Planck's Constant

Purpose:
  The objective of this lab was to verify Planck's constant by essentially messing around with two LEDs.

Note: this lab was completed very quickly after class and the opportunity for photographs was very scarce.  Only one unclear photograph was collected.  

  The wavelength that each LED projects was found.  And the Voltage across each LED was also measured.  Next the wavelength was graphed versus 1/E.     Furthermore, the graph suggest that the slope is h*c , where c is the speed of light.  Thus diving by c should yield Planck's constant. Note multiplying the voltage by charge gives one energy.

For the first LED, 1.95 volts and a wavelength of 590 nm (yellow) were measured.
For the second LED, 2.76 volts and a wavelength of 450 nm (blue) were measured.

 Note this graph is not very good since this lab was done as a class at a very fast pace.  


Since the graph is not very clear, the slope will be calculated manually.
-->  rise/run = h*c --> h = 3.1(10^-15) ev*s.

Multiplying by 1.6 (10^-19) J/1 ev yields --> 4.96 (10^-34) J*s.

This is not too far from the accepted value 6.63(10^-34) J*s.  The percent error is very large at 25%. However the error stems from the equipment.  If more time was given for completion of the lab, more LED lights would have been subject to test.  Inevitably this would have produced a much better slope.  Nonetheless all things considered, the degree of magnitude is the same.

Color and Spectra

The purpose of this experiment is to view the spectrum of colors found in white light and to measure the corresponding wavelengths for each color observed in the spectrum. Moreover, this experiment was is consolidated from three parts.  For part one, two and three, the light source is an incandescent lamp, the light source stems from an unknown source, and the light source emanates from hydrogen gas respectively.  For the second part the unknown will be identified.

Above: The Set up--This set up was used for all cases.


Note two one-meter stick were used and a defraction grating.


The relevant equation used is lambda = D x d/(L^2 +D^2)^(1/2) 
where d is fixed and = 500 slits/mm =2 micro meters and L is the distance from the defraction grating to the light source and D is the distance from the color of interest on the light spectrum to the light source. 



Part 1:

Note: the uncertainty in all the distances D is +/- .1 cm and the uncertainty for L is +/- 1cm

The data collected for the first experiment is presented in the above photograph. The lambda prime equation is derived essentially from solving two unknowns--the slope and the constant--from two equations: lambda prime = m*lambda+ constant for violet and lambda prime= m*lambda + const  for red.  For the first equation, the lambda prime is replaced by lowest boundary for violet--roughly 390nm-- in the visible spectrum for humans and the value for lambda is the one that was calculated for the experiment (see above photograph).  The same idea is used for the second equation. However this time the highest boundary for red in the visible spectrum for humans is used for lambda prime.  


Lambda prime values (nm):
1 = 389 +/- 6.0
2 = 447 +/- 6.5
3 = 540 +/- 7.5
4 = 592 +/-  8.0
5 = 668 +/- 8.8
6 = 749 +/- 9.6    all of these lambda primes are within the uncertainty

Part 2:
For part two, an unknown gas number one was given.

It should be noted that the length L changed and that the uncertainty values for D and L are +/- .1 cm and +/- 1 cm respectively.























All the data is observed in the above photograph.  Moreover, the unknown was determined to be mercury Hg.  Only three colors were observed in the spectrum and they were green, yellow-orange-like color, and blue. These values correspond to lambda-1, lambda-2, and lambda-3 respectively in the above photograph.  Nonetheless, it was determined that the source was mercury through a reliable internet source that had the visible spectrum of mercury that matched the data in the above photograph.  


Part 3:

Note that the uncertainty for D is +/- .1 cm and the uncertainty for L is +/- 1 cm.


There were only three distinct visible lights in the visible spectrum for the hydrogen gas. All the necessary data was found and presented in the above photograph.  Moreover, the corresponding energies E-1, E-2, and E-3 are 4.87(10^-19) J, 4.13(10^-19) J, and 2.98(10^-19) Joules respectively.

CD Diffraction

The purpose of the experiment is to measure the distance d between the grooves on a CD for the Easy Listening Music Studio company.

This is accomplished by allowing a Helium Neon Gas Laser--which has a wavelength of 630 nm--to strike the disk normally.  As a result the first and second order maxima appear on the screen or white board.

(Above: The Set Up)


The distance from the slits or groves to the screen is a distance L 51 +/- 1 cm.  Moreover after the experiment was conducted, the lengths of the first and second-order maxima to the zeroth-order maxima are x-1 22.5 +/- .5 cm and x-2 68.8 +/- .5 cm respectively.  These values were used to find the relative angles, atan (x/L) = theta.


Calculations:

Key Equation: d*sin(theta) = m*lambda , where m is the maxima number and d is the distance between the grooves.

Note that the d set equal to 51 cm in the above photograph is really L, the distance from the slit to the screen or whiteboard.  Also the second equation should be multiplied by two since it is the second-order  maxima, m=2.  

Lenses

The purpose of the experiment is to ultimately verify the relationship between the focal length f, image distance D-i, and object distance D-o by exploring converging lenses.

To begin, one had to measure the focal length of the lense and this was accomplished by using an infinite light source--the sun.  The focal length was distance between the lense and the sharp point on the surface where all the sun rays collected.  The focal length was measured to be 20 +/-1 cm.

Once that was accomplished, the lense was placed four focal lengths--D-o equals 80 +/- .5cm--away from an object--a filament--that had a height of 9 +/-.5 cm ( one should keep in mind that there is a light source emanating from the object. See picture below). 
     

The height of the image H-i was measured to be 3 +/- .5 cm and the image distance D-i equals to 25 +/- .5 cm.   It is very important to note that image being projected is real and inverted.  Nonetheless, the lense was flipped around so the light of the object enters the other side.  This change did not produce any new data that was not already known.  That is to say that the image was still real and inverted. The image distance and height were remained the same as well.  The magnification was .49 +/-.091.

The lense was then moved to 2 focal lengths:
D-o = 40 +/- .5 cm H-o = 9 +/- .5 cm
D-i = 36 +/- .5cm   H-i = 8.5+/- .05 cm
M = .94 +/- .058

*After flipping the lense again, nothing was changed--all numbers are the same and the image was still real and inverted.

The lense was then moved to 1.5 focal lengths:
D-o = 30 +/- .5 cm H-o = 9 +/- .5 cm
D -i = 52 +/- .5 cm H-i = 16 =/- .5 cm
M = 1.78 +/- .153
*Once again the image was real and inverted and numbers did not change after flipping the lense.


The next part of the experiment consisted of applying the same technique of moving the lense to various distances from the object. The object was no longer the fillament but a small key chain that had the form of a bear.  The following data was obtained.

 

Object distance (cm)		Image distance (cm)		object height (cm)	image height (cm)	M	Type of image
5f		23 +/- .5		5.1 +/- .05	1.7 +/- .05	0.333 +/- .0131	real,inverted
4f		24.5 +/- .5		5.1 +/- .05	2 +/- .05	0.392 +/- .0136	real, inverted
3f		28 +/- .5		5.1 +/- .05	2.5 +/- .05	0.49 +/- .0057	real, inverted
2f		32 +/- .5		5.1+/- .05	3.2 +/- .05	0.627 +/- .0159	real, inverted

 

The image is real and inverted.

The graph above is one over the image distance versus negative one over the object distance.  The y-intercept represents one over the focal length 1/f.  Initially the focal lenth was measured to be 20 +/- 1 cm.  However with respect to the graph, the focal length is 19.69 cm.  This yields a percent difference 1.5 percent which is within error range.  More importantly the lab supports the idea that 1/D-i + 1/D-o = 1/f.

Concave and Convex Mirrors

The objective of this experiment is to explore the images formed by convex and concave mirrors.  

CONVEX MIRROR:

For a convex mirror the object appears larger than the image and is upright.  If one moves the object closer to the convex mirror, then the image becomes a bit larger but still smaller relative to the size of the object.  The image also gets a little bit closer.  Moreover if  object moves farther away from the convex mirror, then the image gets smaller, the image is farther away, and remains upright.

The convex diagram is an emblem of the observation made during the activity.

For Convex Diagram (left):
height of object Ho= .031 meters
height of image Hi = .007 meters
Distance of object Do = .062 meters
Distance of image Di = -.019 meters

Magnification M = Hi/Ho = .226

CONCAVE MIRROR:

For the concave mirror, the image is inverted and much larger than the object.  When the object is retreated to a large radius, the object remains inverted and becomes larger.  However if the object is moved close enough to the concave mirror, the image becomes up right and smaller.

This concave diagram is a representation of the observations made during the activity.

For Concave Mirror (left):
Ho = .031m
Hi = -.007 m
Do = .113 m
Di = .023 m

M = Hi/ Ho = -.258

Measuring the Length of a Pipe with Standing Waves?

The purpose of this lab was to measure the length of an open pipe by swinging it around and creating standing waves.  A microphone connected to a sensor was positioned near the swinging pipe and recorded all necessary data--omega specifically.

Although it may not be distinguishable in the photograph, omega equals to 3859 rad/s for the first case. Hence for the first case f-1 roughly equals to 614 hz.

 Again it may not be distinguishable in the photograph, but for the second case, omega = 5068 rad/s.
Therefore f-2 equals 806 hz.

From here, there are three equations and three unknowns.  (eq1) f-1 = (n-1*V)/(2L)
(eq2) f-2 = (n-2*V)/(2L)               (eq3) n-2 = n-1 +1

After solving for the wanted unknown length of the pipe L, one gets .837 meters.

Introduction to Sound

The purpose of the experiment was to become familiar with sound and it's characteristics lambda, period T, frequency f, and amplitude A by saying "AAAAAA" into a microphone connected to a sensor.

After saying "AAAAA" into the microphone, the wave projected has a periodic pattern--see above photograph and in large if needed.  Specifically there were three waves captured by the sensor in the small time interval.  And the period T was determined by diving the time interval by the number of waves.  This number was .009 Sec.  From here the frequency of the sound was determined 1/T = f which was 111.11 hz. The assumption is that the speed of sound is 340 m/s in the room.  From here, the wavelength was determined to be 3.06 meters.  The amplitude A was determined by subtracting the lowest peak of the wave from the zenith point of the pattern and diving that by two.  The highest point was 2.594  and the lowest point is 2.620.  This yields an amplitude of 0.167 m.  Moreover if data was collected for an "AAAAA" sample that was 10-times-longer time interval, the data might change a little bit because of damping.  That is to say that the wave will fall off some because one's "AAAAA" may not be uniform over the entire time frame.  If the "AAAAA" is quieter as time increases, then amplitude will become smaller.  This is the case because the amplitude of the wave is proportional to the magnitude of the noise.      

Standing Waves

The purpose of the experiment is to investigate the ideas of standing waves emanating from an external force. The wave speed v, wavelength, period T will be determined for two distinct cases.  For the first case there will be a hanging mass of 0.200 +/- 0.0005 kg. For the second case, there will a hanging mass of 0.050 +/- 0.0005 kg.  The first seven harmonics will observed for both cases.

The equipment is set up such that one end of the string tied down to the wave driver and the other runs over a pulley where there is a hanging mass attached to it.  

Note: It should be noted that the mass of the string is 0.00286 +/- 0.000005 kg and the length of the string is 2.48 +/- 0.005 M while the length of the string oscillating is 1.60+/- 0.005 M.  





Data for .250 kg hanging mass:


Harmonic   frequency f (hz) wavelength (m)        Period T (s)     1/wavelength (m^-1)
1               13 +/- .00005         3.2 +/- .00005     .077+/-.00005    .3125+/-.00005
2               26+/- .00005         1.6 +/- .00005     .038+/-.00005       .6250+/-.00005
3               39 +/- .00005       1.06 +/- .00005     .026+/-.00005   .9434+/-.00005
4              51.5 +/- .00005  .8 +/- .00005     .019+/-.00005     1.250+/-.00005
5              64 +/- .00005        .64 +/- .00005     .016+/-.00005      1.5625+/-.00005
6             77.2 +/- .00005      .53 +/- .00005     .013+/-.00005   1.8867+/-.00005
7            90.1 +/- .00005     .457+/- .00005     .011+/-.00005    2.1882+/-.00005




The slope of the graph represents the experimental velocity, 40.9 m/s.
The theoretical velocity v is equal to (T/mu)^(1/2) = 46.11 m/s.
This yields a percent error of 12.7 %.






Data for .050 kg hanging mass:

Harmonic       frequency f (hz) wavelength (m) Period T (s)             1/wavelength (m^-1) 1               6+/-.00005   3.2+/-.00005        0.1667+/-.00005 0.3125+/-.00005 2               12+/-.00005 1.6+/-.00005        0.0833+/-.00005 0.625+/-.00005 3               18+/-.00005 1.07+/-.00005      0.0556+/-.00005 0.935+/-.00005 4               24+/-.00005 0.8+/-.00005        0.0417+/-.00005 1.25+/-.00005 5               32.2+/-.00005 0.64+/-.00005      0.0311+/-.00005 1.5625+/-.00005 6               39.2+/-.00005 0.53+/-.00005       0.0255+/-.00005 1.8868+/-.00005 7               46+/-.00005 0.457+/-.00005      0.0217+/-.00005 2.1882+/-.00005




The slope of the graph represents the experimental velocity, 21.5 m/s.

The theoretical velocity v is equal to (T/mu)^(1/2) =20.62 m/s
This yields a percent error of  4.10%

Boiled down to its purest form, the ratio of the experimental wave speeds is roughly 2:1. This inevitably is also the case for theoretical wave speeds--2:1.  Moreover, this lab supports that the nth harmonic frequency is equal to the product nf, where f is the fundamental frequency.  One should consider case one.  For case one, the third harmonic frequency was measured to be 39 hz.  Similarly, applying f-n = nf where n is equal to three since it is the third harmonic and f= 13, one also gets 39 hz.  For the seventh harmonic in the same case, the frequency was found to be 90.1 hz meanwhile the product nf--n is seven this time--yields 91 hz. These numbers do not perfectly emulate one another but this is the small error pronounced in the experiment rooted in the lil-bit-faulty equipment.  Nonetheless, the datable for the second case--the 0.050 kg case--also  supports f-n = nf.        

Microwave Lab

The purpose of this experiment was to place marshmallows in a microwave and let it ride for 30 seconds and measure the frequency of the microwave, how many photons per second are oscillating in the microwave, and how much pressure do the photons exert on the side of the microwave.

Before continuing, it should be noted that a styrofoam cup of 100 grams of water was also placed in the microwave for 30 seconds as a part of experiment--needed to find total energy in the microwave. 

Note that the dimensions of the microwave--length, with, and height--are 35, 35, and 23 cm respectively.

After the marshmallows were microwaved for 30 seconds, the wavelength was measured for this standing wave 12 +/- 1 cm. Therefore one can calculate the frequency of this microwave using f=c/wavelength.

f= 2.5(10^9) hz
Next, energy per photon E-p = hf, where h= 6.636(10^-34) = E-p =1.66(10^-24) J
E = mc(Tf-Ti) = 100(4.186)(34)   (during the 30 sec, the temp elevated from 23C to 57C) 
E = 14232.4 J 
Therefore the number of photons n is equal to total energy divided by energy per photon n = E/E-p.
n = 8.57(10^27)
Moreover dividing photons by time n/t, one gets the number of photons oscillating per second in the microwave.  Photons per second = 2.88(10^26) n/s

Pressure, Pr, = E/(A*C*t) where A is equal to the are of the microwave wall. Plugging in the numbers yields Pressure total = 3.87(10^-7)  pascals
Furthermore pressure per photon Pr-p = Pr/n = 3.87(10^-7)/(8.57(10^27)) = 4.52(10^-35) pascals.

Fluid Dynamics

The aim of the experiment was to fill up a big bucket that has a small hole drilled at the bottom of it with water to a height h of three inches and measaure the time t it takes for the bucket to empty out 16 ounces or approximately 473.18 ml of water.

	1 st	2nd run	3rd run	4th run	5th run	6th run
Time to empty (sec)	40.68	41.16	41.77	41.65	42.94	42.70

"6 t_iàt_ave= 41.81 sec

Calculating theoretical time:
 
The diameter d of the drain hole was measured to be 0.5 cm and so the radius r is .25 cm. Hence the area of the drain hole is 1.96*10^-5 cm^2.

Next,

from the consveration of mass one gets M1= M2= rA1v1t= rA2v2t= A1v1= A2v2 where A*v is equal to volume flow rate V/t .

Now looking at the diagram of the bucket and using bernoulli's equation one derives that v1--the velocity at which the water exits the small hole of the bucket--is equal to (2gh)^(1/2) where h is equal to Y_2 minus Y_1.  

Putting the equations together: V/t = A1(2gh)^(1/2) --> so t = V/((A1)(2gh)^(1/2))

Therefore t-theoretical = 21.05 sec.

Percent error is there for 49.65 percent which is regretably way,way too large.

This inevitalby suggest that the measured diameter is way off.  Therefore rearraning the equation for time and solving for r one gets 0.001772. The diameter is then 0.003544. The error in the diameter is 29.11 percent. 

Furthermore replacing the new area to find the theoretical-time it took to drain the 473.18 ml of water, one gets t-theoretical = 41.83 sec. This yields a percent error in time of 0.04 percent.

Fluid Statics

The objective was to measure the buyont force of a metal cylinder in three distinct fashions. One may refer to the first technique used as the Water Weighing Method, since the metal cylinder was submerged in water to find buyont force. The second was the Displaced Fluid Method where Archimede's principle was used to calculate the buyont force, and the third is the Volume of Object Method where the volume of the cylinder was used to find the volume of the displaced water and multplied by gravity and the density of water to obtain the buyont force, rho*g*V.

Equipment:

1. Force Probe
2. String
3. Overflow Can
4. Beaker to Catch Overflow
5. Metal cylinder
6. Meter Stick
7. Verinier Caliper or Micormeter Caliper

1) The Under water Method:

To measure the buyont force, one initial used a force sensor and measure net to find the weight of the cylinder in air. This value was 1.101 N. Once that was accomplished, the metal cylinder was drowned in water to find the new tension in the spring, which was 0.730 N. After drawing the free-body diagram of the cylinder, one can find the buyont force.

Here one sees F_b = mg-T.  This value is 0.371 N. 

2) Displaced Fluid Method:

In this case, the metal cylinder was dunked into a graduated cylinder filled with water--to the very, very top.  From there the water was allowed to overflow into a large beaker.  The mass of the beaker before the cylinder was dunked was measured to be 0.1282 +/- 0.00005 kg.  Once all the water from the graduated cylinder was done spilling into the beaker, the mass of the beaker plus water was obtained.  This value was 0.1672 +/- 0.00005 kg; hence, the mass of the water was 0.039 +/- 0.0001.

From here one can use Archemides Principle which suggest that the weight of the ater displaced is equal to the buyont force; therefore, the buyont force for the second method was 0.3812 +/- 0.0001 N.

3) Volume of Object Method:

The volume of a cylinder today is defined as pie*r^2*h, where r is the radius of the cylinder and h is the height of the cylinder. These values are 0.076 +/- 0.0005 and 0.025 +/- 0.0005 meters respectively.  Nonethelss these values are used to find the volume of the amount of water that would be displaced if suppose the cylinder was sitting in an arbitrary position in the pacific ocean. The volume is then 3.73 *10^-5 +/- 3.2*10^-6 meters cubed.  Here once can see that the buyont force for this third part is 0.38024 +/- 3.2*10^-6 Newtons by multyplying the volume by the gravity g and density of water rho(note that the density of water is 1000kg/m^3). 

Boiled down to its purest form, the technique that produced the most accurate buyont force was the first method--The Under water Weighing Method--because the error in the force sensors has to be much smaller than the error rooted in humans introduced in the last two methods ideally. So if one had to gamble on either of the three, the first basket should get all of the eggs--or at least the majority of them.