Measuring a Human Hair Lab

Objective: Determine the thickness of a human hair by observing the interference pattern created when a laser beam passes through a single strand of hair.

Lab Equipment: 


- index card
- strand of hair (at least one group member must not be bald)
- helium neon laser
- meter stick
- micrometer

Lab Procedure 


1) Punch a hole in the index card and tape the strand of hair across the hole. Make sure that the hair is straight.

2)  Clamp the card at least 1 meter away from a wall so that it is parallel to the wall.

3) Mount the laser so that the beam is perpendicular to the wall.

4) Adjust the card so that the beam falls directly on the hair. This should produce a diffraction pattern on the wall. Measure the distance between the card and the wall and the spacing between the 2 first order minima of the pattern.

5) Calculate the width of the hair using these measurements. Confirm this result using a micrometer.

Data:


distance between 2 first order maxima = 2y = 4.1 cm (error = 0.04 cm)
distance from card to wall = L = 180 cm (error = 0.5 cm)
wavelength of laser beam = lambda = 632.8 nm (error = 0.1 nm)

d*sin(theta) = m(lambda)

theta = arctan(2.05/180)

d = m(lambda)/sin(theta) ~ m(lambda)(L)/y = 55.5 micrometers (error = 0.73 micrometers)

Using the micrometer we found the width of the hair to be about 60 micrometers. This does not fall within our predicted interval. However, we had trouble figuring out how to use the micrometer and the discrepancy was most likely due to our inability to use the instrument.

Lenses Lab

Objective: Measure the properties of a converging lens observe the characteristics of images created by converging and diverging lenses.

Lab Equipment:

- Socket Lamp with V-shaped filament (filament is asterix-shaped)
- converging lens
- object (large hanging mass)
- masking tape
- lens holder
- large piece of cardboard (or other dark material)
- meter stick

Lab Procedure:

Part 1:

1) Obtain a converging lens and measure the focal length using a light source (the sun) located very long distance from the lens.
2) Place the lens in the lens holder on the table. position the lens holder about four focal lengths away from the filament. Move the cardboard screen in the path of the light which has passed through the lens. Move the cardboard screen back and forth until a sharp image occurs. Record the distance between each object and other observations specified in the lab manual.
3) Reverse the lens and note any changes in the image.
4) Repeat this process with the lens placed 2 focal distances and 1.5 focal distances away from the filament.

Data:

f = focal length, di = image distance, do = object distance, hi = image  height, ho = object height,
M = magnification = (hi)/(ho)
1) do = 60 cm, di = 21 cm, ho = 9 cm, hi = 3 cm, M = 0.333

The image is real and inverted.  When the lens is reversed all previous measurements of the image remain the same.

2) do = 30 cm, di = 31 cm, ho = 9 cm, hi = 9 cm M =1.00

 The image is real and inverted.  When the lens is reversed all previous measurements of the image remain the same.

3) do = 22.5 cm, di = 46 cm, ho = 9 cm, hi = 17.5 cm, M = 1.944

 The image is real and inverted.  When the lens is reversed all previous measurements of the image remain the same.

4) When half of the lens is covered in masking tape the image becomes much dimmer, but the shape and orientation of the image remain intact. We think that this occurs because every individual point on the lens projects the entire image out into the opposite side. covering any part of the lens will simply decrease the intensity of light going through.

Part 2:

1) Place a distinguishable object in front of a light source.
2) Using the same lens as in part 1, place the lens 5 focal lengths away from the object.
3) Measure the image distance, image height, and determine the type of image and its inversion.
4) Repeat for  a lens placed 4, 3, 2, and 1.5 focal lengths away from the object.

Data:

f = focal length, di = image distance, do = object distance, hi = image  height, ho = object height,
M = magnification = (hi)/(ho)

f = 15 cm

1) do = 75 cm, di = 18.5 cm, ho = 11.5 cm, hi = 2.5 cm, M = 0.217

2) do = 60 cm, di = 20 cm, ho = 11.5 cm, hi = 3 cm, M = 0.261

3) do = 45 cm, di = 22.5 cm, ho = 11.5 cm, hi = 4 cm, M = 0.348

4) do = 30 cm, di = 28 cm, ho = 11.5 cm, hi = 7 cm, M = 0.609

5) do = 22.5 cm, di = 35 cm, ho = 11.5 cm, hi = 10 cm, M = 0.870

in each of these trials the image was inverted and real.

when the object distance is decreased to a level below the focal length, the image is virtual and is not inverted.

Swinging Pipe Activity

Objective: Use the frequency of sound emitted by a swinging pipe to determine it's length using sound wave and standing wave equations.

Procedure:

1) Professor Mason will swing the pipe in a circle at a constant angular velocity in order to create a standing wave in the pipe. The frequency will be recorded using a digital microphone.

2) Professor Mason will accelerate the pipe until it emits the next harmonic frequency. This frequency will be recorded using the digital microphone.

3) determine the length of the pipe using these two frequencies assuming that the speed of sound remains constant.

Calculations:

Recorded frequencies: f1 = 614 hz, f2 = 806 hz

n = harmonic number of lower frequency, L = wavelength

(2n + 1)(L1)/4 = (2n + 3)(L2)/4

L1 = v/f1   L2 = v/f2

(2n + 1)(f2) = (2n + 3)(f1)

n = 2.7 ~ 3

Length of pipe = (2n + 1)v/(4f1) = 0.978 (v = speed of sound assumed to be 343 m/s)

Concave and Convex Mirrors Lab

Objective: observe and analyze the images formed by concave and convex spherical mirrors.

Lab Equipment:

- Convex Mirror
- Concave Mirror
- Object (bag of oats)
- Ruler
- Worksheets (substituted whiteboard diagram for ray diagram worksheet)

Lab Procedure:

1) place an object in front of a convex mirror and describe the image characteristics specified in the lab manual.
2) Repeat this process for an object placed in front of a concave mirror.
3) Complete the ray diagrams as instructed in the lab manual.

Observations:

Convex Mirror

a) the image appears smaller than the original object

b) the image is upright in comparison to the original object

c) the image is located somewhere behind the convex mirror

2) When the object is moved closer to the mirror the image appears to approach the object's actual size.

All other aspects of the image remain the same as stated above.

3) When the object is moved further away from the mirror the image appears to shrink in comparison to the object's actual size. All other aspects of the image remain the same.

Convex Mirror Ray Diagram

1) Draw a ray parallel to the optic axis to the mirror. The reflected ray will go in the opposite direction of the focal point.

2) Draw a ray from the object to the center of curvature of the mirror. The reflected ray will go in the opposite direction of the original ray.

3) Draw a ray from the object to the mirror in the direction of the focal point of the mirror. The reflected ray will be parallel to the optic axis.

Measurements: do = 15 cm, ho = 10 cm, di = -7 cm, hi = 5 cm

All of our observations agree with the light ray sketch.

Concave Mirror

(original observations taken at about focal length)

a) The image appears larger than the object.

b) The image appears upright in comparison to the original object.

c) The image is located in front of the mirror.

2) When the object is moved closer to the mirror the image appears to be the same size as the object and the image is located behind the mirror. The image remains upright.

3) When the object is moved further away from the mirror, the image is inverted and becomes much smaller than the original object. The image is located in front of the mirror.

Concave Mirror Ray Diagram

1) Draw a ray from the object parallel to the optic axis. This ray will reflect from the mirror toward the focal point.

2) Draw a ray from the object through the focal point. This ray will reflect from the mirror parallel to the optic axis.

3) Draw a ray from the object through the center of curvature. This ray will reflect   back in the opposite direction from which it came.

Measurements: do = 20 cm, ho = 8 cm, di = 12 cm, hi = -5 cm

our observations do agree with our light ray sketch for objects placed relatively far away from the mirror (more than a focal length).

Microwave Lab Quiz

Lab Quiz: A microwave oven is in the back of the classroom. We have placed marshmallows in the microwave to make some measurements of the standing wave. Determine the frequency of the microwave. From this deduce a range in possible dimensions for microwaves including the smallest possible microwave. We also microwaved a cup of water. What is the total energy content of cavity? How many photons per second are oscillating in the microwave? What pressure do these photons exert on the side of the microwave?

Data:

wavelength = 12 cm (error = 1 cm)

dimensions: height = 23 cm, width = 35 cm, depth = 35 cm (error = 1 cm)

Microwaved cup:
mass = 100 g (error = 1 g)
time elapsed = 30 s (error = 1 s)
initial temperature = 20 degrees Celsius (error = 1 degree Celsius)
final temperature = 57 degrees Celsius (error = 1 degree Celsius)

Analysis:
frequency of microwave = (velocity of light)/(wavelength) = (3.00*10^8 m/s)/(0.12 cm)
                                        = 2.52*10^9 hz (error = 2.1*10^8 hz)

For this frequency the microwave could be any length which sets up a standing wave with a wavelength of 12 cm. The depth and height of the microwave would not matter given that the microwave radiation is emitted uniformly from one side. The width could be any value expressed as a multiple of 12 cm and some positive integer.

total energy for 30 seconds = (Tf - Ti)*(M)*(Cw) = (37)(100)(4.186) = 15500 J (error = 570 J)

Average Power = (total energy)/(time elapsed) = 517 W (error = 20 W)

Photons oscillating per second = (Average Power)/(energy of individual photon) = 517/(h*f)
h = planck's constant = 6.6261*10^(-34) kg*m^2/s, f = frequency = 2.52 Ghz

Photons oscillating per second = 3.12 * 10^26 (error = 2.6 * 10 ^25)

We assume that all surfaces of the microwave are perfectly reflective.
The pressure exerted on a perfectly reflective surface = I/c where I is the intensity given by the pontying vector

Assuming that pressure is exerted on all sides of the microwave equally we can approximate the electromagnetic intensity as the total power in the cavity divided by the surface area of the cavity.

Surface Area = 2WD + 2WH + 2DH = 2(0.23*0.35 + 0.23*0.35 + 0.35*0.35) = 0.5817 m^2
(error = 0.0375 m^2)

Intensity = Power/SA = 517/0.5817 = 894 W/m^2 (error = 93 W/m^2)

Pressure = I/c = 2.98*10^(-6) N/m^2 (error = 3.1 *10^(-7) N/m^2)

Sound Lab

Objective: Observe and analyze the characteristics of different waves created by tuning forks and the human voice

Lab Equipment:

- Lab Pro with Microphone

- computer

- tuning fork

Data Collection:

For this lab we needed to collect five different wave patterns using the Lab Pro setup. The setup consisted of a microphone that detected pressure variations in the air connected to a computer. Three wave patterns were human voice recordings of two members of our group. The other two wave patterns were created by striking a tuning fork near the microphone and recording the pattern. Below are the collected wave patterns and corresponding analysis.

Graph #1

This is a graph of one group member saying "AAAAA" over a collecting period of 0.03 seconds.

Questions:

a) This wave does appear to be periodic since the collected wave has a fairly consistent repeating pattern between the highest peaks.

b) Approximately 3.5 waves are shown in this sample. We determined this number by measuring the length of the total display and the distance between peaks on the computer screen and dividing the first by the second.

c) The length of this data collection period could be compared to the duration of a single frame on a traditional television which lasts about 0.02 seconds.

d) Period = T = (collection time)/(# of wave repetitions) = 0.03/3.5

               = 0.0085 seconds (error = 0.0005 s)

e) Frequency = 1/T = 1/0.0085 = 118 hz (error = 7 hz)

f) wavelength = (speed of sound)/(frequency of wave) = 340/118 = 2.9 m (error = 0.19 m)\

g) Amplitude = ((max wave height) - (min wave height))/2 = (3.5 - 1.8)/2 = 0.85 (arbitrary units of pressure)

h) increase the collections period to 0.3 seconds

Graph # 1h

This is a graph of the same group member's voice over a period of 0.3 seconds

The only part of the wave that seems to change is the amplitude. The wave appears to oscillate about 36 times in 0.3 seconds. This gives us roughly the same period, frequency, and wavelength as in the last trial. The amplitude did decrease to about 0.25, however, this is most likely due to a change in the volume of the student's voice or the distance to the microphone.

Graph #2

This is a graph of another group member speaking into the microphone.

a) # of waves = 4

b) frequency = (#of waves)/(time elapsed) = 133 hz (error = 7 hz)

c) Period = 1/frequency = 0.0075 s (error = 0.0004 s)

d) amplitude = 0.075 (arbitrary units of pressure)

e) wavelength = 2.56 m (error = 0.13 m)

\

Graph #3

This is a graph of a tuning fork after striking a soft object collected over 0.03 seconds

a) # of waves = 7.5

b) frequency = (# of waves)/(time elapsed) = 7.5/0.03 = 250 hz (error = 10 hz)

c) period = 1/frequency = 0.004 s (error = 0.0002 s)

d) amplitude = 0.1 (arbitrary units of pressure)

e) wavelength = 340/250 = 1.36 m (error = 0.06 m)

waves made by the tuning fork appeared to be simple harmonic waves whereas the human voice waves were the superposition of several different sinusoidal waves.

\

Graph #4

This is a graph of the same tuning fork striking a soft object and being slightly muted over a collection period of 0.03 seconds

a) # of waves = 7.5

b) frequency of waves = 250 hz (error = 10 hz)

c) Period of wave = 0.004 s (error = 0.0002 s)

d) Amplitude = 0.02 (arbitrary units of pressure)

e) wavelength = 1.36 m (error = 0.06 m)

this wave was nearly identical to the wave shown in graph #3. The only difference was that the amplitude of this wave was much less than that of the previously collected wave.

Standing Waves Lab

Objective: To gain knowlesge and understanding of standing waves driven by an external force and investigate conditions for resonance on a taught string.

Lab Equipment:

- Frequency wave driver
- long string
- Function Generator
- Weight hanger and slotted weight set
- 2 table clamps
- short rod
- pendulum clamp
- pulley
- digital multimeter
- meter stick

Lab Set-up:

1) measure and record the mass and length of the string
2) tie two loops in the string more than 1 meter apart
3) set up the pendulum clamp and pulley clamp slightly more than 0.7 cm apart
4) attach the string to the pendulum clamp and thread it through the pulley. attach the mass hanger to the hanging end of the string.
5) thread the string through the wave driver. measure and record the distance from the wave driver to the pulley.
6) connect the wave driver to the function generator

Data Collection:
1) adjust the frequency of the function generator until the string oscillates in the fundamental mode. Record the oscillation frequency, the number of nodes, and the length of string participationg in the oscillation. Determine the distance between nodes and wavelength of oscillation.
2) repeat for at least the first 7 harmonics
3) change the tension in the string by changing the hanging mass and repeat steps 1 and 2 for this new tension.

mass of string = 7.5 g (error = 0.02 g)
length of string = 90.6 cm (error = 0.2 cm)
mass per unit length = 0.00826 kg/m (error = 0.0004 kg/m)

Case 1

hanging mass = 0.55 kg (error = 0.001 kg)
Length of vibrating string = 0.61 m (error = 0.005 m)
Tension = 5.4 N (error = 0.01 N)
wave speed = 25.6 m/s (error = 1.0 m/s)

f = frequency, l = wavelength, x = # of nodes, n = x - 1

1) f = 21 hz, x = 2, l = 1.22 m, n = 1

2) f = 43 hz, x = 3, l = 0.61 m, n = 2

3) f = 64 hz, x = 4, l = 0.407 m, n = 3

4) f = 85 hz, x = 5, l = 0.305 m, n = 4

5) f = 108 hz, x = 6, l = 0.244 m, n = 5

6) f = 129 hz, x = 7, l = 0.203 m, n = 6

7) f = 152 hz, x = 8, l = 0.174 m, n = 7

8) f = 174 hz, x = 9, l = 0.153 m, n = 8

9) f = 195 hz, x = 10, l = 0.136 m, n = 9

Plot f vs 1/l and find the slope of the linear fit. compare this value to the predicted wave speed.
Using a stat plot and least squares regression line we find:

y = 26.69x - 1.39 (y = f, x = 1/l)

the slope of the line (26.7) is within the margin of error of our prediction for the wave speed.
Case 2

hanging mass = 0.3 kg (error = 0.001 kg)
Length of vibrating string = 0.61 m (error = 0.005 m)
Tension = 2.94 N (error = 0.01 N)
wave speed = 18.9 m/s (error = 0.9 m/s)

1) f = 16 hz, x = 2, l = 1.22 m, n = 1
2) f = 32 hz, x = 3, l = 0.61 m, n = 2

3) f = 48 hz, x = 4, l = 0.407 m, n = 3

4) f = 64 hz, x = 5, l = 0.305 m, n = 4

5) f = 80 hz, x = 6, l = 0.244 m, n = 5

6) f = 96 hz, x = 7, l = 0.203 m, n = 6

7) f = 112 hz, x = 8, l = 0.174 m, n = 7

8) f = 128 hz, x = 9, l = 0.153 m, n = 8

9) f = 145 hz, x = 10, l = 0.136 m, n = 9

plot f vs 1/l and find the slope of the linear fit. compare this value to the predicted wave speed.
Using a stat plot and least squares regression line we find:

y = 19.65x - 0.367

the line slope (19.7) is within the margin of error of our predicted wave speed.

Additional Analysis:

- the ratio wave speeds obtained from the graphs for the first and second case is 1.36. The ratio of the predicted wave speeds is 1.35. These values are very close.

- The standing wave frequency for each case was approximately equal to n*(fundamental frequency) for each of the harmonics. There were very slight deviations since the frequency generator only increased by unit values.

- compare the corresponding harmonics for the two cases
ratio 1 = 21/16 = 1.31
ratio 2 = 43/32 = 1.34
ratio 3 = 64/48 = 1.33
ratio 4 = 85/64 = 1.33
ratio 5 = 108/80 = 1.35
ratio 6 = 129/96 =  1.34
ratio 7 = 152/112 = 1.36
ratio 8 = 174/128 = 1.36
ratio 9 = 195/145 = 1.34

each of these ratios is very close to the ratio of the wave speeds of the two cases.