An Investigation to discover whether the string length of a pendulum affects the pendulum in any way.

Andrew Lyons 4th December 01

Introduction

A pendulum is a device made up of an object suspended from a fixed point that swings back and forth under the influence of gravity. Pendulums are used in several kinds of mechanical devices; for example, certain types of clocks use pendulums.

The most basic type of pendulum is the simple pendulum. In a simple pendulum, which swings back and forth in a single direction, all the mass of the device can be entirely in the suspended object. The motion of pendulums, such as those in clocks, closely approximates the motion of a simple pendulum. A spherical pendulum is not confined to a single direction, and as a result its motion can be much more complicated than the motion of a simple pendulum.

The principle of the pendulum was discovered by Italian physicist and astronomer Galileo, who established that the period for the back-and-forth swing of a pendulum, of a given length, remains the same, no matter how large its arc, or amplitude. (If the amplitude is too large, however, the period of the pendulum is dependent on the amplitude.) This phenomenon is called isochronism, and Galileo saw its possible applications in timekeeping. Because of the role played by gravity, however, the period of a pendulum is related to the location, because the strength of gravity varies as a cause of latitude and elevation. For example, the period will be greater on a mountain than at sea level. Therefore, the pendulum can be used to determine accurately the local acceleration of gravity.

(Reference: Encarta Encyclopedia 1999)

Planning

For this investigation I choose to investigate what effect the string length would have on the time it takes a pendulum to do one complete swing. This was not the only key factor that I could have changed. I could have investigated whether the weight of the pendulum, the angle you drop the pendulum from, or the material that the string is made from will make any difference.

For my investigation, I predict that the length of string does affect the pendulum. I think that when the string is shortened, the time it takes the pendulum to do one complete swing (there and back again) will be shorter than when it was longer. I think that this will happen because in the formula T=2π√(L /G) length is divided by gravity. This means that changing the length will change the time. I also predict that the length of the string will be directly proportional to the square of the time taken to do one complete swing. I have predicted this because of the formula. There is a square root sign on the right hand side of the equation, which makes me think that you must square the time.

To make this investigation fair, I am going to only change one input variable at a time. In my experiment this will be the length of the string. I will keep the other variables the same. These are the weight of the pendulum, the type of string, the angle that I am dropping the pendulum from, and the force I am releasing the pendulum with (for example, if I push it out of my hand I should push it with the same force each time). I will also time the swings with the same stopwatch and stop it after ...

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I carried out some preliminary work at home before the investigation. I took a piece of string and tied a coffee mug to the end of it. I then held it out at a non-specific angle and let it swing back and forth. I watched it swing back and forth and then shortened the rope as it was swinging. The mug began to swing back and forth faster than before. I carried out this work to give me some idea as to whether my prediction was right or not. I also found out a formula to use to find out what the actual time should be, so I would be able to check my results against it and see if I my results were correct. I also collected some general data on pendulums as an ‘introduction’ to the investigation. This can be found in the introduction section at the start.

The following is my method of what I am going to do:

I will set up the apparatus as in the diagram shown on the separate sheet.

I will check that the string on the pendulum is 100cm to the pivot so that I will have accurate results.

I will draw an angle of 20 degrees on a piece of paper using a protractor. I will use this to make sure that when I hold back the string, before I release the pendulum, I am bringing it back to 20 degrees. This will make sure that it is a fair test and that my results will be correct.

I will line up the pendulum with the 20˚ mark on the piece of paper, and have a stopwatch ready to begin timing.

I will release the pendulum and start the stopwatch timing.

I will wait for the pendulum to do 10 swings, and then stop timing. As I only want the time of one swing, I will divide this time by ten. I will measure it at ten swings, as it will be easier to measure than starting and stopping the stopwatch very close together. I am hoping that it will be more accurate this way.

When I have stopped the watch, I will record my results in a table.

I am going to repeat each length three times, so that I can get more accurate results. I may have gotten an anomalous result the first time and this will let me see if I did. It will also give me enough numbers to work with to form an average time.

9) When I have repeated the experiment two more times, I will shorten the pendulums rope. I am going to shorten the rope by 10 cm’s each time. By the end I should have recorded results from 100, 90, 80, 70, 60, 50, 40, 30, 20 and 10 cm’s. I will go down in even steps, as this will make it a fair test and easier to work out any sums with.

10) At each length I will repeat the steps 4 – 7.

11) I will work out the averages and record them, I will also work out what the correct answer would have been using this formula: T=2π√(L /G)

12) I will put the equipment away.

The Formula

The formula T=2π √(L /G) is used to find out what the time to complete the swing should have been if the experiment was flawless. T is the time it takes to complete the swing, L is the length of the string at the time, and G is gravity, which is 9.87m/s².

To make this investigation safe, I will put some weight on the base of the clamp stand. I will do this in case the table is knocked and the stand falls over. The weight should hopefully stop this happening. I will also make sure no one is standing near by, so that the pendulum does not hit him or her when it is swinging.

Results Table

The degrees change ½ way through the experiment because of a mistake on my part. When drawing the 20-degree angle on a sheet, I accidentally drew it at 30 degrees. However I do not believe that this will make a difference as the angle you hold the pendulum at was not in the formula.

I got the calculated time by using the formula T=2π √(L /G), where L is the length (m) and G is gravity, which is a constant at 9.87 m/s².

I considered whether the first result would be zero or not, and decided not to show it on my table or graphs. I decided this because although you could say that the pendulum does zero swings in zero seconds, you could also argue that it does an infinite number of swings in that period of zero seconds. I thought that this would be too complicated to show on my graph and results, so decided not to include it.

Conclusions

From doing the investigation and drawing out a graph (see separate sheet) I have found out that as you change the length of the string of a pendulum, the time it takes the pendulum to do one oscillation is also changed. This was what I had predicted earlier on, so that part of my prediction is correct. This is shown on my graph showing the average time that I calculated. The line on the graph gets gradually higher. I also predicted that as you shorten the string, the time taken to do one oscillation was also shortened, and this is also shown to be correct from my results. For example, when the string length was at 100cm, I measured the average swing time at 2.01 seconds. Then at 90 cm, I measured the average swing time at 1.92 seconds, and at 80 cm, I measured it at 1.76 seconds. The time is getting lower, and the string length with it. My final prediction was that the length of the string would be directly proportional to the square of the time taken to do one complete swing. This prediction also seems to be true, by looking at my graph. The graph shows a reasonably straight, continuous line, except for the point that is below the straight line of the others. This shows a direct proportion between them and backs up my final prediction.

Evaluating Evidence

During this investigation, I met two difficulties. These were that I found it difficult to measure the angle at which I should pull the pendulum back to. I could not quite get the pendulum to line up exactly each time with the line on my paper. This could be one reason why some of the results were not what they were meant to be from using the formula. The other reason was that, as the investigation was cast over two lessons, when I re-drew the angle on the paper, I drew it to 30º instead of 20º like I was meant too. Although I did this, I believe that it did not change my results, as the angle that you hold it from, was not one of the variables in the formula.

I think that my results were actually very accurate. The result that was furthest off was the final one at 10cm, and that was only off by 0.07 seconds. However when I drew my graph, the result with string length 80 seemed to be the most off, as it did not lie in a straight line with the rest of the points. I think that the reason that this result was off was because the pendulum was moving very quickly here, and I could not stop the watch at the exact right time.

I have already mentioned two possible sources of error, however I do not think that one would have made any difference. Another reason could be that instead of releasing the pendulum gently, I could have gave it a slight push, which may have affected the results slightly.

My results lie very close to the line of best fit on my graph. I think that the evidence is sufficient and reliable enough to make a firm conclusion. There were also no anomalous results which shows that the measurements were reasonably accurate, or that I was the same amount off every time for some reason.

To improve my experiment, I could have used a different timing method. Timing by hand is all right, however it is not completely accurate. To improve it I could use a clock, which begins timing at the moment that the pendulum is released. This way, my results will be more accurate. I could also use a type of string, which has less friction on the pivot. The friction on the pivot may be slowing the pendulum down faster, and may account for some inaccurate results. Another way would be to do the experiment in a vacuum. This would be difficult to do, however there would not be any air resistance on the pendulum, and it would, once again, stop it from slowing down as fast and I would get more accurate results.

As an extension for this investigation, I could see what effect different types of string will have on the pendulum swing. I believe that it will make a difference as some strings, such as fishing wire, have less air resistance and will create less friction in the pivot than the string that I used for this investigation.