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Physics 2107 Experiments with Sound Experiment 3

Background/Setup – Elastic Deformation Stretching, Compression and Young’s Modulus

We all know that a spring returns to its original shape when the applied force that compresses or stretches it is removed. In fact, all materials are distorted when they are squeezed or stretched and many materials return to their original shape when the squeezing or stretching is removed. Such materials are termed elastic and this behaviour has its origin in the forces between atoms in the body. The inter-atomic forces that hold a solid together are particularly strong and it is necessary to apply a reasonable force in order to stretch or compress a solid object. If we consider a rod as shown in Figure 1 and if we assume that the stretch or compression of the rod, ΔL, is small compared to its original length, L0, then the magnitude of the applied force, F, is given by:

,0

ALLYF ⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ= (1)

where Y is a material-dependent constant called Young’s Modulus and A is the cross-sectional area of the rod.

The units for Young’s modulus are -2Nm . The cross-sectional area need not be circular but can have any shape (e.g. square). The value of Young’s modulus depends on the nature of the material under study e.g. brass has a much higher Young’s modulus1 of -210 Nm1002.9 × as compared to mohair which has a value of

.Nm109.2 -29× Note that the value for brass can vary according to the composition of the brass. Brass is an alloy of copper and zinc and the amount of zinc varies from 5% to 45% to create a range of brasses each with unique properties (and unique values for Young’s modulus).

One condition that arises due to the variation in Young’s modulus according to material is the frequently disastrous deterioration to the condition of human bone when metal implants are used in hip replacement surgery. It can be shown that the velocity of sound in a material is related to Young’s modulus

for the material by: ρYvsound = , (2)

1 The values given here for brass are based on a 67% copper and a 33% zinc mixture.

Figure 1: In this diagram F

denotes the stretching force. [1]

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where ρ is the density of the material.

Shear Deformation and the Shear Modulus An object can be deformed in ways other than stretching and compressing. For example, if you push a book while it is lying on a surface the top is shifted relative to the bottom. Such a deformation is termed a shear deformation and is a result of the applied force F

and the force applied by the table to the book, .F

−

Figure 2: An example of a shear deformation. [1]

An alternative form of shear deformation can be defined if we consider a solid rod with equal but opposite torques applied to at the two ends. Provided that the elastic limit is not exceeded it can be shown that the shearing force is given by: ,φηAF =

where η is the shear modulus of the material, A is the cross-sectional area and φ is the shear angle of the object. Note that the shear modulus is sometimes called the modulus of rigidity. For brass2 .Nm105.3 -210×=η Note that this is an approximate value due to the variations in composition of brass.

Figure 3: A shearing force applied to a rod by equal and opposite torques at the ends of the rod.

It can be shown that the velocity of sound in a material is related to the shear

modulus by: ρη

=soundv . (3)

In this experiment you are required to determine Young’s modulus and the shear modulus for brass by setting up vibrations in a brass rod and determining the fundamental frequency of vibration. In addition, you must measure the speed of sound in air using Kundt’s dust tube (c.f. Appendix A). The idea is to induce longitudinal waves in air by generating longitudinal waves in a brass rod and using these to drive the waves in air.

2 Based on a 67% copper and 33% zinc mixture.

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Part 1: Identifying the Fundamental Mode of vibration using the FFT By pulling a cloth containing resin longitudinally along the brass rod set it into vibration. In generating the longitudinal waves in the rod, it is important to use the minimum pressure when drawing the cloth along it. If you pull too hard, you will generate a lot of harmonics and it will be difficult to determine the fundamental frequency. For best results grasp the rod ~half way along its length before steadily pulling the cloth. The sound time-series is recorded using a microphone interfaced with the PC. This time series can in turn be analysed using a FFT (see Experiment 2). Note: make sure that the data acquisition is properly triggered (click on “data collection”, “triggering” and then enable the triggering). By following this procedure identify the frequency of vibration, f.

Part 2: Speed of sound in a brass rod and Young’s modulus for brass The speed of the sound wave travelling through the brass rod is related to the frequency f and the wavelength λbrass of the wave in the rod via vbrass = f λbrass. (4)

Hence, to calculate vbrass we need to measure λbrass.

Measuring λbrass

Depending on the mode of vibration you set up in the rod, you will hear either a lower frequency (http://astro.ucc.ie/py2107/lo_f_sound.mp3 ) or a higher frequency (http://astro.ucc.ie/py2107/hi_f_sound.mp3 ) tone.

Because the rod is clamped in the middle (x=0 in the figure below), the amplitude of the vibration at this location must be zero. For maximum amplitude at each end, then the following scenarios apply: For the lower frequency vibration, we have

from which λbrass/2 = L => λbrass=2L.

In contrast, for the higher frequency vibration, we have

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(here we have plotted the displacement at a time t = 0 (red) and t = P/2 (blue), where P=1/f ). In this case (3/4)λbrass = L/2 => λbrass = (2/3)L. Measuring vbrass and Y for the brass rod

Depending on the frequency you measure, measure λbrass: using equations (4) and (2), calculate the speed of sound in the rod, and Young’s Modulus for brass, respectively. Take ρbrass = 8.44×103 kg m-3 .

Compare your values with those in the literature, and discuss why any discrepancy occurs. Ensure you make a reasonable estimate on the maximum error in each of your measurements.

Part 3: Determination of the shear modulus for brass The shear modulus is determined by rubbing the cloth on the brass rod in a rotational manner and following the procedure in Part 2 to determine the fundamental frequency of vibration and, hence the speed of sound in brass.

Figure 4: Rotational induction of sound in the brass rod.

Use Equation (3) to calculate the shear modulus, η. Again, ensure your answer contains suitable errors.

Part 4: Determine the ratio of the velocity of sound in brass to that in air. Using the Kundt’s tube set the brass rod into vibration, as before.

To measure the speed of sound in air the wavelength is determined by measuring the distance between successive nodes as seen in the dust powder in the tube. The distance between two successive nodes is 2λ . In order to increase the accuracy of your measurement, determine the distance between a number of nodes (e.g. 10 nodes will yield a distance of λ5 ). By measuring the frequency f as described in Part 1 above, the speed of sound in air can be determined from vo = f λ.

clamp

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In some situations, lower amplitude nodes may be visible between higher amplitude ones: in this case, use the distance between the higher amplitude nodes to measure λ.

Comment on how your measured value of ov compares to the known speed of sound in air of (331 + 0.6 T) ms-1, where T is the room temperature in °C. Remember to include maximum error estimates in your answer.

Finally, determine the ratio of the speed of sound in brass to the speed of sound in air, obrass vv .

References

1. “Physics” by Cutnell and Johnson.

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Appendix A

Kundt’s Dust Tube

Sound in air propagates by means of longitudinal waves. In a solid, e.g. a brass rod, sound can propagate by either transverse or longitudinal waves. Kundt’s dust tube was devised by Karl August Kundt (1838-1894) for investigating vibrations in columns of air. It displays the nodes and allows one to measure the wavelength in a tube and, hence, the velocity of sound and modulii of elasticity.

Figure 5 shows a schematic representation of Kundt’s dust tube, which consists of a glass tube containing dust held

horizontally on a table, one end closed and the other end open. A brass rod is clamped in the middle to ensure that the tube is airtight. Sound waves are produced in the brass rod by rubbing it longitudinally with a chamois to generate the fundamental frequency and higher harmonics.

Figure 5: Kundt’s dust tube

Since the rod is clamped at its centre, this represents a node since there can be zero amplitude of motion at this point. The ends which are free to vibrate therefore are the antinodes and vibrate with maximum amplitude. If the rod is set vibrating in this manner the fundamental frequency has been generated and the wavelength of the standing wave in the rod

Longitudinal wave in rod

Wave in air

λ

2λ

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is twice the length of the rod. The vibrations in the rod are then transmitted to the air in the glass tube and must propagate through the air at the same frequency as the waves in the rod. The waves are reflected at the closed end of the glass tube, thereby creating a standing wave in the tube, characterized by a series of nodes and antinodes i.e. areas of maximum and minimum disturbance. These nodes and antinodes can be observed by looking at the dust figures in the tube. Neighbouring nodes denote half a wave, and it is therefore possible to measure the wavelength.

Once the fundamental frequency and wavelength have been determined, it is possible to determine the speed of sound in air at room temperature using the wave relationship .λfv =

A video demonstration of the generation of nodes in dust particles using sound propagation in a metal rod and Kundt’s tube can be watched at:

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/kundtosc.html

The cartoon below shows an example of how to generate dust patterns in a Kundt’s tube.