Relatavistic Time and Lenght

Purpose:

            The object of both of these experimental activities is to gain a further and deeper understanding of the relativity aspect of length and time of object that are moving at speeds comparable to that of the speed of light. The online program of activity physics will be used to demonstrate how time and length changes with velocity of an object reaching speeds near that of the speed of light. Primarily, this lab will be dealing the equations of time dilution and length contraction.

Data analysis:

            Part 1: Relative Time

Question 1: Distance traveled by the light pulse
How does the distance traveled by the light pulse on the moving light clock compare to the distance traveled by the light pulse on the stationary light clock?

-          The distance traveled by the moving clock is further than the distance traveled by the stationary clock. 

Question 2: Time interval required for light pulse travel, as measured on the earth
Given that the speed of the light pulse is independent of the speed of the light clock, how does the time interval for the light pulse to travel to the top mirror and back on the moving light clock compare to on the stationary light clock?

-          Since the distance traveled is longer, the time required for the light to bounce off of the mirrors must be longer than for the stationary clock. 

Question 3: Time interval required for light pulse travel, as measured on the light clock
Imagine yourself riding on the light clock. In your frame of reference, does the light pulse travel a larger distance when the clock is moving, and hence require a larger time interval to complete a single round trip?

-          No, the time measured with the clock moving in the same frame takes the same amount of time. 

Question 4: The effect of velocity on time dilation
Will the difference in light pulse travel time between the earth's timers and the light clock's timers increase, decrease, or stay the same as the velocity of the light clock is decreased?

-          The difference will decrease as the velocity also decreases. 

Question 5: The time dilation formula
Using the time dilation formula, predict how long it will take for the light pulse to travel back and forth between mirrors, as measured by an earth-bound observer, when the light clock has a Lorentz factor (γ) of 1.2.

-          Using t=γt_o, the answer comes out to be: t = (1.2)(6.67) = 8.00s

Question 6: The time dilation formula, one more time
If the time interval between departure and return of the light pulse is measured to be 7.45 µs by an earth-bound observer, what is the Lorentz factor of the light clock as it moves relative to the earth?

-          t = t_0γ

γ = t/t₀ = (7.49)/(6.67) = 1.12

Part 2: Relativistic Length

Question 1: Round-trip time interval, as measured on the light clock
Imagine riding on the left end of the light clock. A pulse of light departs the left end, travels to the right end, reflects, and returns to the left end of the light clock. Does your measurement of this round-trip time interval depend on whether the light clock is moving or stationary relative to the earth?

-          No, it does not matter if the clock is moving and you are on it. 

Question 2: Round-trip time interval, as measured on the earth
Will the round-trip time interval for the light pulse as measured on the earth be longer, shorter, or the same as the time interval measured on the light clock?

-          Longer, since the light appears to travel a greater distance. 

Question 3: Why does the moving light clock shrink?
You have probably noticed that the length of the moving light clock is smaller than the length of the stationary light clock. Could the round-trip time interval as measured on the earth be equal to the product of the Lorentz factor and the proper time interval if the moving light clock were the same size as the stationary light clock?

-          The length has to be shorter so that the time difference factor, gamma, cane the same for both. 

Question 4: The length contraction formula
A light clock is 1000 m long when measured at rest. How long would earth-bound observer's measure the clock to be if it had a Lorentz factor of 1.3 relative to the earth?

-          l = l_o/γ = 1000/1.3 = 769m

Conclusion:

            For the first part of this experiment, we explored the properties of time when an object is traveling at comparable speeds to that of light. When an object is moving at such speeds, two different times are recorded, proper time, and diluted time. The proper time, as we learned, is the time interval in which is in the frame of that of the object moving. That is, when light has appeared to leave and return in the same spot. The diluted time is just the opposite. It is the time in which the light is emitted at one point and returns at another different from the original. This is called diluted time because it is not possible and would not make sense for there to be an instance in which there is contracted time.

            For the second part of the experiment, we explored the properties of length as speeds reach that close to light. The main factor learned is that the length is contracted whenever an object is moving at very fast speeds. Meaning that a stationary observer viewing an object moving at speeds comparable to the speed of light, sees the length of the object as shorter than what it is in actuality. This is called length contraction. The proper length, is the length that is measured from the frame that is moving. A person moving close to the speed of light in that reference is the same as the object, will measure the proper length.

CD Diffraction

Purpose:

            The primary purpose of this experiment is to use a simple apparatus to determine the spacing of the lines within a compact disk. This experiment will explore the properties of diffraction gradients using everyday items and a laser of known wavelength. The laser will be used to measure the spacing of the lines by using the relationship learned of light diffraction. Once the spacing have been determined, the obtained results will be compared to the factory requirements of 1600nm and evaluated. In addition to the CD, a DVD will also be evaluated and its spacing will also be compared to the factory specifications of 740nm.

Procedure:

            First, the following supplies were gathered: CD, DVD, piece of paper, meter stick and a laser. The paper was first taken and a hole was placed in the middle of it. Once that was done, the laser was placed on the table so that the beam would strike through the hole created. On the opposite side of the paper, the CD was placed so that the beam would reflect off and bounce diffractively back at the paper. 

Once this was completed, the spacing of the dots were measured and recorded. The length from the CD to the paper was also recorded to be used in later calculations. This entire process was then repeated with the DVD to determine the differences.

Data Analysis:

CD:

            The following data chart displays the raw recorded data for the CD measurements with x representing the distance to the first maxima, l indicating the distance from the paper to the CD, and lambda representing the wavelength of the laser:

Trial	x(cm)	l(cm)	λ(nm)
1	11.5 ± .5	24.3 ± .5	633
2	14.8 ± .5	32.4 ± .5	633
3	16 ± .5	34.8 ± .5	633

The following sample calculation represents the work done to determine the spacing between each line for the CD:

Sinθ = λ/a = x/(x²+l²)^1/2

a = λ(x²+l²)^1/2/x

a = (633nm)(11.5cm²+24.3cm²)^1/2/11.5cm

a = 1480nm

The proceeding table includes the calculated spacing for all three trials:

Trial	Spacing(nm)
1	1480
2	1520
3	1520

DVD:

            Unfortunately, the DVD experiment was only done once due to time restraints. The following table represents the raw data achieved for the DVD, again the same values represent the same quantities as the CD observations:

Trial	x(cm)	l(cm)	λ(nm)
1	9.1  ± .5	5.5  ± .5	633

Using the same calculations as for the CD, the spacing for the DVD is as followed:

Trial	Spacing(nm)
1	739  ± 89

Conclusion:

            After completing all necessary calculations, the spacing for both the CD and DVD were found. The computed spacing of the CD were found to be 1480 nm and 1520 nm. When these values were compared to the factory standard of 1600nm, the range of errors ranged from 5.0% to 7.5%. Considering the small amounts of distances being evaluated, these numbers are rather large. It would be recommended that the factory consider the way their CDs are produced to ensure that the standards are more closely achieved. As for the DVD, the spacing was found to be close to 740nm. When comparing this to the standard of 760nm, the error of 2.6% is not as far off as from the CD. It would not be recommended to the factory to consider the manufacturing process of their DVDs since the error is relatively small.

Measuring a Human Hair

Purpose:

            The primary purpose of this laboratory experiment is to determine the diameter of a human hair using two different methods. The first method that will be used is with the use of a laser with a known wavelength. The second method, is to use a micrometer to determine the diameter of the hair.

Procedure:

            Part 1:

            First, the laser was obtained as well as a note card, a white board, and the hair that will be measured. The note card was punched through with a hole punch to allow for the gap where the hair would be placed. The hair was then placed vertically in the hole created. Once that was completed, the laser was positioned directly behind the hair so as that the hair would split the beam.

 

Once this was complete, the white board was then placed a distance away from the laser and the gap between the hair and the board was measured.

 

The light of the laser was then turned on and the created interference pattern was observed. This pattern was tehn marked using a marker to be measured at a later time.

 

Once this was completed, the values were recorded and logged to be used to calculate the diameter of the hair at a later time.

            Part 2:

            For this part, the micrometer was obtained and placed on the table along with a light source to enable a better viewing of the hair. The hair was then placed under the scope (while still being attached to the note card) and was then looked for through the lens.

 

Once the hair was found, the cross hairs of the microscope where then placed on one side of the hair. The knob on the side of the micrometer was then turned until the cross hairs reached the other side.

 

Once this was completed, the measurement on the side of the micrometer was measured and recorded as the diameter of the hair.

 

Data Analysis:

            Part 1:

The following chart depicts the recorded data for the laser experiment:

Trial	y (cm)	m	λ laser (nm)
1	1.85 ± .05	3	633
2	1.83 ± .05	3	633

 Since the distance from the board to the hair was measured to be 100 ± 1cm, we can use the following equation to determine the diameter of the hair, d.

d = mλL/y

Upon doing this, we get the following values for the diameter of the hair:

Trial	Diameter (mm)
1	.103 ± .038
2	.104 ± .038

          Part 2:

The following table shows the measured values obtained for the diameter of the hair using the micrometer:

Trial	Diameter(mm)
1	.18 ± .01
2	.13 ± .01

 Using these two numbers, the average of the two was found to be .155 ± .02 mm.

Conclusion:
            According to online sources, the actual diameter of a human hair is about 180 micrometers, or .180 millimeters. It is clear that the use of the micrometer is the best and most accurate way to measure the hair but the laser, did in fact come close, at least certainly within the orders of magnitude. To conclude, the seemingly difficult task of measuring such small distances can be made simpler with the use of a laser, or better yet, a micrometer, fairly easily.