Relativity of Time and Length

Purpose:
The purpose of this activity is to understand how the time interval between the two events and the length of the objects in spaces depending on the inertial frame of reference. Based on the relativity, the clock of an object moving with the constant velocity runs slower compared with the rest one. This is demonstrated by the relationship between the time interval of the moving and stationary light clock:

∆t = γ∆t₀,     

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 length differs depending in the inertial frame of reference. An object moving with constant velocity sees the length of the object to be contracted compared to the object in the rest frame. The relationship is shown by the equation:

l= l₀/γ 

Where l is the length in the moving frame, and l₀ is the length in the rest frame

Questions and images:

Part 1: relativity of time

Question 1: The distance traveled by the light pulse on the moving light clock is longer than the distance traveled by the pulse on the stationary light clock

Question 2: The time interval for the light pulse to travel on the moving light clock is longer compare to on the stationary light clock.

 

Question 3: If the time is measured in the moving moving light clock, it takes the same amount of time interval to complete a round trip.

Question 4: When the velocity of the light clock is decreased, the difference in the light pulse travel time between the earth's timer and the light clock's timer will be decreased.

Question 5: when the light clock has a Lorentz factor (γ) of 1.2, it will take Δt = 8.00 µs for the light pulse to travel back and forth as measured on the earth. 

Question 6: Calculate the Lorentz factor (γ) if Δt = 7.45 µs. Lorentz factor (γ) = 1.116

 Part 2: Relativity of length

Question 1: As a pulse of light travels back and forth the light clock, the time interval of this round-trip does not depend on whether the light clock is moving or stationary relative to the earth.

Question 2: The round-trip time interval for the light pulse as measured on earth is longer than the time interval measured on the light clock.

Question 3: Yes, the length of the moving light clock is contracted by the Lorentz factor (γ) in order to obey the time-dilation relation.

Question 4: The light clock is 1000m long when measured at rest, it would be 769 m when the light clock had the Lorentz factor of 1.3.

Summary:
      The first part explains the concept of relativity of time. As shown in the pictures, the distance of the light pulse traveled on the moving light clock is lager the stationary light clock. Because the moving light clock moving with constant velocity, the light pulse leaved the light clock and returned at the different position in space. The path of the light pulse traveled is the hypotenuses in the moving light clock. Because its path is hypotenuses, it takes longer time for the moving light clock to complete a round-trip than the stationary light clock. If the observer is moving with the moving light clock, the time interval for a round-trip of the light pulse does not change, because the observer is moving the same speed with the light clock. In that case, the light clock is always stationary in the observer's frame of reference.
      The second part explains the concept of length contraction in space. If the observer is riding on the left end of the light clock, the time interval for the light pulse travels a round-trip is the same whether the light clock is moving or stationary relative to the earth. This is because the observer is moving with the light clock, in the observer's frame, the light clock is stationary. However, if the light clocked is measured on the earth, the light clock is moving with the a constant velocity relative to the earth frame. The light pulse travels a longer distance than the object. Thus, it takes longer time interval for the light pulse travel back to the left end. As shown in the pictures, the length of the object is smaller in the moving light clock than in the stationary light clock. If the length is the same, the time interval as measured on the earth is not equal to γ∆t₀

. 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. The length of the moving object is equal to l₀/γ 

Experiment 12: CD Diffraction

Purpose:
The purpose of this experiment is to use the concept of light interference and diffraction when a light beam goes through a grating to measure the distance between the grooves for a CD and a DVD. When the grooves spacing for a CD and a DVD are experimentally found, the results will be compared with the actual values are computed by the factory. The standard grooves spacing for a  CD is 1600nm and DVD is 740nm.

Procedure:
1. Obtained a meter stick, a laser beam, a CD, a DVD, and a screen with a small center hole.
2. Set a laser beam pointing through the hole of a screen to the CD or DVD.

3. Adjusted the disks so that the zero order maximum reflected light strikes through the hole of the screen and the first order maxima appeared on the screen.

Data Analysis for the CD:

	λ (nm)	Distance (l)  between the screen and the disk (cm)	Distance (x) between the  hole and the bright fringe (cm)
Trial 1	633	32.4 ± 0.5	14.8 ± 0.5
Trial 2	633	34.8 ± 0.5	16.0 ± 0.5
Average	633	33.6 ± 0.5	15.4 ± 0.5

Results:

Experimental grooves spacing (nm)	Standard spacing  (nm)	% error	% uncertainty error
1519 ± 112	1600nm	5 %	7 %

Data Analysis for the  DVD:

	λ (nm)	Distance (l)  between the screen and the disk (cm)	Distance (x) between the  hole and the bright fringe (cm)
Trial 1	633	5.5 ± 0.3	9.1 ± 0.3

Results:

Experimental grooves spacing (nm)	Standard spacing  (nm)	% error	% uncertainty error
739 ± 89	740nm	0.14 %	12 %

Sample calculations:

how to calculate the grooves spacing

 uncertainty calculation

Summary:
According to the results, the % error between the experimental grooves spacing of the CD and DVD comparing to the standard values are 5% and 0.14% respectively. These % error are within experimental uncertainties. The error could be from the scratch on the surface of the disks and some uncertainties of the measurement.
The grooves spacing depends on the wavelength of the laser light they use to shine the disk. Therefore, the shortest wavelength they can use now is the the gamma ray which should give the spacing in order of

10^-12 m. One way to improve the capacity of a DVD or CD is to add more layers. However, these layers have to have different pitch so the incoming light will not be destructive interference.

Experiment 11: Measuring a Human Hair

Purpose:
The purpose of this experiment is to measure a human hair by two method. The first method is using the concept of light interference when a light ray goes through two small (separated by a distance d) slits . The second method is using a micrometer. Two measurement will be collected and compared to see which method is better to measure a small object.

Procedure:

Using a laser
1. Obtain a hair and tape it across the hole in the 3x5 note card.
2. Clamp the card so that it is parallel to the white board a distance 1 m.
3. Pointing a laser toward the board through the hair.
4. The distance between the first 3 fringes is recorded.

The laser pointing through the hair

Diffraction pattern formed by the light interference

Data:

Trial	y (cm)	m
1	1.85 ± 0.05 cm	3
2	1.83 ± 0.05 cm	3

where: y = distance between 3 adjacent diffraction
λ = 633 nm  (wavelength of the laser beam)

L = 100 ± 1 cm (distance between the board and the note card)

d = diameter of the hair

d = mλL/y = 0.103 ± 0.038  mm

Using a micrometer
Place the note card under the micrometer and measure the diameter of the hair

The hair is seen through the micrometer

Data: 

Trial	d (mm)
1	0.18 ± 0.01
2	0.13 ± 0.01
Average	0.155 ± 0.02

Data Analysis:
% error = (0.103 - 0.155)/((0.103 +0.155)/2) = 4.03%

Summary:
           As the results, the diameter of the hair could be from a range 0.103 ± 0.038  mm  - 0.155 ± 0.02 mm, and the percent error is 4.03 %. The two methods give the results within their uncertainties. Therefore, the two methods are the appropriate ways to measure a small object such as a hair.
           Base on the observation of the experiment, light interfered constructively and destructively depends on the distance of the light source. At the middle, the light interfered brightest which the distance of the 2 light rays through the slits were equal. When the distance traveled was half a wavelength, there was no light. Therefore, the darkness was between the two bright fringes. Because the distance between the white board and the hair was a lot longer than the distance of 3 fringes, the equation d = mλL/y can be used. Otherwise, the equation cannot be used in this case. Moreover, the laser beam had to be perpendicular to the note card and pass through the hair to get the interference. The light interference method is more accurate than the micrometer method to measure the hair or the width of the slits. The micrometer had a big uncertainty in measurement. However, using the micrometer could be done faster because we don't need to worry about the distance between the hair and micrometer.