We felt that we could improve our accuracy further by performing a couple of trial experiments to see how we could improve upon our initial ideas. Our trials did not suggest any obvious improvements so we continued with the above method and recorded the following results.
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.
We acknowledged that k should be the same for each mass and noted that k was similar for each result, being approximately 24.76 N/m.
However, we need to consider the following errors.
Errors - first experiment
Note that the meter rule has not been included in the above error table as we did not use it to take measurements and no measurements of length from the meter rule were used in any calculations based on the results of this experiment.
After considering the above table of errors, we amend the natural resonance of the spring recorded from the first experiment as follows:-
k = 22.42 N/m +/- 7.14% = 22.42 +/- 1.60 N/m
This would mean that the top end of our range is k = 24.02 N/m. The second experiment gave us k = 24.76 N/m. However, we need to take into account the error of the meter rule.
Errors – second experiment
So, according to our second experiment, k = 24.76N/m +/- 0.42% = 0.10N/m
This means that the bottom end of our range is 24.66N/m.
So the smallest difference between the value for k from the first experiment and the value for k from our second experiment is 0.64N/m. Whilst this means that the value of the second experiment is out of range, the difference between the first 2 figures is only 2.66% of our top end value for k from the first experiment.
It therefore seems fair to say that our graph appears to support the hypothesis that resonance occurs when the driving frequency is equal to the natural frequency. I feel that further readings would improve the graph and perhaps yield a more accurate value for k.
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,
