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Mass on a spring - and investigation into resonance

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Introduction

An Experiment to Investigate a Mass on a spring as an Example of Resonance

Method

We set up the apparatus as shown below.  We also included a meter rule to the left of the spring so that we could see the size of the oscillations.

image00.png

We set the signal generator to produce a sine wave output and set both the frequency and amplitude to a minimum.  We switched on the signal generator and set the amplitude to its middle setting.  We pulled down gently on the load and allowed the spring to oscillate.  We slowly increased the frequency, monitoring the amplitude of the oscillations of the load by reading from the meter rule placed next to the apparatus.  We noted when the amplitude appeared to be at its largest and took this frequency to be the resonant frequency.  We repeated the experiment using different masses and decided to repeat each experiment 3 times for comparison.

Measurements

Before commencing the experiment, we considered what precautions we could take to ensure accuracy.

...read more.

Middle

1.61

0.499

2.00

1.2

1.2

1.2

1.20

1.44

We noted that, in general, that frequency decreased with mass.  

Theory

The theory is that resonance occurs at the point where the natural frequency of the spring system is equal to the frequency of the signal generator.

We know that the time period for a mass on a spring is given by

T= (2π)(√(m/k)) but we also know that f = 1/T so

f = 1/T = 1/ ((2π) (√(m/k))

  = (1/(2π)) (1/(√(m/k))

  = (1/(2π)) (√(k/m)

So

f² = ((1/(2π)) (√(k/m))²

   = (1/(2π)² (√(k/m))²

   = (1/4π²)(k/m)

   = (k/4π²)(1/m)

I have included columns in the results table for 1/m and f² as f²= (k/4π²)(1/m) which is in the form y=mx+c.  This means that a graph of f² plotted against 1/m should give us a straight line with a gradient of k/4 π², which means we will be able to find the spring constant k (See graph 1).

From my graph the gradient = 0.568 so

0.568 = k/4 π² which means

k = 0.568 x (4 π²) = 22.42 N/m

In order to verify this we performed a further experiment.  Using the equipment set up in its original format, we taped the string to the meter rule in order to keep the spring stationery.  We then measured the extensions of the spring, at rest, firstly without weights and then with the individual weights previously used.  We recorded the following results.

Mass (Kg)

Spring extension (m)

Spring Constant K

(kg/m)

Spring Constant k (N/m)

0.107

0.041

2.61

25.6

0.205

0.081

2.53

24.82

0.303

0.120

2.53

24.82

0.400

0.161

2.48

24.33

0.499

0.202

2.47

24.23

...read more.

Conclusion

In order to improve the experiment I would attempt to measure the natural resonance of the spring using a stop watch and the meter rule for comparison purposes.  We could also add a Perspex tube in which to place the spring and load to prevent the spring from swinging.  However, we would need to ensure that the spring did not hit the side as this may affect results.

Ideas for further research

We could research whether or not a spring moving in any direction other that up and down, ie swinging during the experiment would materially change the results.  We could also investigate what would happen if we damped the oscillation by repeating the experiment with the load suspended in water.  Initial thoughts would be that the velocity of the oscillations may be reduced but we would be more concerned with whether or not the amplitude of the wave had changed and thus the frequency of the natural resonance,

...read more.

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