Experiment # 7: Introduction to Reflection and Refraction

Using a light box and a half cylinder of acrylic, it is possible to observe light's behaviour when it changes mediums.

 

In this experiment, the acrylic object is rested on top of a paper protractor. Then the incident and refracted angles are recorded.

As always, the data associated with this experiment can be found in my Google documents folder here. The uncertainty associated with each angle measurement is plus or minus a half degree.

From the data, it appears that there are two observable trends as illustrated by the following graphs.

These graphs represent a plot of the refracted angle (Theta 2) vs. the angle of incidence (Theta 1). The graph which is titled air to acrylic represents the data collected when the light hits the flat part of the acrylic first, and the chart titled acrylic to air represents when the light hits the rounded side of the acrylic first.

First thing of note, both of these graphs have a relatively linear trendline, as demonstrated by their regression numbers(R^2 on the graph). Unfortunately, they both have non-zero intercepts with the vertical axis. Also, both appear to have some curvature in their data points as the angle increases.  This leads me to believe that the linear relationship between the refracted angle and the incident angle only applies when the small angle approximation can be used. (angles less than 30 degrees)

Additionally, it is interesting to note that the slopes of both graphs are nearly the inverse of each other.v(within 10 percent)

When the Sine of the angles are plotted against each other, a clearer picture begins to form.

 

 This relationship has a demonstrably more linear relationship, because the regression numbers are much closer to unity than the straight angle to angle relationship. Additionally, the intercepts with the vertical axis are almost zero, which supports the experimental data. (In both data sets, the point 0,0 appears).

Also, the slopes of these lines are closer to the inverse of each other.(2 percent deviation)

In titling these charts, it is important to note that the description of the interface applies to when the light contacts the linear part of the acrylic disk. When the light enters or exits the acrylic disk through the rounded portion, no refraction occurs because it always has an angle of incidence of zero. This is assuming that the light enters the disk aimed at the mid point of the linear portion of the disk, which is the case for the entirety of both data sets.

Observations:

The slope of the graph was greater when the light travelled from the acrylic to the air. Additionally, the slopes of each graph are very nearly the inverse of one another. This leads me to believe that there is some constant which represents a ratio between the sine of the angle of incidence and the sine of the angle of refraction.

According to Sears and Zemansky's University physics, the relationship between the angle of incidence and the refracted angle does depend on the sine of both angles, and is modified by 2 constants. It takes the following form:

where v1 is the velocity of light in medium 1 v2 is the velocity of light in medium 2 and n1 and n2 are some constants which describe the refractive property of a given medium.

Air is said to have a refractive index of nearly 1, so it is apparent that the acrylic disk used for the experiment has a refractive index of 1.466. Furthermore, this measurement is consistent with reported indexes of refraction for acrylic glass which fall in the range of 1.490 - 1.492. (error =1.6-1.7%)

TEST OF MATLAB PUBLISHING

Contents

• LOADING THE RELEVANT DATA
• PLOTTING THE RAW
• PLOTTING THE MEAN
• PLOTTING THE RUNNING AVERAGE

%practice script from the matlab interactive tutorial
%loading data into the matlab editor

LOADING THE RELEVANT DATA

load annual_temps
%LOADING THE RELEVANT DATA

PLOTTING THE RAW

plot(year,annual)
title('HadCRUT3 Temperature Anomaly Measurements')
xlabel('Year')
ylabel('Temperature Anomaly (°c)')
% PLOTTING THE RAW

PLOTTING THE MEAN

annual_avg = mean(annual);
figure
area(year,annual_avg)
title('Average Annual Temperature Anomaly')
xlabel('Year')
ylabel('Temperature Anomaly (°c)')

 

PLOTTING THE RUNNING AVERAGE

k = 1/5*ones(1,5);
five_year_avg = convn(annual_avg,k,'valid');
plot(year,annual_avg,'--o')
hold on
plot(year(3:end-2),five_year_avg,'r')
legend('Annual Avg', 'Five Year Avg')

Published with MATLAB® 7.13

Wave length, frequency, tension lab

Data for all of the following graphs can be found here.


In the previous experiment, a relationship was found between wavelength and period. In the following experiment, we establish that this relationship also applies to frequency, and there is a relationship between these variables which depends on the tension in the vibrating medium.

The experimental apparatus can be observed in the following video.

The set up consists of a mechanical oscillator, a function generator, a length of string, a pulley and a weight.

The string is attached on one end to the mechanical oscillator, and across the pulley to the weight on the other end. The weight serves to maintain a nearly constant tension for the duration of the experiment.

The function generator is set to generate sinusoidal voltages of varying frequency. This frequency is adjusted until a given number of anti-nodes is established in the pattern of oscillation.

For a given tension, at least 6 different standing waves were tested by varying the frequency, these standing waves represent a specific wave length as measured by comparing the space between nodes to a meter stick. A table with the relevant data is available at the top of this post.

Observations: A plot of the frequency vs. wavelength results in a plot which resembles functions of the form;
frequency = some constant divided by wavelength.

when plotted against the reciprocal of wavelength, a linear relationship is observed.

when a linear regression is added to the chart, it is evident that in order for the units in the function to make sense, the coefficient of x must have units of meters per second. (owing to the fact that 1/wavelength has units of meters, and frequency has units of 1/seconds.)

Thus, the coefficients from the linear regression can be converted to velocities. It is important to note that the units from the above graph are in cm not meters.
These velocities can be plotted against the corresponding tensions in order to establish a trend. 

In the above graph, it is evident that the blue points, (the plot of velocity vs. tension) very nearly adhere to the function f(x)=30.47x^.48.

The equation which models the velocity of a wave through a medium is:

where v is the longitudinal(parallel to the medium) velocity of the wave, and mu is the linear density of the medium. The linear density of the string was determined to be .00119 kg/m.

Also visible in the graph is a plot of the theoretical velocities which correspond to the given tensions. The function which most closely matches those values is f(x)=28.99x^.50.

The equations are similar in that the coefficient of x differs by approximately 5 percent, and the power of x differs by approximately 4 percent. This indicates a strong relationship between the measured data and the theoretical model.

A comparison of the ratio between the real velocities and the predicted velocities can be observed in the aforementioned spreadsheet.

For each step of the harmonics measured, it was observed that the required frequency was always nearly a whole multiple of the first measured frequency. (15,30,45,60...) This indicates that once the first harmonic of the apparatus is discovered, finding the requisite frequencies for any other harmonic is as simple as multiplying the first frequency by the number of the harmonic desired.

wavelength lab

The fundamental relationship between wavelength and period can be studied with a length of spring and a stopwatch.

This movie shows a standing wave set up in a spring. The benefits of using a standing wave to study wavelength are that the distance from person to person is a good approximation for the wavelength, and frequency measurements can be simplified.

For this experiment, our procedure is as follows:

• two students stand approximately 4 meters apart
• each student takes one end of a 4 meter spring and allows it to hang between them
• one student sets up an oscillation in the spring until a given number of anti-nodes are established
• a stopwatch is used to record the time for the standing wave to oscillate ten times
• the time is divided by ten in order to determine the period of one oscillation

The relationship between period on wavelength was determined for 4 data points.
all wavelengths are accurate to plus or minus .1 meter
all times are accurate to plus or minus .15 seconds
wavelength              period
8 meters                  1.04 seconds
4 meters                  .587 seconds
2.66 meters             .350 seconds
2 meters                  .262 seconds

these data points, when plotted, imply a linear relationship between wavelength and period.

This supports the formula for a standing wave, where the velocity of a wave in a given medium is a constant based on its linear density and the tension in that medium, and that velocity is equal to the wavelength divided by the period. This relationship is of the form;

Put a Hole in the Bucket Dear Liza...

Experiment 2: Fluid Dynamics

The equation which models the behaviour of a laminar, non-viscous, incompressible fluid flow is as follows:

This equation can be used to determine the theoretical velocity of water exiting a hole in a bucket.

For this experiment, we measured the rate at which water flowed out of a bucket, by recording the amount of time it took to fill a 1000 mL beaker. Data follows:

Bucket Diameter = 25 cm
Initial height of Fluid (above hole) = 12.4 cm
Area of drain hole = .31669 cm^2
Volume emptied = 1000 mL

Trial 1 = 30.01 s
Trial 2 = 29.35 s
Trial 3 = 29.59 s
Trial 4 = 29.39 s
Trial 5 = 29.82 s
Trial 6 = 29.38 s

This data indicates that the time to empty 1000 mL of water should be 29.59 ± .25 seconds. This equates to a flow velocity of:
1000cm^3/.31669cm^2=3157.66cm
3157.66cm/29.59 s=1.06m/s

The predicted velocity, from Bernoulli's equation should be close to the square root of 2*g*height of the water.
that results in a theoretical velocity of:
1.559 m/s

This velocity predicts that the volume of water should be emptied in 20.24 seconds, which represents nearly a 33% deviation from the measured time.
Even if the height of the water is adjusted to reflect the average height over the duration of the experiment, the theoretical time to completion only changes by 1.25 seconds.

These findings lead me to believe that some part of our model is incorrect, for example, the size of the drilled hole may not be exact.
The given data predicts that the cross sectional area of the hole may be in error by as much as 30 percent. This could be caused by the plastic deforming during the drilling process, and leaving a lip which blocked some of the flow.