Variables:
Independent – The independent variable in this experiment is the length of the string
Dependant – we will be measuring the period of the pendulum, this is also the time taken to make one full swing. This measuring will be done by using a stop clock and will rely greatly on our vision and reaction time. We could not find an apparatus that would automatically record this dependant variable; however a stop clock is sufficient.
Control – The controls in this experiment are the mass of the pendulum and the angle at which it is dropped from, both of which will be kept constant. We will keep the mass constant simply by never changing the bob on the pendulum, and the we will keep the angle relatively controlled by measuring the same angle before every pendulum swing. The acceleration due to gravity, g, will be assumed to remain constant around 9.81 ms-2 because we will perform the experiment and all trials in the same room.
Apparatus:
- Meter Stick
- Stopwatch
- Protractor
- String (non extensible)
- Pendulum bob (minimal air resistance; metal)
- Lab stand
- Cartesian graph paper
Method:
- Set up the simple pendulum. Check to make sure the string is tightly tied around the lab stand and does not slide back and forth. Choose an angle, preferably not too big, that will remain constant throughout the experiment and will be the angle you always release the pendulum from.
- Investigate if the period is dependant of the length of a pendulum. Choose 5 lengths (e.g 100, 80, 60, 40 and 20cm) of string and make sure each type is identical in material and quality.
- For each length, swing the pendulum bob several (preferably 45or so) times from the same angle and record the five trials in a table
Collection and Analysis of Data:
- You should have now obtained a table with the trails for each of the five given lengths. Now you should compute the average period for the length and also place it in a table.
- Now compute the theoretical period for each pendulum length using the expression previously mentioned and place the results in a new data table.
- Calculate the percent error between the experimental and the theoretical values of the period for each pendulum length and record in a table.
Data Collection:
Mass of pendulum bob: 2.4 grams (± 0.1 gram).
Angle at which bob released: 45º (± 1.0º)
Data Table 1: varying length and raw period times
Data Processing:
Now in the following table I will first calculate all the averages for the different lengths by simply adding up the period times and dividing by the amount of trials:
Sample calculation for 0.4m: 1.2 + 1.3 + 1.2 + 1.2 +1.3 = 6.2, divided by 5 = 1.24. limited to 2 decimal places, = 1.2 s
Then I will calculate the theoretical textbook values by using the formula I introduced at the start of this lab:
Sample calculation for 0.4m: where L is 0.4 and g is 9.81, T = 1.26874… limiting to two decimal places = 1.3 s
Finally I will find the percent error between the experimental average and the theoretical value for the period by finding the difference between the two, dividing by the theoretical value and finally multiplying by 100 to a percentage:
Sample calculation for 0.4m: (1.3 – 1.2) / 1.3 = 0.07692… ∙ 100 = 7.69%
Data Table 2: varying length and avg. period times compared to theoretical period times
Also in the table above I have found the values for T squared, or the period squared. This was done because the theoretical formula shows that there should be a linear relationship between the length and the period squared.
Data Presentation:
For my data I will construct two graphs, the first one will display simply the length against the time period. This graph will allow us to see how exactly one factor directly affected the other. On the x-axis the uncertainty is ± 0.01 of a meter, as we had set before. On the y-axis the total uncertainty is ± of a second
The other graph will be plotting L versus T2, because as the theoretical formulae shows, there should be a clear linear relationship between the two. This graph is perhaps more relevant to us because from my hypothesis I know that this should be a straight line.
Conclusion
Coming back to our hypothesis, I was correct to suggest that length would affect the time period of a simple pendulum. Moreover, after plotting the graphs and interpreting the results, it can be seen that there is a linear relationship between the length and the time period squared. This can be justified because if we square both sides of the equation, we get
or
This equation represents a hyperbola in the form of y = ax2. So in theory, once plotted, we should have obtained a somewhat parabolic curve. Unfortunately when I plotted L vs. T, a clear parabolic curve could not be seen. This is not to say that our data is incorrect, however it is possible that we would need to expand the investigation to look at greater lengths until we finally see a parabola emerging.
Furthermore, in the general straight line formula y = ax, if x was assigned the value of T2 then we would obtain a straight line with the gradient a being g/4π2. Luckily, our data coincides with this reasoning. When I plotted T2 versus L, there was certainly a linear relationship between the two.
Evaluation
By looking at the percent error between the real theoretical values and the experimental values I myself obtained, it appears as if the procedures chosen and limitations implemented worked well. Although for all but one of the experimental values the percentage of error was zero, it does not necessarily mean our experiment was flawless. One must keep in mind that when we were timing with the stopclock, we only recorded to 0.1 of a second, which may seem imprecise, however it is known that human’s reaction time is around 0.1-0.2 seconds. As a result of this our times were limited to 0.1 of a second, and when comparing the experiment and theoretical values, that made a big difference because it made it seem as if there was absolutely 0% error, which is not necessarily true. Nonetheless, our graphs show that we were indeed accurate enough to notice the correct correlation.
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 measure greater lengths of string since I was unable to find a place to suspend the string from. This limited us to only investigating lengths of 1 meter and less. Weaknesses included being unable to have a steady angle measurement from where to suspend the pendulum bob and also getting the oscillation to happen in a flat plane.
Obvious sources of error would include having tools that were difficult to measure precisely with (the ruler and the protractor) and also getting the pendulum to suspend symmetrically from one point. Also as I previously mentioned, the stop watch is only accurate to about one tenth of a second. On top of that, we had one member of the group releasing the pendulum, and another activating the stop watch, therefore their actions did not coincide directly which could have led to a delay in some results.
Improvements
Realistic improvements that could cut some weaknesses of the procedure could include using greater lengths of string 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 lengths shorter than a meter, 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.
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 we had time to do 10 trials instead of 5, which would be been better. To improve the investigation I would also suggest having the person who releases the pendulum start the timer as well. This is because he/she will have a better understanding and faster impulse as oppose to someone who is waiting for the pendulum to drop
