slinky machine paper - final

8/14/2019 Slinky Machine Paper - Final

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Physics 211 Vibrations, Waves and Optics

Project 3 Presentation The Slinky Machine

The Slinky Machine Experiment : Analysis and Understanding

Ilia Papa

Abstract

The following article is an analysis and explanation of the dispersion relation for

a mechanical wave on a slinky. The remarkable aspect of the article is the way in which

the waves frequency is correlated to its wavenumber and hence, to its wavelength. In

order to do that we set up the experiment in which a slinky was attached on a rod by

several strings and we started driving it at different rates. After we conducted the

experiment and we got our results, we plotted the graph of angular frequency vs. the

wavenumber which also helped us estimate the cutoff frequency. Also, from the

theoretical analysis we managed to determine the equations for the group and phase

velocity and the cutoff frequency. We compared the cutoff frequency with the

experimental one that we determined from the graph and we were finally able to draw

some conclusions for all the possible errors.

I. Introduction

The term dispersion refers to a mechanical wave which results when different

frequencies propagate at different velocities. The dispersion relation can be illustrated

in a plot of vs. k, where is the angular frequency and k is the wavenumber.

Therefore, from the graph we can see how the frequency depends on the wavenumber

(=(k)). We distinguish the velocity of the wave into two different ones which

represent the high-frequency carriers propagation (phase velocity) and the low-

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frequency envelopes propagation (group velocity). In order to write down the exact

formula for phase and group velocity we first need to define (k), something which is

shown below in the theoretical analysis. Moreover, it is feasible to obtain both the

phase and the group velocity from the graph, besides calculating them with the use of

formulas (the way is explained in more detail below). Furthermore, before proceeding

to the experimental procedure, an understanding of the wave cutoff frequency is

necessary. If the strings supporting the slinky were too short then it would be

impossible to observe a wave propagation for a very low frequency 1. Therefore, the

critical frequency which determines whether there will be propagation or not is called

cutoff frequency.

Finally, it is reasonable to assume that the motion of the slinky satisfies an equation

which will be similar to the wave equation

0

2 22

2 2

y yv

t x

=

(1).

However, since there is presence of gravity and the strings, the wave equation that

satisfies the above motion would be

0

2 22 2

02 2

y yv y

t x

=

(2)

in which

2

0

s

g

L = (3)

where 0 is the angular frequency, g is the gravity of Earth and Ls is the length of the

string,

0

2 T

v = (4)

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where v0 is the velocity of the wave, T is the period and is the mass density,

provided that sy L=1

II. Theoretica l Analysis

The next step that would complete our understanding of the dispersion relation is to

theoretically analyze the aspects of this experiment based on equation 2. The

difference between equation 2 and the wave equation is that in this case we take into

consideration the presence of gravity and the strings. As it is explained in the Wave

cutoff on a suspended slinkyarticle, the string connecting the slinky with the rod has a

mass of dx and experiences a restoring force of Td(dy/dx), where T is the tension

and y is the displacement. Additionally, there is a restoring force due to gravity

which is illustrated as (dx)gy/L1. The results of the addition of these two forces are

equations 3 and 4, while the wave equation takes the form of equation 2. Nevertheless,

the real challenge of the theoretical aspect of the experiment lies on the fact that there

is need to find the equations which will determine the phase and group velocities. For

this reason, we need to guess a solution which we will derive and plug in the equation

2. After that, by solving in terms of , we will find the dispersion relation of (k).

At first we have to guess a solution. Since we are dealing with waves, the most

appropriate equation which could illustrate a wave propagating to the right would be a

sinusoidal function of the form2:

( , ) sin( ) y x t A kx t = + (5)

Before we plug it in equation 2 we need to calculate the second partial derivative with

respect to time and space. Hence, we get:

With respect to time t:

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Taking into consideration the above information and before we proceeded to our

experiment the main purpose was to show that the dispersion relation would appear as

a hyperbolic function from which we would be able to calculate the phase and group

velocity. Furthermore, when we observed the motion of the wave we wanted to make

sure that the periods for each trial seemed consistent with the rate at which we moved

the slinky. Thus, trying to be as careful as possible while taking our measurements, our

expectations were to show that the frequency of the wave was positively correlated to

the wavenumber through a hyperbolic function which would allow us also determine

phase and group velocity.

IV. Experimental Procedure and Results

The first step for generating a precise experiment was to make sure that the slinky

was set up in the right way. We measured the length of the strings connecting the

metal rod with the slinky to be 35cm. In order to calculate the period of each trial we

used three different stopwatches for more accuracy. We drove the slinky at different

rates for 5 complete oscillations during each trial while we were taking a picture of it

and measuring its period. We conducted the same procedure 10 times having a

respective period and picture for each trial, which would allow us to identify the

different frequencies and wavelengths.

After we collected our data the first thing we did was to examine the pictures and

determine a wavelength for each wave. Using the formula

2k

= (12),

we calculated the corresponding wavenumber for each wavelength. Also, we calculated

the respective average frequencies from each period based on the formula

1f

T= (13),

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and then we found the angular frequencies using the formula

2 f = (14).

Continuing, we plotted vs. k and the graph we got was the following:

Figure 1: Plot of angular frequency (k) vs. the wave number k for a slinky which is

connected with a string of length Ls=35 cm.

Comparing the above graph with a second graph where the length of the strings

was longer, we can see that even if the values are different, the trend of the points is

similar, which means that the corresponding function would be of the same hyperbolic

form:

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Figure 2: Plot of angular velocity (k) vs. wave number k for a slinky which is

connected with a string of length Ls=72 cm.

At this point its convenient to explain how phase and group velocity can be

obtained from the graph. Phase velocity can be determined at any point of the graph

from the division of (k)/k, that is the division of the (k) and k values that represent

that specific point on the graph. Group velocity can be determined graphically from the

slope of the function at any given point since it is theoretically determined by the

derivative d/dk.

Finally, in order to complete our understanding around dispersion we need to find

and compare the cutoff frequency for both graphs, experimentally and theoretically.

Experimentally, the cutoff frequency can be estimated from the graph at the point

where k=0. So, for the first one the cutoff frequency is approximately =3.5 rad/sec

and for the second one is =2.8 rad/sec. These values are approximations since the

line used for the best fit does not represent a hyperbolic function but an exponential.

However, it looks similar enough with a hyperbolic function in order to fit between the

points and to give us an decent approximation. Thus, these estimations are reasonable

enough because the first experiment was conducted with a slinky attached to shorter

strings, which prevents it from propagating as much as the slinky with the longer

strings when both of them are driven at a very slow rate. As a result, the frequency in

which we would not be able to observe a successful propagation would be higher for

the first experiment. Theoretically, since we assume that there isnt any damping, we

can say that =0, and in order to calculate the cutoff frequency we can use equation

3. Thus, for the first experiment we have:

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0

9.81

0.35

5.29 / sec

s

g

L

rad

= =

=

=

For the second experiment we have:

0

9.81

0.72

3.69 / sec

s

g

L

rad

= =

=

=

Therefore, comparing these two values with our estimations from the graphs, the

resulting error we get for the first experiment is 34% and the for the second one is

24%.

V. Conclusion

The purpose of the experiment was to illustrate and analyze the dispersion relation

between the angular frequency and the wavenumber of a slinky. As we showed in the

article, the aim of the experiment was succesfully reached. We managed to plot a

graph which, besides any errors during the measurements, accurately shows the

dispertion realation as a hyperbolic function. Furthermore, the way in which the phase

and group velocity can be determined from the graph and theoretically was analyzed

and explained.

However, even if we consider the above experiment and its analysis succesful there

is an issue that needs to be pointed out. The experiment wasnt conducted in the best

possible way since there wasnt any driver for the motion of the slinky which would

precisely give us the period and therefore the frequency of the propagation. For this

reason, the driving of the slinky was conducted manually, something which led to a

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cutoff frequency when the motion was performed at a really slow rate. Thus, when

examining the picture for the low frequencies we couldnt observe a perfectly

propagated wave. The errors for the two experiments were 34% and 24%, which

justifies the above problem we had during the experiment. This issue was the major

problem which had a negative impact on the data analysis and therefore on the

resulting graph.

VI. References

1. Wave cutoff on a suspended slinky, G. Vandegrift, et al., American Journal of

Physics, v. 57, p. 949-951 (1989)

2. The Slinky as a model for transverse waves in a tenuous plasma, J. Blake and L.N.

Smith, American Journal of Physics, v. 47, p. 807-808, (1979)

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slinky machine paper - final

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