Knowledge Transfer Challenges

We know that an isolated system vibrates with its natural frequency.

What happens when two such isolated systems are coupled together?

The presence of coupling affects its amplitude and frequency of oscillation. We expect that the motion may not remain simple harmonic.

Does this mean that for a coupled system we cannot define the period of oscillation? To answer this and other related questions we consider a system of two identical coupled oscillators. The apparatus needed for this purpose is listed below:

Apparatus

Two identical hacksaw blades (1/2" or 1" width and 12" length), two vices, rubber bands/soft springs/a pair of strong bar magnets, and a stopwatch.

Fig. 4.1: A coupled oscillator system

Set the apparatus as shown in Fig. 4.1. For the success of this experiment you should note that both oscillators (in this case hacksaw blade) should be in the same plane and act as identical oscillators. That is, the time periods for both these oscillators should be the same. To ensure this, you should use a stopwatch with good accuracy. Note the least count of the stop watch and record it in Observation Table 4.1.

Next displace one of them from its respective mean position and then release it. It begins to oscillate. You should ensure that the oscillations are free. To begin with, you should count time for 10 oscillations. Enter your data in Observation Table 4.1. Calculate its time period by dividing the measured time by N, the total number of oscillations counted.

Observation Table 4.1: Time Period of Isolated Oscillators Least count of stopwatch = ... s

S.

No.

No. of Oscillations

(N)

Time for N Oscillations (s)

Time Period (s)

1st Oscillator

2nd Oscillator

1st Oscillator

2nd Oscillator

1 10

2 20

3 30

4 40

5 50

Repeat the same procedure for the other hacksaw blade. Compare their time periods. Are they same? We expect these to be same. If not, then load the blade with larger time period with wax. Alternatively you can file the blade which has smaller time period. You will require considerable experimental skill of measuring time and practice to achieve exactly same values of time periods. You should repeat this process till you get identical time periods. In case you fail to do so repeat the procedure till the difference between these periods is not more than 0.1 %. Next you should measure time for 20 oscillations and repeat the above-said procedure. To get more precise results you can work with 30, 40, 50 or more oscillations. Let us denote the time period byT0. Now couple these two oscillators by putting a rubber band or a spring near the fixed end. In this way you obtain a mechanically coupled system. Alternatively you can use a pair of strong bar magnets. Is there any difference between these two types of couplings? We expect that the system will display similar behaviour in both cases. You can, therefore, use either of these arrangements for this experiment.

In the course Oscillations and Waves, you will learn that the motion of a coupled system is not simple harmonic. However it can be analysed in terms of normal modes. For two coupled simple pendulums two normal modes are shown in Fig. 4.2. Consider the transverse motion and first excite the in-phase normal mode by equally displacing the two oscillators (hacksaw blades) in the same direction (Fig.4.2a). You

should ensure that the two oscillators always oscillate in phase. As such, this is somewhat tricky and you will need some practice. When you are finally convinced, measure time for 30 oscillations and calculate the period. Let us denote it byT1. It is important that the amplitude of oscillations be small.

Fig. 4.2: (a) In-phase, and (b) out of phase normal modes

Now you make the system to vibrate in the out of phase normal mode without changing the position of the coupling system (spring/magnet/rubber-band). This can be done in two ways, as shown in Fig 4.2(b). You can choose to work with the case in which two oscillators are drawn closer. Repeat the above procedure and determine the time period for this case. Let it be T2.. Are T0, T1and T2 the same?

We expect them to be different. What do you conclude from this? This only means that coupling is effective.

Next, you move the coupling arrangement away from the fixed end by 1 cm. This will bring about a change in the coupling. Another way of changing the coupling strength will be to change the quality of rubber band or take springs of different spring constants. Repeat the above procedure and record your data in Observation Table 4.2.

Table 4.2: Effect of Coupling on the Period of Normal Modes Least count of stop watch = ...s

No. of oscillations (N) = ...

S.

No.

Distance of rubber band from the fixed end (cm)

Time for N Oscillations (s)

Ist Normal Mode

IInd Normal Mode

T1

(s)

T2

(s)

1.

2.

3.

4.

5.

6.

Are time periods influenced by changing the position of the rubber-band?

Repeat the experiment for other positions of the rubber band and enter your readings in Observation Table 4.2.

Calculate the corresponding angular frequencies using the relation,

T / 2π

ω = . (The difference in the frequencies of two normal modes is known as frequency splitting. We denote it by the symbol ^∆v and it is given by^(ω2 −^ω1^)/2π ). Do these frequencies vary as position of the rubber band is changed? A variation in their values suggests that coupling has an influence on the motion of the system. To clarify this further, you can plot angular frequency as a function of the distance of the rubber band from the fixed end. Is the relation linear? Discover the functional dependence between the two quantities by following the procedure outlined in Experiment 1. Discuss your results with your counsellor.

Conclusion: The angular frequency varies... with the distance of the rubber band from the fixed end.

SELF ASSESSMENT EXERCISE 1

Is there any damping in the system? How will you account for it?

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SELF ASSESSMENT EXERCISE 2

Choose two widely separated points on your angular frequency versus distance of rubber-band graph and correlate frequency splitting to the coupling constant.

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