15. 5. LONGITUDINAL OSCILLATION OF SPRING
15.5.1. Introduction
15.5.2. Experimental section 15.5.2.1. Installation
15.5.2.2. Procedure 15.5.3. Result 15.5.4. Reference
Equipment 1. Vibrator
1.1. Vibrator on stabilizer alumunium base x 1 1.2. RCA cable x 1
2. Demo object 2.4. Spring x 1 4. Accessories
4.1. Pulley ∅=50mm x1
4.2. U-Shaped aluminium weight holder 20 g x 1 4.3. Alumunium weight 10 g x 4
4.4. Three-way connector x 1
4.5. Support rod holder with table clamp x 1 4.6. Support rod l=600mm x1
4.7. Support rod extender 5. Optional
5.1. Sine wave generator x 1 5.2. Power cable 11 A / 125 V x 1
15.5.1. Introduction
Features of standing waves such as nodes and antinodes can also be observed in a spring that is fixed at one end and connected to a vibrator at the other end. A longitudinal wave propagating down the length of a spring can be seen as a zone of compression followed by a zone of rarefaction. The wave will reflect off at the other end of spring and travel back and the two waves will interfere with one another. If one end of the spring is vibrating at certain frequencies, a standing wave (or resonance) appears in the spring. As the frequency of oscillation approaches another resonant frequency, standing waves appear again in the spring.
Figure 13. Longitudinal wave in spring.
The longitudinal waves produced in helical springs have a speed v given by:
v=L
√
mk Equation. (4) Where L is the length of the vibrating part of the spring, k is spring constant and m is the mass of spring.The frequency of the standing waves that have nodes at both ends is given by fn=nv
2L(n=1,2,3, …) Equation. (5)
When combined.
fn=n
2
√
mk Equation. (6) The frequency of the standing wave is dependent on the spring constant rather than the length of the spring.15.5.2. Experimental section 15.5.2.1. Installation
1. Take one end of the spring that has a loop and hang it vertically from the metal rod extenderas shown in Figure 14.
Figure 14. Experiment setup
2. Make sure to use the screw at the end of the extender to clamp the top loop of spring, so it doesn’t move during the experiment.
3. Connect the sine wave generator to the vibrator using the RCA cable.
4. Turn on the sine wave generator and adjust the frequency and amplitudo to zero.
15.5.2.2. Procedure First part
1. First, determine the spring constant of the spring. Without connecting the bottom of the spring to the vibrator, measure the length of the spring without any weight.
2. Then hang a weight of 10,20,30, and 40 g sequentially using the aluminium hanger and the measure the displacement x of the sprig by measuring the length again.
3. Calculate the spring constant with the experimental data using Hook’s law (F=−kx ; F=mg). Be sure to include the mass of the aluminium hanger in the calculation.
Second part
1. Now connect the bottom hook the spring to the vibrator by the hook on top of the three-way connector. See Figure 15.
Figure 15. Exsperiment setup.
2. Start increasing the frequency slowly and adjust the amplitudo until a standing wave forms. Record the resonance frequency.
3. Under different fruquencies, you will find that some parts of spring seem to be static (the node), but some are vibrating (the antinode). When the frequency increases, the number of nodes and antinodes also increases.
4. It may be necessary to lower the amplitude of the driving wave when the spring recahes resonance so you can observe the standing wave more clearly.
NOTE: When observing the nodes and antinodes, it is recommended to have a bright background, such as light-colored walls. The background will help to distinguish the nodes and antinodes, especially at higher frequencies.
5. Calculate the harmonic frequency using the spring constant obtained in the first part using Equation 6. Compare the calculated frequency to the obeserved frequency and determine how much the result differ.
Third part
1. Now change the length of the spring by placing the bottom hook of the spring to a higher position in the spring so the vibrating part will become shorter.
2. Repeat steps 5-7 from the second part of the procedure, and observe and record whether the resonance frequency is different or not when the same number of nodes is generated.
15.5.3. Result
First part: Determination of spring constant For F=−kx∧F=mg ,then k=mg
x
Mass of the spring =9,44 g mload
(g)
mg (gm/s2)
mg (N)
Displacement x (cm)
Displacement x (m)
Spring constant k(N
m) 1
2 3
4
Avarage k
Second part: Determination of resonance frequency.
n Measured frequency (Hz)
Frequency calculated withk(Hz)
Measurement deviation %
1 2 3 4 5 6 7
8
Deviation %= |Measured Frequency−Frequency calculated with k|
Frequency calculated with k x100 %
Third part: Harmonic frequency of spring stretched to various length
n Frequency (Hz)
L = cm L = cm L = cm 1
2 3 4 5 6 7 8
15.5.4. Reference
First part: Determination of spring constant For F=−kx∧F=mg ,then k=mg
x Mass of the spring =9,44 g
mload (g)
mg (gm/s2)
mg (N)
Displacement x (cm)
Displacement x (m)
Spring constant k(N
m)
1 30 294 0,294 6,1 0,061 4,82
2 40 392 0,392 8,3 0,083 4,72
3 50 490 0,49 10,4 0,104 4,71
4 60 588 0,588 12,4 0,124 4,74
Avarage k 4,75
Second part: Determination of resonance frequency.
fn=n
2
√
mk for example , when n=1will be f1=12
√
0,009444,75N/kgm=11,21Hzn Measured frequency (Hz)
Frequency calculated withk(Hz)
Measurement deviation %
1 11 11,21 1,91
2 23 22,43 2,54
3 33 33,64 1,91
4 45 44,86 0,32
5 57 56,07 1,65
6 68 67,29 1,06
7 80 78,50 1,91
8 90 89,72 0,32
Deviation %= |Measured Frequency−Frequency calculated with k|
Frequency calculated with k x100 %
Third part: Harmonic frequency of spring stretched to various length
n Frequency (Hz)
L = cm L = cm L = cm
1 11 11 11
2 23 23 23
3 34 35 35
4 45 46 46
5 57 58 58
6 68 69 68
7 80 80 80
8 90 91 91