The Affect of Mass on the Period

Colin Wick 2/17/09

Period 4 IB Physics

How does the mass affect the period of a pendulum? We believe that if the mass of a pendulum were increased, then the period of the pendulum would decrease, because as the mass is increased so is the downward force of gravity acting on the pendulum, causing the acceleration of the pendulum to increase. This experiment will be conducted by taping a “light”, measured string to a table so that it hangs 90 degrees to the horizontal. Measured masses will then be attached to the string for each data point, and the resulting period recorded for five trials. The gathered data will be graphed in order to visually represent the affect that mass has on the period of a pendulum. The only variable that will be manipulated in this experiment will be the mass on the end of the pendulum, while the angle of release and the length of the string will remain same. The angle of release and string length will be controlled, because by changing either of them, the distance the pendulum must travel is also changed. Therefore, both controlled variables could conceivably affect the period of the pendulum. If our hypothesis is correct, then the period of the pendulum will have an inverse response to the mass, meaning that as the mass increases, the period of the pendulum should decrease.

The materials utilized in our experiment are listed in bullets below and set up according to the following diagram:

Pre-measured 20 gram mass
Pre-measured 50 gram mass
Pre-measured 100 gram mass
Pre-measured 200 gram mass
Pre-measured 250 gram mass
“Light” string (34cm)

Meter stick to measure string

Protractor
Stopwatch
Table
Tape

To conduct our experiment we set up the materials as illustrated in the previous diagram. We taped a light string and a protractor to a tabletop so that he string hung at a 90-degree angle to the horizontal floor. We then used the meter stick to measure off 34cm of the string (including a loop to connect masses to) and cut off the excess. Then, we measured all of the “pre-measured” masses using an electronic scale and obtained the following results: the actual mass of the 20g mass was 20g, the actual mass of the 50g mass was 50g, the actual mass of the 100g mass was 100.2g, the actual mass of the 200g mass was 200.2g, and the actual mass of the 250g mass was 250.3g. We then proceeded to collect data for a total of twenty-five trials for five data points (five trials per data point). For the first five trials of data point one, the 20 gram mass was attached to the end of the 34cm string. The mass was released from rest at 45 degrees to the vertical, as time collection began simultaneously using the stopwatch. When the mass reached its original position after being dropped (one full, “back and forth” swing), time was stopped and recorded, giving the time for one period. This same process was repeated four more times for the four other trials for data point one (20g mass). Five trials for data point two were then conducted and recorded in the same fashion using the 50g mass. We then conducted five more trials for data point three, using the 100g mass (actual mass was 100.2) in the same manner, and recorded the data. A 200g mass (actual mass 200.2g) was used to collect and record data for the five trials for data point four using the exact same method of experimentation used to collect information for data point one. Finally, the five trials for data point five were conducted and the results recorded using the 250g mass (actual mass 250.3g). All twenty-five trials were conducted and recorded exactly the same as the first trial for data point one (explained above).

The raw data that was collected is duplicated below as well as the analyses of the raw data, including a graph, justifications for our uncertainties, and explanations and examples of the calculations used during this lab.

Raw Data and Accompanying Graph:

This graph illustrates that there is no relationship between the mass and period of a pendulum because the slope of the graph is so close to zero. Therefore, the period constantly stays ...

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Raw Data and Accompanying Graph:

This graph illustrates that there is no relationship between the mass and period of a pendulum because the slope of the graph is so close to zero. Therefore, the period constantly stays exceptionally close 1.1s despite the change in mass. The uncertainty in mass is not represented in the graph because it is so miniscule that it cannot be seen without misrepresenting data.

The calculations that were done in this lab are represented below through explanations and examples (all examples use data from data point three), as well as the various uncertainties used.

Uncertainty of Mass:

The uncertainty of the mass is plus or minus 0.2g. The Uncertainty is very low because the measurements were done using an electronic scale. Therefore, there is only a very small margin of error involved.

Uncertainty of the Period:

The uncertainty of the period is plus or minus 0.1s for each of the trials and was based off of the human reaction time. Any deviation from the true value would be a result of human error (the time it takes to start/stop the stopwatch after the true start/stop time has passed).

Uncertainty of String Length and Release Point

Both the uncertainty of the string length (0.2cm) and the uncertainty of the release point (one degree) are based off of the inability of humans to be exact in reading measurements on tools. They are also based partly on possible defects of the protractor and meter stick utilized in the lab.

Average Period:

Add up the total of all the trials and then divide by the number of trials (5)

Example: (1.1s+1.1s+1.1s+1.1s+1.1s)/5=1.1s

Average Period Uncertainty:

Find the difference between the trial that is farthest away from the average period and the average period. Then, add that difference to the established uncertainty of each trial—in the case of this lab that uncertainty represents the human error or stopping/starting the stopwatch and is 0.1s.

Example: 1.1s-1.1s=0s 0s+0.1s=0.1s

In completing this lab, we found that there is no clear relationship between mass and the period of a pendulum. This is seen when analyzing the graph of the average period in relation to mass. The slope of the graph is 0.0001, which is extraordinarily close to a slope of zero. Because the slope is so close to zero, when the lines of the maximum and minimum gradients are factored in, we can see that a line with a slope of zero lies in between those two gradients. Therefore, with a slope of zero, it is made obvious that there is no visible relationship between the mass and the period of a pendulum because any mass times zero will always be zero, so the resulting period will never change. In the case of our data, the resulting period of a 34cm string stayed a fairly constant (remember, slope was not exactly constant zero due to most likely a small amount of error) 1.0707s. This 1.0707s period is the constant of the equation of the graph, or the y-intercept. This lack of a relationship that is represented by our data could be a result of two things. The first being that data we collected was in some way faulty due to error. The second and what we see as being a more reasonable explanation, is that there is simply no correlation between mass and period, and that the two variables are independent of each other. We believe that the mass of a pendulum does not affect its period for two reasons. The first is that we believe our data is fairly reliable because all of our trials for each data point had had very similar results. The second is that the idea that mass would not affect the period is a very reasonable theory. Although the force of gravity increases with the mass, which would theoretically increase acceleration, acceleration does not increase because it takes more force to move the larger mass from rest. Therefore, the increase in force and the increase in mass cancel each other out causing the acceleration to stay constant. This idea can be represented with the following equations where m=small mass, M=large mass, f=small force, F=large force, and a=acceleration: F/m=a and f/M=a, therefore F/m=f/M. Therefore, we can confidently say that mass is independent of the period of a pendulum, which proves our hypothesis incorrect.

The lab as a whole was conducted fairly smoothly and without any large incidents. The only possible systematic error could have been the air resistance that each mass encountered. Because the larger masses had a larger surface area, they encountered more air resistance. This error would cause the period to not only be larger overall, but also for the period to increase a little as the masses increased, which is what most likely accounts for the small slope of 0.0001 instead of zero. The inevitable random error that occurred in the lab, which also may account for the slope, could have come from a number of places. It could have easily originated from the error in the human reaction time, in other words our inability to stop and start time at exactly the right moments. This error could cause both an increase or decrease in the period because the button of the stopwatch could have been pressed early or late. Despite this random error that may have had a human origin, there was a more significant event that may have caused some error in the data as well. During the fourth trial of the third data point, the increase of the mass and the resulting increase in downward force caused the tape to pull off of the table and the string and mass to fall. Although when the string was re-taped it was again measured to the same length, it was most likely not exactly the same as before. This again could have caused the rest of our data to either increase or decrease, depending on whether or not the string was accidentally lengthened or shortened during the reassembly of the lab.

There are a number of improvements that could be made to the lab that would decrease, and in some cases eliminate both the systematic and random error that was encountered. The first improvement would almost entirely eliminate the systematic error of air friction by conducting the trials inside a vacuum. By eliminating air friction, we would make the surface area of the masses obsolete. The second enhancement to the lab that would make the gathered data more reliable, would be to instead of tape the pendulum to the table, to tie it to a bar bridging two tables. Therefore, the bar would be able to hold the added mass successfully, unlike the tape, allowing us to gather data without having to occasionally reset the lab, lessening the chance that the string length would change. Finally, we could set up an infrared timing system where a beam would be aimed at the top of the pendulums swing on one side and when allowed to connect would start time (the mass would be in the way at the start), and when broken again (when the mass has completed one period) would stop time. This would make the collection of time much more accurate, eliminating the human error of pressing the start and stop button on the timer at exact moments.

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