Mass vs. Period (spring IA)

Research Question
In this investigation the following question will be explored; how does mass effect the period of a spring?
IV and DV Statements
The independent variable is the mass that was added to the end of the spring, measured in grams. The dependent variable is the resulting amount of time it takes for the mass to complete one period, this will be measured in seconds.
Purpose
The purpose of this experiment is to determine whether or not the amount of mass added to the end of a spring will affect the length of the period. If the mass is found to effect the period, then the relationship between the variables will be explored.
Diagram: Apparatus Set-Up
The following is a diagram of the apparatus used in this investigation.
**Paper copy of diagram available**

Controlled Variables/ Constants

In order to increase precision and reliability- the same metal stand, spring, and meter stick(shown above) were used in each trial. The period was measured each trial from when the mass was dropped  to when the mass returned to that same spot. This start/stopping position was held constant at 40 cm above the table, and was clearly marked my the meter stick adjacent the metal stand. Additionally, the bottom of the spring, rather than the bottom of the mass, was used to judge starting and stopping position- so as to keep drop height consistent even though the masses used were slightly different shapes and lengths.  The meter stick also remained in the same place throughout the trials. The same spring was never removed from the apparatus during the investigation, therefore ensuring a consistent position of the spring, weight of the spring (47 grams), and k-constant. The units of measurement for time and mass (seconds and grams, respectively) remained the same. Finally, the technique for dropping the weight was held constant  by having the same experimenter drop the weight for each trial. For example, no force was added to the mass or the spring- it was simply released.

Justification: Range of IV and Number of Trials  

The levels of the independent variable range between 70 grams and 212 grams. The first IV level was set at 70 grams because when trials were attempted at a lower mass (50 grams) the period was to short to reliably time. The upper limit was set at 212 grams due to the fear that more mass would over-stretch and permanently damage the spring. There were five trials to test the dependent variable. Five trials were done to increase reliability of the data, in that the precision of the data can be judged. Additionally, if an outlier is produced taking the average of all five trials will increase the accuracy of that data point.

Procedure
 In order to collect the data below, the apparatus is set up as shown above in the diagram. A metal stand is placed near the edge of a flat table, to ensure the mass can complete a full period without hitting the table. The meter stick is secured, parallel to the metal stand, to mark the start/stop point. The spring is secured to the end of the metal arm, and the masses are weighed and hooked to the bottom of the spring.
The point where the bottom of the spring meets the top of the mass is lined up at the 40 cm mark. The first experimenter drops the weight, careful to simply release the mass without adding any additional force to the mass or the spring. As the weight is released, a second experimenter starts the timer, stopping it when the mass returns to the initial position. The time is recorded. This is repeated five times at each IV level.
Data & Processed Data
                                                                                                                                          
|                                   |       Period ( +/- 0.005 seconds)           |              |  Statistical    |
|Mass (+/- 1.5 grams)| Trial 1| Trial 2| Trial 3| Trial 4| Trial 5| Average | Uncertainty |
|           70                    |  0.54   |  0.60   |  0.54   |  0.49   |  0.58  |  0.44      |    0.05           |
|          100                   |  0.63   |  0.69   |  0.72   |  0.62   |  0.74  |  0.68      |    0.05           |
|           141                  |  0.95   |  0.87   |  0.81   |  0.95   |  0.84  |  0.88      |    0.07           |
|           170                  |  1.00   |  1.09   |  1.02   |  1.00   |  1.01  |  1.02      |    0.05           |
|           212                  |  1.15   |  1.11   |  1.26   |  1.19   |  1.28  |  1.20      |    0.09           |

Uncertainty
    The measurement uncertainty for the mass is set at (+/-) 1.5 grams. This was done because each mass was measured individually prior to starting the trials. The scale was correct to the nearest gram, meaning it had a measurement uncertainty of (+/-) 0.5grams. In order to add the maximum mass(212 grams) to the spring three smaller masses were attached together at the end of the spring. When the uncertainty of all three smaller masses is added together, the measurement uncertainty of 1.5 grams is found.
    The measurement Uncertainty for the period is set at (+/-) 0.005 seconds because the timer recorded the time to the nearest hundredth of a second; Therefore, it can be assumed the timer has a measurement uncertainty of (+/-) 0.005 seconds. However, the statistical uncertainty -ranging from (+/-) 0.05 seconds to (+/-) 0.09 seconds- is much larger than the measurement uncertainty. This uncertainty most likely was caused by human error (slow reaction time in starting/ stopping the timer. The statistical uncertainty is found by finding the range of the data and dividing it by two. For example:
               Sample Calculations (Statistical uncertainty for the 212 gram IV level)
        1) The maximum (1.28 seconds) and the minimum (1.11 seconds) values are found.
        2) The range is found by subtracting the min. value from the max. value: 1.28s-1.11s= 0.17s
        3) The range (0.17s) is divided by two: 0.17s/2= 0.85s
        4) The uncertainty is rounded up (0.9 s), so it can be reported as only one significant figure
Raw Data Graph

This graph displays the average period at each IV level, no relationship has been found yet.

Linear Fit

 A linear fit (shown above) does not work for this data because while it fits easily through the error bars in the y-axis, it does not go through all the error bars in the x-axis. Additionally, with the linear fit there appears to be a y-intercept- suggesting that when there is no mass attached to the spring a period of approximately 0.12 seconds would still exist. The relationship is clearly not a linear one, and the graph appears to curve slightly as more mass is added. The horizontal parabolic shape lead to an attempt to linearize the data by taking the square root of the mass.
Sqrt(Mass) vs. Period

     In taking the square root of the mass that was attached to the spring, there is a clear relationship to the duration of the period. The uncertainty in the x-axis was reduced from the original (+/-) 1.5 grams, to (+/-) 0.75 grams. This is because taking the square root is equivalent to raising the masses to the 0.5 power. Thus, the uncertainty is multiplied by 0.5.
     The mathematical relationship between Period and Mass is that:
                    Period=  0.1206[seconds/sqrt(grams)] x [Sqrt (Mass)]

  Conclusion

     The period is proportional to the square root of x The slope of the linear line of best-fit in this graph means that approximately 0.12 seconds is added to the length of the period for every one (square root of gram.). In other words, as time increases, mass increases by a little less each time.
The y-intercept is assumed to be zero because the min/max slopes surround the origin. This makes sense because the square root of the mass is at zero when the mass is at zero,meaning there is no mass attached to the spring; and when there is no mass it is logical that the period is also at zero seconds. Hence, with no mass to apply force to the spring, it does not  stretch and there is no period.
     The biggest source of error was definitely human error in the reaction time when starting and stopping the timer. Much of the mistakes in timing showed up as random errors throughout the data. However, a source of systematic error in this investigation may arise in the IV levels. In the future, the spring's 47 grams of mass should be considered when graphing the mass against the period. Upon realizing that only the mass added at the bottom of the spring was counted in calculations, the label on the x-axis of the graphs was changed from simply "mass" to "mass attached to spring"
      In the future, it would be very beneficial to use a stronger spring that could withstand more mass. Thus allowing for a greater range in the IV and making the parabolic shape of the graph much more prevalent from the beginning. Additionally, the speed at which the masses completed the period made it  impossible to take accurate data. In the future the trials could be video taped and, using video analysis, the trial could be slowed down so more accurate readings could be taken. With a stronger spring and video analysis of trials the range in the IV could be much larger and the data collected (even on the very small masses) would be much more accurate.