回首頁 | 中興大學物理學系

一維運動 | 向心力 | 扭擺與轉動慣量 | 楊氏係數測定 | 繩波實驗(中) | 繩波實驗(英) | 熱功當量(中) |表面張力測定 |
黏滯係數測定| 凱特可倒擺

 

Data Analysis(PDF)
Appendix(PDF)
Format for data recording(WORD)
Unit conversion
For exploring this webpage, you may need to download the following software

Objectives / Methods / Principle / Equipment/ Steps / References

Experiment objectives :

Top
This experiment studies the standing waves on a rope and tries to figure out the relation between tension, wave length and the linear mass density of rope.

Experiment methods:

Top

A rope, with one of its end, is attached to a vibrator driven by a function generator. Another end of the rope is suspended with a weight for yielding a tension on the rope.By changing the vibrator frequency, the rope length and the suspended weight, standing wave with different wave lengths can be produced.

▼Standing wave

Source:http://www.walter-fendt.de/ph14e/stwaverefl.htm




Principle:

Top

1、Standing wave
We use or to represent a simple wave, where is the wave number and the angular frequency. If the sign of is minus, the wave is propagating along the positive direction of x-axis. Vice versa , if the sign of is positive, the wave is propagating along the negative direction of x-axis. is the initial phase.For simplicity of calculation, we also use an equivalent function to represent a simple wave ( , , ) .

The physical quantity can be simply extracted from the real part of the function. When two waves of same amplitude and frequency travel in opposite direction along a rope, standing wave may form by superposition of the two waves under particular conditions. We are going to discuss the conditions of forming a standing wave on a stretched rope with two fixed ends.

Suppose a traverse wave is traveling from x=0 toward the positive direction on a rope of length (see figure 1).

The mathematical form of the wave is given by              (1)


Where is the amplitude. A reflected wave forms when the wave reaches the fixed end at .

Suppose there is no energy adsorption at the point of reflection thus the amplitude and frequency of the reflected wave remains the same as the incident wave. However the phase of the wave may alter. So the reflected wave can be written as

                                (2)

The superposition of the incident and reflected waves is 

                       (3)

Using the boundary condition at where

we have,and get

                                   (4)

Now, we can rewrite equation (3) as                             

                             (5)

Since the rope is also fixed at implying that.

So ,                               (6)

Equation (6) is the condition for forming a standing wave on a rope with given length of .

Since ,the condition also can be read as which means that the length of the rope is always equal to the half of the wavelength multiplying by an integer.

he wave function of the standing wave is the real part of that is

                          (7)

The value of in equation (7) is the maximum amplitude of the standing wave. The static points on the rope are called nodes. According to equation (7) the nodes happen at positions . While those points oscillate with maximum amplitude are called antinodes. Alphabets N and A in figure 2 indicate the positions of nodes and antinodes.



Does standing wave benefit to the propagation of signal ?

2、The relation between wave speed and tension

Let us consider a pulse wave propagating on a string. Assume that the pulse wave is propagating toward the direction of the right hand side when the string is at rest relative to an observer. If the string length is infinity and the string is moving at the same speed of the pulse wave but in opposite direction, the pulse wave becomes static in space relative to the observer(see figure 3).

Let be the mass per unit length of the string and F is the force that maintains the segment of the string moving in a circular orbit of radius R. The centrifugal force can be approximated by because the angle is infinitely small or .

According to Newton’s second law of motion () , we have . is the mass of the segment of the string. From figure 3 we know the relation . So we have and the propagating speed of the wave can be obtained from the speed of circular motion.

                                     (8)



3、The relation between vibration frequency and string tension

Suppose a standing wave forms on a string of length Equation (6) gives . From the relation , we get

                             (9)

According to the equation (9) if the frequency of vibration f and the mass per unit length of the string are both constant then

                                 (10)

The ratio of tension and the square of the wave length is a constant. This relation can be indentified in the experiment.



Equipments of experiment

Top
Pulley Vibrator Function generator
Triangular base Weights Weight hanger
Measuring tape Transmission lines Electronic Balance

The metallic sheet on the vibrator will oscillate at a frequency same to the alternating signal from the function generator.



Video download ( Introduction to experiment equipments. )




Experiment steps

Top

Experiment setup
1.Cut a suitable length of string( not less than 1.3m). Calculate the linear mass density after measure the length and the mass of the string.
2.Setup the experiment equipments as figure 4 shown.
a.Fix the pulley at the edge of table and locate the vibrator at a suitable distance from the pulley.
   The hole on the metallic sheet of the vibrator should at a same height to the pulley.
   The metallic sheet direct toward and parallel with the pulley.
b.Tight one of the end of the string on the metallic sheet of the vibrator(*Note*).
  Lay the string on the track of the pulley and hang the weight hanger at another end of the string.
c.Connect the BNC head to the “OUTPUT” of function generator and the banana plug to the “INPUT” of vibrator.
   Do not forget to connect the power of function generator also.




影片下載 (儀器架設)

3.Measure the length of the string L(Starting from the knot at the metallic sheet to the contact point between the string and pulley).
4.The control panel of function generator is shown in figure 5. Press the “POWER” button to turn on the machine. Select the sinusoidal wave「」and frequency range 100「」(or 10「」).



Experiment (I) Parallel vibration- find out the relation between f and v.

1.Put a 50g weight on the hanger and adjust the frequency button on the panel of function generator. Find out the corresponding wave number of standing wave for each frequency. Record the values of tension F, wave length and frequency f .
2.Using the data obtained in step 1 to calculate the wave speed v at different frequency f and plot a v-f graph.


Video Download (Part I)

Experiment (II) Vertical vibration

1. Turn the direction of the metallic sheet of the vibrator to perpendicular to the string. Remain the other conditions unchanged and repeat step 1. in experiment (I).
2. Compare the results with experiment (I). What is the frequency of vibrator when the wave lengths of the standing wave in the two experiments are the same?


Video Download (Part II)


Experiment (III) Fix frequency – find out the relation between wave speed and tension

1. Change the metallic sheet direction back to parallel direction.
2. Adjust the “FREQUENCY” button on the panel of function generator to 80Hz.
3. By changing the weight F on the hanger, record the value of corresponding wave length for each F.
4.Calculate the wave speed by equation and plot the graph on a logarithm paper. Use function to fit the data and determine the best values for and . Find out the error of between theory and experiment. Calculate the linear mass density from the value of and compare it to the value obtained from electronic balance and give the percentage of error.



Video Download (Part III)

Experiment (IV) The relation between linear mass density and wave speed.

1. Use a string with different material. Measure its length and mass and calculate the linear mass density of the string.
2. Repeat all the steps in experiment (III).


Video Download (Part IV)

**NOTE**
When you tie the string on the metallic sheet, please make sure that the knot is as simple as possible. As the figure 6-left shown, the knot must be a good node instead of a bad node shown in figure 6-Right. In order to avoid the happening of bad node, please
1. Tie the knot as simple as possible and avoid distortion at the knot.
2. If the bad node still happens after method 1 has been used, try to adjust the weight on hanger, the string length or the frequency to realize a good node.
3. If both method 1 and 2 fail to remove the bad node, discuss the error may arise in the experiment due to the bad node.





References:

Top
    1.Fundamentals of Physics, Fifth Edition (by David Halliday, Robert Resnick, Jearl Walker)

    2.University Physics, 12th ed (H.D. Young & R.A. Freedman)

    3.http://en.wikipedia.org/wiki/Portal:Physics