Experiment 3: Mechanical Waves

In this lab we were using mechanical waves to examine the relationship between the wavelength(λ), The period (T), and the frequency(f). We used a spring that was had a length of 1.0m + 0.1m. We made the spring produce a wave over an interval of 5 seconds + 0.5s. During this time, we counted the amount of waves that passed a point. This would give us the number of waves per second.

In our first trial of this lab we noticed 13 waves in 5 seconds, This results in a frequency of 2.65 Hz + 0.2 Hz. The second trial and third trial had 11 waves in 5 seconds, this gives us 2.25 Hz + 0.2 Hz. With this value of the frequency we can determine the velocity of the wave. The velocity can be found from this equation, v = fλ. 

The velocity from the first trial is 2.65 m/s + 0.593 m/s and the second and third trials have a velocity of 2.25m/s + 0.503 m/s . 

The error from the velocity can be determined by partial derivatives. 

ΔV = sqrt ((∂f/∂v)^2*Δf + (∂λ/∂v)^2*Δλ)) =  0.593 m/s

The error of this lab came from the number of waves determined from the human eye. The waves were moving at a fast enough rate for us to easily lose count. 

The graph of these values will be on a vertical line. The slope cannot be evaluated at this point because all of the lab times were taken at the same time.

Experiment 6: Speed of Sound

In this experiment our goal is to measure the speed of sound in air. We will be doing this by using hollow tubes and measuring the sound that comes out of them. The hollow tube will be spun around fast enough until it produces a noise. We will record two different noises at different octaves with the tube and record both 

sounds. After analyzing the sound vs. time and approximating it with logger pro we determined that the angular frequency of the two different sounds were
ω_1 = 3859 rad/s  + 0.4736 rad/s, ω_2 = 5068 rad/s + 1.106 rad/s.
Using this formula ω = 2πf, where f is the frequency, we can rearrange this equation into f = ω/2π and we can determine that f_1 = 614 Hz + 0.753 Hz and f_2 = 806 Hz + 0.176Hz.
We can also determine the wavelength by using the formula v = fλ. In this equation v is the speed of sound and λ is the wave length. When we rearrange this equation we have v/f = λ. λ_1 = 0.554 m + 0.005m and λ_2 = 0.422m + 0.005m.
We also know that λ_1/2 * (2n+1) = L and λ_2/2 * (2n+3) = L, n is the number of harmonics and L is the total length of the wave. To find n we can rearrange the two equations together and it results in λ_1/λ_2 = (2n+3)/(2n+1).

Conclusion

We began by finding the angular frequency with loggerpro and find the frequency from that information. Once inputting our values we can determine that n is 2.694 which we can round up to 3.
To determine the length of L we can use  λ*n/2 = L
L = 0.831m + 0.015m
We could not measure the actual length of the pipe so percent error cannot be calculated. The error was determined by adding up the associated errors.

Experiment 9: Measuring a Human Hair

In this experiment we were measuring the thickness of a human hair. This task is tough to do by hand so we found out the thickness by using a laser. We were using HeNe lasers during this experiment.To begin this experiment we used one strand of hair and taped it to a 3x5 card with an opening in it.

We shined the laser through the hole in the card and hit the strand of hair to produce different interference patterns. The light will be the strongest in the middle and dissipate as the number of minima increase. We can measure the distances of the gaps in the interference patterns to help us determine the thickness of the hair. We initially measured the distance of interference patterns while using a marker.

We measured the distance between the different minima with a much finer tool than a marker. We changed the tool of measurement to a pen and measured the distance between multiple minima and divided it by the total number of minima we measured across. 

                                                     

We tested this with two different people's hair. We first tested this with a male student's strand of hair. We can analyze the data with this formula.  λ = dy/L. In this equation λ is equal to the wavelength of the laser. d is the thickness of the hair. y is the distance between the minima and L is the distance of the laser to the whiteboard. We rearrange this equation for d because it is the value that we are looking for and the result is d = λL/y. λ = 632.8 nm This laser is calibrated under extreme conditions so we assume that there is no uncertainty. L is 1.00m + 0.05m. y for the first strand of hair is 0.00516 m + 0.0002m. The value of d for the first trial is 122.6 μm + 30.8 μm The uncertainty for this result was determined by using partial derivatives. 

Δd = + sqrt((∂d/∂λ)^2*(Δλ)^2 + (∂d/∂L)^2*(ΔL)^2 + (∂d/∂y)^2*(Δy)^2) 

= + sqrt((L/y)^2*(Δλ)^2 + (λ/y)^2*(ΔL)^2 + (-λL/y^2)^2*(Δy)^2) 

This formula reduces to + sqrt((λ/y)^2*(ΔL)^2 + (-λL/y^2)^2*(Δy)^2)) because Δλ = 0. 

Δd = 30.8 μm 

In the second trial of the experiment we used a female students hair instead. 

Only the value of y changed. y = 0.0025 m + 0.0002m.

d was determined to be 253.1 μm + 63.6 μm. 

Conclusion

The hypothesis for the strand of hair being much thicker in this is case is because the female student used more hair care products than the male student. When measuring objects with a really small thickness, it is best to measure them with a laser. When a laser is shined on the object it will produce interference patterns on a screen and the distance of the maxima or minima can be measured. This is more accurate than measuring the object by hand.

Experiment 7: Concave and Convex Mirrors

In this experiment we analyzed the differences between concave and convex mirrors. We will begin to analyze them by looking at how images appear when looking through the mirrors. We looked at convex mirrors first. Convex mirrors are shaped as if they were pushed outward from a thin straight glass plate.

In the convex mirror the image appears smaller than it's actual size. The image in the mirror still appears upright after looking in the mirror. The image in the mirror appears to also be further away than when the object distance from the mirror. When the object is moved closer to the mirror it appears closer to its true size and is slightly curved. 

The ray diagram agrees with the observations that we had. The images look smaller the further away the object is. The image height is much smaller than the original object height in the ray diagram. The top ray bounces off at an angle and the reflection goes through the focal point of the mirror. The second ray goes through the center and reflects back in the same direction it came in at. The third ray goes down and bounces off parallel to the optic axis. It also reflects in the mirror parallel. These three rays meet and create an image that is smaller than the actual object. These are the measured values on the ray diagram.
h_0 = 2.9 cm + 0.1 cm. h_i  = 0.7 cm + 0.1 cm d_0 = 5.9 cm + 0.1cm d_i = 1.9 cm + 0.1 cm.
The M calculated would be 0.241 + 0.2.

Convex Mirror Ray Diagram

After analyzing convex mirrors, we turned our attention to concave mirrors. Concave mirrors are curved inwards from the shape of a glass plate.

The image in a concave mirror behaves differently than the convex mirror. The image appears larger than the original. It is also inverted if you are far away but the picture becomes upright when you get close to the mirror. When you are relatively close to the mirror about under 1 meter, The object appears larger than it actually is. When you are far from the mirror, the object is inverted but looks like its true size. 

A ray diagram can also describe the phenomena of the concave mirror. 

Concave Mirror Ray Diagram

This ray diagram also agrees with the observations. In this diagram the top ray bounces off the mirror and goes through the focal point. The second ray goes through the focal point and reflects back parallel to the optical axis. The third ray goes through the radius of curvature and goes straight through. The place where the three rays meet is where the image is created. These are the measured values on this ray diagram.
h_0 = 3.1 cm + 0.1 cm h_i = 0.8 cm + 0.1 cm d_0 = 11.6 cm + 0.1 cm d_i = 2.5 cm + 0.1 cm.
The calculated M would be 0.258 + 0.2

Conclusion

Convex mirrors and concave mirrors behave differently. When objects are inside the focal length of a concave mirror, they are large and inverted. When objects are far from the concave mirror, outside of the focal length, they appear inverted and smaller. Convex mirrors produce real images but they are not inverted. The also produce virtual images inside of the mirror.

                                                     

Experiment 8: Lenses

In this experiment we will observe how lenses behave under certain situations. We began by getting a lens from a box but the focal length was unknown so we had to measure it. When we measured the focal length we used the sun as an infinite light source and measured the distance from the lens to the focal point observed on the ground. We observed that the focal length was 9.5 cm + 0.3 cm.

Measuring the Focal length

Once we obtained the focal length. We placed the lens four focal lengths away and observed the image that was projected by it. This distance was 38.0 cm + 0.2 cm. The distance of the image from the lens was measured to be 19.0 cm + 0.2 cm. The height of the filament remained the same throughout the experiment but it was measured to be 8.8 cm + 0.1 cm.  The image height was measured to be 4.4 cm + 0.1cm. From this information we can determine the magnification of the lens at this distance. The formula for magnification is, M = h_i/h_0, or M = -d_i/d_0 where h_i is the height image and h_0 is the height of the object. d_i in this formula is the distance from the lens to the mirror and d_0 is the distance from the light source to the lens. When we calculated all of our magnifications we used the ratio of the different heights. M = 0.50  + 0.2  from this trial. The magnification lower than 1 provided a sharper image. The was inverted and it was a real image. When the lens was reversed the results were the same, it remained inverted and real. The error obtained from these measurements was determined by how inaccurate we believed our measurements to be. In the calculation for magnification we added the errors together from the previously measured objects. This process was used for error analysis throughout the entire lab.

Magnification at 0.50 

In the next trial we moved the lens so that it was two focal lengths away. This distance was 19.0 cm + 0.2 cm. The distance to the image from the lens was 26.0 cm + 0.2 cm. The height of the image was 14.5 cm + 0.1 cm. The magnification was calculated to be 1.648 + 0.2. Because the magnification was larger than 1, the image appeared less focused. The image was also inverted and real in this trial. When the lens was reversed the results were the same.

Next we then moved the lens one and a half focal lengths away from the filament which was 14.3 cm + 0.2 cm. The distance measured from the lens to the image was 23.45 cm + 0.2 cm. The height of the image was measured to be 16.5 cm + 0.1 cm. The magnification calculated from this trial was 1.875. The image was inverted and real and appeared to be unfocused because of the magnification. We also covered the lens halfway with masking tape and observed the differences. It uses the remaining lens to project the image. The image was still completely projected onto the whiteboard but it was dimmer than the previous image because there is less lens for the light to go through.

Magnification at 1.875

Masking tape at 1.875 magnification 

We repeated this process two more times. The distance used was three times the focal length.

d_0 = 28.5 cm + 0.2 cm, d_i = 16.5 cm + 0.2 cm, h_0 = 8.8 cm + 0.1 cm. h_i = 6.6 cm + 0.1cm. 

M = 0.75 + 0.2. The image was inverted and real.   

Magnification at 0.75

The next trial we used the distanced to be five times the focal length.

d_0 = 47.5 cm + 0.2 cm, d_i = 16.5 cm + 0.2 cm, h_0 = 8.8 cm + 0.1 cm, h_i = 3.3 cm + 0.1cm.

M = 0.375 + 0.2. The image was again inverted and real. 

Magnification at 0.375

On our last trial we moved the the lens to half of its focal length. When we moved it to 4.25 cm + 0.2 cm the image was no longer visible. The image could only be seen through the lens and was not inverted.

After all of the trials the data was graphed the first graph is d_i vs d_0. 

The next graph we created was the inverse of d_i vs. the negative inverse of d_0. 

The slope of this graph is 1.5984 and the y-int 0.1225. The y-int represents the length of the virtual image that is created.

Conclusion 

When the lens was placed at different distances from the filament it different effects on the image. When the lens was in between 1 and 2 focal lengths the magnification was greater than one which made a bigger, less focused, image. When the lens was placed further than 2 focal lengths,the magnification was less than one and it produced a much sharper image. When the images were real, they were inverted. The image was virtual when the lens was placed at half of the focal length and would be virtual if it is placed at a distance under the focal length. 

Experiment 5: Introduction to Sound

In this experiment we recorded different sounds and looked at the graphs of their sound pressure vs. time. This helped us examine the different patterns of different sounds. We began by recording my voice along producing the noise "AAAAAAAAA".

This graph represents the sound pressure created when this noise is produced over an interval of 0.03s. This wave is periodic because the period of the wave dissipates over time. There are also three waves shown in this picture because there are three peaks shown that dissipate over time. This interval of 30 times per second is equivalent to how fast a hummingbird can flap its wings. This depends on the species of the hummingbird, some can go faster or slower. The period of these waves are 0.01s +/- 0.005s. This was determining by examining the graph and looking at how long it takes for the graph to do one full cycle. The frequency of the wave can be determined by this equation. f = 1/T. f is the frequency and T is the period. When we input the value for the period the frequency is 100 Hz +/- 200 Hz. 200 Hz uncertainty was calculated by converting the error from the period. We can determine the wavelength of the wave by using this equation. v=fλ, v is the speed of sound, the frequency of the wave which also be expressed as 1/T. λ equals the wavelength so the final equation is. vT=λ and λ = 3.4m. +/- 0.01m. This value of uncertainty was determined by partial derivatives. Error = sqrt(∂T/∂v^2 + ∂v/∂T^2)  ∂v/∂T in this case is zero because the speed of sound is absolute. Something in the classroom that is about that size would be the size of the keyboard on a laptop. The amplitude of the wave cannot be measured in this experiment.

One of my lab partners then recorded his voice and we also analyzed the data. 

The sound waves of his voice dissipate slower than the waves from my voice did and there are also more waves when he recorded his voice. This wave is also periodic. There are four waves analyzed in this time interval as opposed to three. The period of a wave would be 0.0075s +/- 0.005s. The frequency of a wave would be 133 Hz +/- 200Hz. The wavelength of the wave would be λ = 2.55m +/-  0.01m. This value was obtained the same as before.                  

We also analyzed the sound of a tuning fork. 

This data was produced when the tuning fork was producing a loud noise. The tuning fork produces a wave that looks like an ideal sine wave. It vibrates at a steady rate as opposed to the human voices. There are seven waves in this data of the tuning fork. The period is 0.004s +/- 0.005s. The frequency of the tuning fork is 250 Hz +/- 200Hz. The length of the wave is λ = 1.36m +/- 0.01m

We also recorded the tuning fork when it rang at a softer noise. We produced a softer noise by striking the tuning fork with less force. The period of this noise is the same as above, 0.004s +/- 0.005s. The frequency  and wavelength are also the same at 250 Hz, and λ = 1.36m +/- 0.01m

The amplitudes of these two waves vary because they are vibrated at two different intensities.

Conclusion

A certain sound like "AAAAAAA" has the same shape no matter who makes the sound. The number of the waves created in a small interval varies based on the frequency of the person voice. A person with a deeper voice will produce less waves because it is at a lower frequency as opposed to someone with a higher pitched voice. The tuning fork however produces the same amount of waves not dependent on how hard it was struck. The amplitude of the waves differed depending on how loud the tuning fork was ringing at.