Oscillating Mass

Dependent: The period T of the oscillating mass; the time taken to make one full swing. This         

measurement will be done by using a stop watch and will rely greatly on my vision and reaction time.         I could not find an apparatus that would automatically record this dependant variable. However, a         

stop clock is sufficient.

Controlled: Length of elongation l (if m is the independent variable) and the mass m of the object (if         length l is the independent variable). The mass m of the object is kept constant by not switching masses         in the process, and the length of elongation will be controlled by displacing the spring by the same         

length while using different masses. The spring constant k of the spring is also controlled by using the         same spring throughout the experiment.

2        Planning B

2.1        Apparatus

1. 1 elastic spring
2. 1 Meter stick
3. 1 Analytical balance
4. 1 Stop watch
5. 1 Ring stand
6. 1 each of the following masses: 0.400kg, 0.600kg, 0.800kg, 1.00kg, 1.20kg, 1.40kg, 1.60kg.

2.2        Method

Part I        Mass m as a Variable

1. Determine the spring constant k of the spring used. Attach a mass m to the spring on the ring stand         and measure the displacement x of the spring relative to its equilibrium position. The value where          will give you the spring constant k in .
2. Accurately measure the mass m of the object and attach it to one end of the spring.
3. Set the meter stick parallel to the ring stand, and elongate the spring 10cm.
4. Let go of the spring as you start timing on the stop watch. Determine the amount of time it takes for         the mass to complete ten full oscillations. This is the period T of the oscillating mass. Repeat this

process 3 times to obtain an average period T of one full oscillation. Timing 10 periods lowers         

uncertainty. Record mass-period pair values for the mass.

5. Repeat steps 2-4 for the remaining masses while making sure not to switch springs in between to         

keep k constant: 0.600kg, 0.800kg, 1.00kg, 1.20kg, 1.40kg, 1.60kg. Observe whether the spring is         

oscillating slower or faster.

Part II        Length l as a Variable

1. Keeping the mass constant at 0.400kg this time, time ten full oscillations of the mass T while         

6. elongating the spring at 5cm. Repeat this process 3 times to obtain an average period T of one full         oscillation. Timing 10 periods lowers uncertainty.record length-period pair values for each length.

2. Repeat step 1 for the remaining lengths while making sure to keep the mass constant: 10cm, 15cm,         20cm, 25cm, 30cm, and 30cm. Record length-period pair values for each length. Observe whether the         spring is oscillating slower or faster.

2.3        Safety

1. Swinging masses on springs are dangerous. Avoid getting hit by them during the experiment.
2. Do not elongate the spring too much for small masses as the mass may swing out and pose a threat to         others.

3        Collection and Analysis of Data

3.1        Data Collection

Part I        Elongation of the spring:

Length of spring at equilibrium:

Data Table 1: Varying mass of oscillating object and raw period times of ten oscillations.

Qualitative Observations made: The 1.60kg mass oscillates significantly slower than the 0.400kg. The ring stand also wobbled a little during the experiment, which is a source of uncertainty.

Data Processing:

Now in the following table I will first calculate the time for one oscillation in each trial, then I will add up the period times from the three trials and divide them by the amount of trials:

Sample calculation for 0.400kg:

Using the data above, I am also able to calculate the spring constant k

After doing the above calculation for all masses and elongations above and averaging them all, I've found the spring constant of the spring I used to be

Data Table 2: Varying mass of oscillating object and average period of one oscillation. 

In the table above I have also found the values for , or the period squared. This was done because I predicted using the formula derived in the hypothesis that there should be a linear relationship between the mass and the period squared.

Data Presentation

        For the above data I will construct two graphs, one that will display the mass against the period, and one of the mass against the squared period. The first graph will allow us to see how exactly mass directly affected the period. On the x-axis the uncertainty is negligible. On the y-axis the uncertainty is ±0.01s.

        The other graph will be plotting mass m versus T2, because as mentioned before in the hypothesis, there should be a clear linear relationship between the two. On the x-axis the uncertainty is negligible. On the y-axis the uncertainty is ±0.01s2.

Part II        Mass of oscillating object: 0.400kg (uncertainty negligible)

Length of spring at equilibrium:

Data Table 3: Varying elongation of spring and raw period times of ten oscillations.

Qualitative observations made: The mass seemed to oscillate at around the same period in every trial done.

Data Processing:

Now in the following table I will first calculate the time for one oscillation in each trial, then I will add up the period times from the three trials and divide them by the amount of trials:

Sample calculation for 5.00cm:

Data Table 2: Varying elongation of the spring and average period of one oscillation. 

Data Presentation

        For the above data I will construct a graph of the elongation against the squared period. The graph will show how much, or how little elongation's effect has on the period. On the x-axis the uncertainty is ±0.05cm. On the y-axis the uncertainty is ±0.01s.

4        Conclusion and Evaluation

4.1        Conclusion

        Coming back to my hypothesis, I was correct to suggest that mass would affect the time period of an oscillating mass, but the length of elongation would not. This is shown in graphs 1 and 3. Graph 1 depicts an exponential relationship between period and mass. Therefore, after straightening the graph, I was able to interpret that there is a linear relationship between the squared period and the mass. This can be justified because if we square both sides of the equation derived in the hypothesis, we get

 

Where k is a constant, therefore being consistent with my thesis.

        This equation represents a relationship between the two variables.  So in theory, once plotted, we should have obtained a somewhat exponential curve.  When I plotted T vs. m, a clear exponential curve was seen.  This says that my data is correct, to a certain degree, however it would be ideal if I had expanded the investigation to look at greater masses until I saw an even more evident curve.  

Furthermore, in the general straight line formula y = ax, if x was assigned the value of T2 then we would obtain a straight line.  Luckily, as shown in graph 2, my data coincides with this reasoning. When I plotted T2 versus m, there was certainly a linear relationship between the two.

In my hypothesis I predicted that the length of elongation would not effect the period, and the evidence is shown in Graph 3. The best-fit line drawn through the data points is almost flat, with the slope being 0. This is also consistent with my hypothesis.

4.2        Evaluation

        Judging by the difference between maximum and minimum slopes in Graph 2, 0.18, I conclude that the error for that portion of the lab was minimal. But one must keep in mind that when I was timing with the stop watch, I only recorded to 0.1 of a second, which may seem imprecise at first. However, it is known that human’s reaction time is around 0.1-0.2 seconds.  As a result of this, my times were limited to 0.1 of a second. Repeated trials were done to reduce error. Nonetheless, our graphs show that I was indeed accurate enough to notice the correct correlation in the graph drawn. To further improve accuracy, more repetitions could have been done. In the future I may do 5-7 trials of the same situation.

        The procedure I planned out allowed me to test my hypothesis adequately. Limitations in the procedure include the fact that it was difficult for me to experiment with greater masses since I was unable to find a place high enough to allow to mass to oscillate. This limited me to only investigating masses under 1.60kg.  Weaknesses included being unable to have a steady ring stand (despite many efforts to force it down) on which the mass could oscillate freely without wobbling from the stand. Getting the oscillation to happen in a flat plane also proved difficult.  

Obvious sources of error would include having tools that were difficult to measure precisely with, namely the meter stick and the stop watch, and also getting the spring to suspend symmetrically from one point. Also as I previously mentioned, the stop watch is only accurate to about one tenth of a second.

4.3        Improvements

        Realistic improvements that could cut some weaknesses of the procedure could include using greater masses to obtain a larger range of results and doing the procedure in a greater space. The reason for this change would be to increase the time of oscillation because when we used smaller masses, often the time of the period was so short it was very difficult to start and stop the timer. If the period becomes slower and takes up more time, it could make timing more precise. Furthermore, more data points would increase the likeliness of an obvious correlation.

Moreover, I believe the more trials done, the better chance you will get of obtaining the best average value. The procedure itself is not that complicated, and I believe if I took the time to do 5-7 trials instead of just 3, the data would be more accurate. To improve the investigation I would also suggest using photo gates instead of stop watches to obtain time measurements more accurately and precisely.