17.00 ± 0.25		4.40 ± 0.25	Inverted, real
3f	28.5 ± 0.75	18.00 ± 0.25		6.60 ± 0.25	Inverted, real
2f	19.0 ± 0.50	25.50 ± 0.25		14.30 ± 0.25	Inverted, real
1.5f	14.3 ± 0.38	23.25 ± 0.25		16.50 ± 0.25	Inverted, real

Table 2: Comparison of magnification from distance ratios and height ratios

Object distance as a multiple of f(cm)	Md (di/d0)	Mh (hi/h0)	% difference between Md and Mh (%)
5f	0.347 ± 0.011	0.375 ± 0.030	5.10
4f	0.447 ± 0.013	0.500 ± 0.032	7.33
3f	0.632 ± 0.019	0.750 ± 0.036	11.1
2f	1.34 ± 0.04	1.63 ± 0.05	12.6
1.5f	1.63 ± 0.05	1.88 ± 0.06	9.28

Graph 1: Object distance vs. Image distance

Graph 2: Inverse image distance vs. Negative inverse object distance

Conclusion

According to table 1 and graph 1, as the object distance decreased, the image distance increased. Therefore, the image and object distance had inverse relationship as shown in graph 1. However, at 1.5f, even though the object distance decreased, the image distance decreased. This was because the image became dimmer and unclearer as the object distance decreased. Hence, uncertainty became greater since it became harder to see the actual height of the image. Another relationship observed in this experiment was that the image height increased as the object distance decreased. This was because the image distance got smaller. Besides, the image observed was always inverted. Since the image formed by the converging lens was inverted, the image was real. This idea was explained in experiment 9. As shown in table 2, the magnification obtained from distance ratio and height ratio increased as the object distance decreased. This agreed with the observation since the image got larger as the object was nearer to the lens. However, most of these values were not within the uncertainties of each other. This was possibly because the focus was not as accurate as it should have been. Unclear images at small object distances also contributed to this error.

According to graph 2, the inverse image distance and inverse object distance had linear relationship. This relationship actually described the relationship between the image distance, object distance, and the focal length, which was 1/d₀ + 1/d_i = 1/f. The y-intercept 0.08178cm^-1, obtained from the graph was the inverse of the focal length of the lens. By taking the reciprocal of the y-intercept, the graphical focal length was computed to be 12.23cm. However, the measured focal length was 9.5cm. This large difference in focal length was contributed by the image distance at 1.5f. As mentioned earlier, this was the only data set that deviated from the image-object distance relationship. This smaller image distance with larger inverse value lower the slope, hence, smaller y-intercept and larger focal length.
When half of the lens was covered, the image became dimmer since the light passed through the lens was decreased by a factor of 2 due to halved-lens. Besides, as the object distance became closer to the lens, the image size and distance became larger. At 0.5f, there was no image because the image was between the vertex and the focus; hence, the image became virtual.

Experiment 9: Concave and Convex Mirrors

Introduction

                The purpose of this experiment is to understand how images are formed differently in a plane, a concave, and a convex mirror. In this experiment, a concave, a convex, and a plane mirror are used, and the changes in image at different object distances are observed. A light ray diagram corresponding to each case is also constructed to confirm this observed behavior. The magnification of the concave and convex mirrors are also computed using the light ray sketches. 

Figure 1: Convex mirror

Figure 2: Concave mirror

Diagrams

Diagram 1: Light ray diagrams which show how the image in convex mirror changes depending on the object distance

Diagram 2: Light ray diagrams which show how the image in concave mirror changes depending on the object distance

  

Diagram 3: Light ray diagrams which were used to compute magnification of the mirrors

Data

Table 1: Comparison of magnification

Magnification ratio	Magnification of the mirror
	Convex	Concave
hi/h0	0.318 ± 0.024		-0.339 ± 0.009
di/d0	-0.304 ± 0.023		0.297 ± 0.008

Sample Calculation

Conclusion

                According to the experimental observation, the image formed by the convex mirror appeared to be always smaller than the object, and it is always upright. As the object was moved further away from the mirror, the image also appeared smaller, and it became larger as the object was closer. The reason that the image appeared as it was, was illustrated by diagram 1. When a ray of light hit the convex mirror, the light was reflected, and these reflected rays were appeared as they were radiated from the image. Since the reflected rays of light did not pass through the image, the image was a virtual image. Based on this observation, it was concluded that a convex mirror always formed upright, virtual image which were smaller than the object.

                As for concave mirror, the image was observed to be inverted when the object was placed further away from the mirror. As the object got closer, at a certain distance, the image became erect. This was because the object distance determined how the image was formed. As shown in diagram 2, when the object distance was larger than the focal length, the image was always inverted. Once the object distance equaled the radius of curvature, the object distance and the image distance became the same, but the orientation was different. But the image disappeared if the object was at the focal length since the image was at infinity. On the other hand, when the object distance was smaller than the focal length, the image was always virtual and erect since the actual light rays did not pass through the image. Therefore, the observation was agreed with the light ray sketch illustrated in figure 2.

                In plane mirror, the image and the object always showed the same size and distance no matter where the object distance was. In addition, the images formed in plane mirror were virtual and erect images.

                In the last part of the experiment were the mirror magnification were to be computed based on the sketch, it was found that the magnification computed from the object and image distance ratio was agreed with that from the height ratio as shown in table 1. However, their signs were opposite. This was because, in convex mirror, the image was a virtual image, but both the object and image distances were erected. On the other hand, in concave mirror, the image is real and inverted. Therefore, the image height was negative.  These sign determinations were shown in diagram 3.