Processed Data:
Sample Calculations:
Extension:
= (Length After Extension) - (Length Before Extension)
= 4.85 - 3.85
= 1.0 cm
Extension Uncertainty:
= Length After Extension Uncertainty + Length Before Extension Uncertainty
= 0.05 + 0.05
= 0.1 cm
Average Force:
= (Trial 1 Force + Trial 2 Force + Trial 3 Force) / 3
= (0.490 + 0.458 + 0.491) / 3
= 0.480 N
Average Force Uncertainty:
= (Maximum Force - Minimum Force) / 2
= (0.491 - 0.458) / 2
= 0.02 N
The equation of the line is:
F = (0.291 ± 0.007)x + 0.1
The slope comes out to be (0.291 ± 0.007) N/cm. Therefore, the spring constant, k, is equal to (0.291 ± 0.007) N/cm.
Sample Calculations:
Slope Uncertainty:
= (Maximum Slope - Minimum Slope) / 2
= (0.3085 - 0.2950) / 2
= 0.0135 / 2
= 0.007 N/cm
CONCLUSION:
As seen within the data collected and processed above, as the extension of the spring increased, the force exerted by the spring increased proportionally.
The results of this experiment seemed to mostly agree with my hypothesis. The relationship between the average force and the extension of the spring was a direct relationship, allowing the slope to be calculated and the spring constant to be found.
The spring constant, k, was calculated to be (0.291 ± 0.007) N/cm.
However, my hypothesis predicted that the linear regression line would pass through the origin. In the actual results, the linear regression line passed above the origin, which hints towards the existence of systematic errors within the data, as will be discussed within the evaluation.
EVALUATION:
There were obvious places of weaknesses and limitations within this lab that could have been improved to increase the accuracy and precision of the data.
Systematic Errors:
As mentioned previously, the fact that the linear regression line did not pass through the origin of the graph and, instead, passed through the y-axis at about 0.1 N simply demonstrates how systematic errors definitely existed within this lab.
The most obvious source of systematic error would have been from the force sensor. This sensor was very sensitive to any slight touch or pull to the sensor. Therefore, it is very likely that the sensor was not properly zeroed, thus causing this inaccuracy in data.
Random Errors:
Also, from the data points on the graph, it can be seen that the points are not very precise. Some points stray farther from the regression line than others, and some points' error bars do not touch the line of best fit, the maximum line, or the minimum line. This can all be due to random errors within the experiment.
While measuring the length of the stretched spring, it was very hard to obtain an exact value of the full extension of the spring. This was due to the fact that the spring was not on a flat surface, and, instead, suspended in the air. Therefore, the ruler had to be held beside the spring by a person but could not actually touch the spring as this would affect the force sensor's results. Thus, it is likely that random errors existed due to the weaknesses present with this method of measuring spring length.
Reliability of Results:
Ultimately, the equipment used was very precise and, therefore, allowed relatively low uncertainties to be developed. However, there were possibilities of inaccuracies within the equipment and obvious sources of random error from the procedure. None of the possible errors listed earlier, however, could have been severe enough to completely alter the data. Therefore, the results received were quite reliable and allowed sufficient conclusions to be made that supported the original hypothesis.
Anomalous Data:
Some data, such as the point with extension 1.0 cm and average force of 0.480 N, strayed far from the line of best fit and minimum and maximum lines. Points such as these are considered anomalous and could have been due to the multiple errors listed above.
Improvements:
Ultimately, to improve the weaknesses, limitations, and possible errors listed above a few courses of action would have had to be taken.
First, zeroing the force sensor and ensuring that the force sensor was zeroed before collecting data for all new spring lengths could have gotten rid of the systematic error present within the results and would have allowed the linear regression line to pass through the origin as expected.
Similarly, setting up the equipment so that the spring had to be stretched out on a flat surface, and not while in the air, could have allowed the spring to be held more steadily while taking measurements. Also, clamping the ruler down onto a flat surface instead of, also, holding it up in the air by hand could have, again, lowered the random errors that occurred because of the weaknesses collecting the data for the extension of the spring.
Ultimately, making these improvements could have lowered the random and systematic errors that existed, thus making the data more accurate and precise. This, in turn, could have reduced the anomalous data and created results that more thoroughly portrayed what was expected.