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The Frequency of a Vibrating Metal Rod

Name:

Date:

Objective:

Data Tables

Predicted Frequency:

Harmonic

L

#

Measured frequency:

Harmonic

t1

#

ÃŽÂ»

t2

ÃÂ

Y

t

#

intervals

v

T

f (pred)

f(meas)

Calculations:

Sources of Error:

Answer to Question:

Conclusion:

Attach Photo of you holding the aluminum rod

Extra credit: Include a video of you producing the standing waves in the rod.

% Diff

Frequency of a Metal Rod Instructions

Objective: To predict the frequency of standing wavs in a metal rod and the sound

waves they produce.

Materials and Equipment: Long aluminum rod, violin rosin, tape measure, app on

phone

Theory: Imagine dropping something and hearing the sound it makes as it hits the floor.

Why does it make a noise with that particular pitch/frequency? We will answer that

question for one particular object Ã¢â‚¬â€œ a metal rod.

Standing waves in any medium produce, at certain frequencies, a large amplitude at the

certain locatioins and zero amplitude at others. These are determined by the end

conditions of the vibrating system. For example, a string tied at both ends must have

nodes there. This means that waves for which multiples of oneÃ¢â‚¬â€˜half wavelength fit onto

the length of the string can produce standing waves, and only these waves. This yields

the result that the harmonics are integer multiples of a fundamental.

The waves traveling through the metal rod in our experiment are longitudinal waves,

i.e., sound waves. At the ends of the rod the medium is not restrained. In fact, the

medium is most free to vibrate there. So instead of there being nodes at the ends, we

will have antinodes there. The simplest standing wave has antinodes (A) at both ends

and a node (N) at the center, as in the diagram below.

Since the distance between two antinodes is one-half of the wavelength, the wavelength

is twice the length of the rod: ÃŽÂ» = 2L.

Since f = v/ÃŽÂ», we have for the frequency of the fundadmental of this rod

f=v2L

Actually, when an object drops and hits the floor, it will produce a set of frequencies

simultaneously and a complex sound. The one we find today is only one of the possible

standing waves in this set.

The existence of a node at the center is helpful. Holding the rod at that location means

that we will not dampen the wave Ã¢â‚¬â€œ the medium doesnÃ¢â‚¬â„¢t move there anyhow. Holding

the rod at another position will dampen the wave quickly.

We will need to know the velocity of the sound waves traveling through the metal rod to

determine the frequency. The speed of sound through solids is given by the formula:

v=YÃÂ

where Y is YoungÃ¢â‚¬â„¢s Modulus and ÃÂ is the solidÃ¢â‚¬â„¢s density. YoungÃ¢â‚¬â„¢s Modulus is a measure

of how much force is necessary to stretch a material by a certain distance and so plays

a similar role to the spring constant k. It tells us essentially how Ã¢â‚¬Å“springyÃ¢â‚¬Â the material is.

Its units are Pa = N/m2, the same as for pressure. The density is a measure of inertia

with units kg/m3. Knowing these two constants will allow us to calculate the speed of

sound in that medium.

Procedure:

1. Practice producing longitudinal waves on the aluminum rod. These can be

produced using two methods (see video below):

a. Put a little rosin powder on your fingertips. Balance the rod on one finger of the other

hand to find its center and hold it there. With the hand with the rosin, squeeze and pull

along the length of the rod, letting your fingers slide over it. This will take a little practice,

but when you get the right combination of speed and pressure, the rod will produce a

dramatically loud standing wave. This seems to work best with aluminum rods that are

not too thick. You will probably have to sqeeze the rod somewhat more than you find

comfortable because of the friction. And your fingers get a little dirty from the rosin. But

it washes off easily. This method is the most fun.

b. Find the rodÃ¢â‚¬â„¢s center as in method a above and hold it securely there. Tap one end of

the rod with a hammer or other similar instrument. With this method, there is usualy a

Ã¢â‚¬Å“clangÃ¢â‚¬Â sound at the beginning that obscures the true resonant frequency weÃ¢â‚¬â„¢re looking

for. (ThatÃ¢â‚¬â„¢s because of other partials being produced at the same time.) You have to

wait a couple of seconds for that clang to fade out. This is not a problem with method a.

Video Link

(Links to an external site.)

2. Measure the rodÃ¢â‚¬â„¢s length and use it to determine the wavelength of the sound

wave in the rod.

3. Find the necessary constants and calculate the speed of sound in that

medium. Be careful with the scientific notation and exponents.

4. Calculate the frequency of vibration (f pred) for the rod.

5. Produce the standing waves by either of the two methods descibed above.

Measure the frequency of the sound using the Hokusai Audio Editor app on

your phone (downloaded for Lab 5). Anyone using a desktop computer with a

microphone could also try the Audacity app (download free) which is

sometimes clearer. Refer to the Lab 5 Instructions to refresh your memory of

how this is done.

6. Compare the two values of frequency using percent difference.

7. Where would a node for the second harmonic be? Measure this position on

the rod and produce the standing wave. You will probably need to use

method b (see above) to produce this wave. Now perform steps 2 through 5

for this rod. (Careful: the relationship between length and wavelength is

different here.)

Question: If this had been a steel rod thatÃ¢â‚¬â„¢s twice as long, what fundamental frequency

would it resonate at? Show your calculations in the calculation section.

Density and YoungÃ¢â‚¬â„¢s Modulus for Some Common Metals

Metal

Density (kg/m3)

YoungÃ¢â‚¬â„¢s Modulus (Pa)

Aluminum

2712

7.0 x 1010

Copper

8930

11 x 1010

Steel

7850

20 x 1010

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