The period verses mass for vertical oscillations on a hanging spring

 
Research question
 
What is the relationship between the period of oscillation T and mass m, when the mass is in vertical oscillation on a light, hanging, steel spring.
 
Hypothesis
 
It was noted in preliminary trials that the period increased as more mass was added. It is expected that T is proportional to the mass, m.
 
Apparatus and procedures
 
A mass was hung on a spring and displaced vertically the same distance each time. The period of the oscillations after release was measured with a hand held stop watch by timing three oscillations. In all five masses were used, and the period was measure five times for each mass.

Data
 
The mass was found with an electronic balance to within ± 0.5 g. The period was measured five times for each of five masses. The error in the period listed in Table 1 below is half the range.

Table 1

Analysis

Graph 1 shows the period plotted against the mass.

Graph 1: the period is proportional to the square root of the mass within errors.

 

In Graph 2 the period squared is plotted against the mass.

Table 2 The errors in period squared have been calculated in Logger Pro by entering maximum and minimum values in Table 2.

Graph 2: period squared versus mass.

 

Graph 2 is a straight line through the origin within errors. The period squared is proportional to the mass.

T² = km ... where k = 2.05± 0.05 s²/kg

The error in k has been determined by fitting extreme lines to Graph 2.

Discussion

The period T has been shown to be proportional not to the mass as expected, but to the square root of the mass since the line of best fit on Graph 2 passes the through the origin.

Evaluation

Errors in the period T were disappointingly large at ~ ±5% and the corresponding error in T ²was ~±10%. It was difficult to record exactly when the mass was passing the equilibrium position (a place where more accurate time values can be obtained than at the points of extreme displacement).

In any futire work the equilibrium level could be marked on paper directly behind the spring .. but ... the large percentage errors were due mainly to timing only three periods because the motion was strongly damped. If the spring had been longer the motion would have been more persistent, and 10-20 longer periods could have been timed with a consequent improvement in accuracy. As an alternative the motion could be driven gently by hand to maintain the initial amplitude.

Measurements could be made to determine whether the period is independent of the amplitude, and whether driving the motion affects the period in any measurable way. Graphs 1 and 2 have only five data points. Normally 6-7 data points are required to establish a relationship.

 

Editor's comments The title is improved. The student seems to have used a template with section numbers supplied by the teacher. The numbers add nothing and have been deleted. The hypothesis remains weak and is presented without explanation. A diagram or photograph in the procedures section would be helpful. Table 1 appears to contain real measurements, but the selected spring was unsuitable. Placing three or four springs in series would have permitted larger amplitude oscillations and the timing of 10 or 20 periods. The student could have established the amplitude/period independence before proceeding, or they could have driven the oscillations gently by hand to maintain the amplitude. Typical electronic balances come with errors specified as ±x% plus two digits. The half-the-smallest-scale-division rule of thumb for an an analog scale (like a ruler) is not appropriate. Half the range (by inspection) is a simpler way to find the error in the period. The student's averages at least were well done, except for the exclusion of the outlier in the last set. The original graphs were of little value. (If a proper graphing program is not available graphs must be done properly by hand.) The error bars in the originals were of no value at all. A curve fit to the first graph in Logger Pro allows the second linear graph to be plotted without guessing. If a proper graphing program is not available a log-log graph will disclose a power relationship. If the spring constant is to be measured the relationship should be included in the introduction. The student was taught to start and stop the watch as the mass passed the central position but they failed to get accurate data for other reasons. They failed to plan an investigation worthy of an IB report. They failed to write in the normal third person passive construction. They failed to establish appropriate procedures (to measure 10 periods) but they did collect real data. They failed to present clear well labeled graphs with at least 6-7 points. They failed to put correct error bars on the graphs. They failed to measure the spring constant and complete the analysis. The student was struggling - working to much for to little.