In this Coursework, we were given the task of investigating some factors which affect the period of a simple pendulum.
PLAN
INTRODUCTION:
A pendulum is a device which consists of an object suspended from a fixed point that swings back and forth under the influence of gravity. This effect is known as gravitation. However simple it may seem, this structure is very beneficial in our everyday life for it is used in several kinds of mechanical devices such as the all popular grandfather clocks. In addition to this, a pendulum could determine the local acceleration of gravity. This is the case, as the strength of gravity varies at different latitudes and as gravity is one of the main forces acting on the pendulum, the acceleration of gravity could be noted. Further uses of the pendulum are found in the field of astronomy for some have been used to record the irregular rotation of the earth as well as to detect earthquakes and others are used to demonstrate the rotation of the earth.
The pendulum and its applications were first discovered by Italian physicist and astronomer Galileo, who established that the period for the back-and-forth oscillation 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.
A pendulum can be seen as a device whose energy is continually changing. When the pendulum swings to & fro, its energy changes from gravitational potential energy to kinetic energy and so on. This diagram of a simple pendulum clarifies this:
AIM:
In this Coursework, we were given the task of investigating some factors which affect the period of a simple pendulum. A period, is one complete oscillation. Possible factors that effect the period of a pendulum are:
- Mass of object suspended
2- Length of string
3- Angle.
However, due to limitations of time, and the complexity of carrying out some of the factors mentioned above, I will only look at 2 of the factors to investigate. They are:
- Length of string
2- Angle.
Before I give my predictions, I will give a brief outline for the experiments I shall be carrying out to allow the reader to understand my predictions.
- VARYING THE LENGTH: In this experiment, as the title suggests, I will be varying the length of the string the object is suspended from. The range will be between 0.1 M and 1 M & I will be going up in fixed intervals of 0.1 M. I will then measure the time it takes for 10 complete oscillations to occur at each length.
2- VARYING THE ANGLE: In this experiment, I will be varying the angle. The range will be between 10 & 80 degrees and I will be going up in fixed intervals of 10. I will then measure the time it takes for 10 complete oscillations to occur at each angle.
VARIABLES:
Before going on to explaining the experiments I will be devising in detail & my predictions, I must distinguish between three types of variables:
- INDEPENDENT VARIABLE: This can be considered as the input of the experiment. It is therefore deliberately changed, in order for us to see the effects of its change and obtain the results.
2-CONTROL VARIABLE: This is the factor which insures that the experiment is a fair test. It is always kept constant while the independent variable changes.
3- DEPENDANT VARIABLE: This can be considered to be the output of the experiment. It depends entirely, on the variation of the Independent variable.
In the experiment where I am varying the length, the variables are as follows:
Independent variable: The length of the string. I will be testing my experiment when the length of the string is at 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 & 1 metre.
Control variable: In order to maintain a fair test in this experiment, I shall control the following variables:
- The mass of the suspended object which is 21 grams.
2- The angle which will be 10 degrees.
Dependant Variable: The dependant variable is the time it takes 10 periods to occur.
In the experiment where I am varying the angle, the variables are as follows:
Independent variable: I will be testing my experiment when the angle is at 10, 20, 30, 40, 50, 60, 70 & 80 degrees.
Control variable: In order to maintain a fair test in this experiment, I shall control the following variables:
- The mass of the suspended object which is 21 grams.
2- The length of the string which will be 20 centimetres.
Dependant Variable: The dependant variable is the time it takes 10 periods to occur.
PREDICTIONS & THEIR EXPLANATION:
EXPERIMENT No. 1- VARYING THE LENGTH:
- As the length of the string increases, the period increases.
2- Their will be a proportional relationship between length of string & the period squared.
- As the length increases, the time taken for a period to occur will increase. This can very easily be justified by looking at the nature of arcs. As it is known, the object suspended from a simple pendulum moves from A to B through an arc. By increasing the length of the string, one side of the arc i.e. the radius increases and hence the object has to travel through a larger arc and a greater distance provided that the mass of the object and the angle remain constant.
The distance an object suspended from a pendulum travels to reach from A to B could be worked out. I will compare 2 situations where the length of the string varies and hence prove that the time taken for a period to occur will increase as the length increases.
Situation no. 1: When the length of string i.e. radius is 1 m and the angle is 10 degrees, we could use the formula X/360 * ?D, to work out the circumference of the arc. It will equal:
20 / 360 * ? * 2 = 0.35m
Situation no.2 : When the length of string i.e. radius is 2 m and the angle is 10 degrees, we could use the formula X/360 * ?D, to work out the circumference of the arc. It will equal:
20 / 360 * ? * 4 = 0.7m
The two situations prove that if the length of string (radius) increases, then the arc increases. i.e. the distance between A & B will increase. This will undoubtedly lead to an increase in the time it takes 1 complete oscillation to occur, and thus I could conclusively say that the period will increase as the length increases.
2- Initially, I thought that there will be a proportional relationship between length of string & period because when looking at the two situations above, we find that when the radius - length of string - is doubled, the arc also doubles in length. Hence, I thought that the period will double as the distance of the arc will double when we double the length of the string.
After further reading to this topic, I found that this will not be true. This was discovered, after reading the formula which relates period of the pendulum to ...
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2- Initially, I thought that there will be a proportional relationship between length of string & period because when looking at the two situations above, we find that when the radius - length of string - is doubled, the arc also doubles in length. Hence, I thought that the period will double as the distance of the arc will double when we double the length of the string.
After further reading to this topic, I found that this will not be true. This was discovered, after reading the formula which relates period of the pendulum to the length of string used. The formula is:
T = 2?? l / g
T = period
l = length of string
g = gravitational acceleration (approx. 9.81)
This formula does not provide a proportional relationship due to the existence of the ?. Therefore with this formula, a graph like the one below will be established:
A proportional relationship is present however between Period squared and length of string. The formula connecting this is:
T2 = 4? l / g
From this formula, we can derive the proportional relationship between period squared and length. The following graph should then be achieved to show this proportional relationship:
EXPERIMENT No. 2- VARYING THE ANGLE:
- I predict that as the angle increases, the period will remain the same. i.e. angle does not have an effect on period.
- As the angle increases, I presume that that there will be no changes to the period. Although, when I first thought about it, I thought that an increase in the angle will undoubtedly lead to an increase in the period as the distance the suspended object has to travel is greater with a greater angle. However, after I thought about it more carefully, I thought otherwise. It is true that the distance the suspended object has to travel is greater with a greater angle, but another factor must be considered. It is the amount of energy in the bob. We must remember that as the angle increases, the bob is raised higher and therefore the gravitational potential energy in the suspended object increases. Gravitational potential energy is proportional to velocity hence: The angle increases, leading to greater potential energy in the bob leading to the bob travelling much faster from point A to B than if the angle is less. I could say that although an increase in the angle means that the suspended object has to travel further than that of a smaller angle, its higher velocity compensates for it & visa versa. I therefore think that the advantage of the smaller angle having a smaller distance and having a smaller period is cancelled out by the increase in Kinetic Energy leading to greater velocity, found in larger angles. The following graph is a simple sketch of my predictions.
APPARATUS:
The following illustrates the apparatus used in my experiment:
METHOD:
Having shown my predictions, variables and the apparatus I will use for the experiment, I can now go on to explaining my method of carrying out the experiments. As is mentioned above, I will be carrying out two experiments. One where the independent variable is the length of the string, and the other where the independent variable is the size of the angle. Although they are two different experiments, they were both set up in the same way. The following diagram shows how they were set up:
The diagram above shows that a G- clamp, stand, cork, protractor, stopwatch and string were used to set up the experiments. The G- clamp was used to hold the stand firmly to the working bench. The stand was adjusted to hold a cork (see diagram above) which from it we suspended a bob from a length of string. Also attached to the cork, was a protractor, which was very carefully attached so it can be directly in line with the string. Now that I have explained how the apparatus was set up, let me go on to explain the procedure for each experiment.
EXPERIMENT NO 1: VARYING THE LENGTH
The following provides a step by step guide as to how this experiment was carried out:
- 1 metre length of string was measured using a meter rule from the point of suspension to the centre of mass of the bob. It was adjusted to 1 metre by means of releasing the xxxxxxx so the slit in the cork widens and the string can be adjusted. Once the string is adjusted to the fixed length, the cork is tightened and the string is held firmly to maintain the required length of string.
2- The bob is then elevated to an angle of 10 degrees. This can be measured by means of the protractor which is attached to the cork.
3- The bob is then released, and at the same time the stopwatch is turned on.
4- We allow 10 complete oscillations to occur and stop the stopwatch when they occur. This gives us the period of 10 complete oscillations. This is recorded on the table.
5- We repeat steps 1-4 three times and calculate the average period of 10 oscillations. This is recorded on the table.
6- We divide the average period of the 10 oscillations by 10 to find the period of one oscillation. The period is then squared. This is recorded on the table.
7- Steps 1-6 are repeated with 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 metre lengths of string.
EXPERIMENT NO 2: VARYING THE ANGLE
The following provides a step by step guide as to how this experiment was carried out:
- 1 metre length of string was measured using a meter rule from the point of suspension to the centre of mass of the bob. It was adjusted to 0.2 metres by means of releasing the xxxxxxx so the slit in the cork widens and the string can be adjusted. Once the string is adjusted to the fixed length, the cork is tightened and the string is held firmly to maintain the required length of string.
2- The bob is then elevated to an angle of 10 degrees. This can be measured by means of the protractor which is attached to the cork.
3- The bob is then released, and at the same time the stopwatch is turned on.
4- We allow 10 complete oscillations to occur and stop the stopwatch when they occur. This gives us the period of 10 complete oscillations. This is recorded on the table.
5- We repeat steps 1-4 three times and calculate the average period of 10 oscillations. This is recorded on the table.
6- We divide the average period of the 10 oscillations by 10 to find the period of one oscillation. The period is then squared. This is recorded on the table.
7- Steps 1-6 are repeated with the angles 20, 30, 40, 50, 60, 70 & 80 degrees.
SAFETY PRECAUTIONS:
Through carrying out this experiment, I took a number of safety precautions. They were:
- I used a G - clamp. This held the stand with the weights firmly in its place, preventing the stand from falling over and hurting anybody.
2- I carried out my experiment in a place which is inaccessible of moving people. This is because when the bob moves in an oscillation, it has high momentum and if it were too hit someone's eye, it will be a problem.
3- I cleared the area I carried out my experiment in a place where there were no obstacles in the way, thus reducing the chance of any danger.
OBSERVATION
RESULTS:
The following table includes my results for the experiment, where length was the independent variable:
Length of String (M)
Period of
0 complete
Oscillations (s)
Average of 10 T
T
2
T
T1
T2
T3
0.1
6.5
6.5
6.5
6.5
0.65
0.42
0.2
8.9
9
9.1
9
0.9
0.81
0.3
1.1
1.1
1.1
1.1
.11
.23
0.4
2.7
2.7
2.7
2.7
.27
.61
0.5
4.2
4.3
4.4
4.3
.43
2.04
0.6
5.5
5.7
5.6
5.6
.56
2.43
0.7
6.7
6.9
6.6
6.8
.68
2.82
0.8
7.8
7.8
7.9
7.8
.78
3.17
0.9
8.8
8.9
9
8.9
.89
3.57
20
9.9
9.9
9.9
.99
3.96
The following table includes my results for the experiment, where the angle was the independent variable:
Angle (?)
Period of
0 complete
Oscillations (s)
Average of 10 T
T
2
T
T1
T2
T3
0
9
9
9
9
0.9
0.81
20
9
9
9
9
0.9
0.81
30
9.1
9.2
9.1
9.1
0.91
0.83
40
9.3
9.3
9.3
9.3
0.93
0.86
50
9.5
9.5
9.5
9.5
0.95
0.9
60
9.5
9.6
9.5
9.5
0.95
0.9
70
9.7
9.8
9.8
9.8
0.98
0.96
80
0
9.9
0
0
ANALYSIS
(I have not drawn the line of best fit on any of my graphs, so I could compare my results to the linear results)
- VARYING THE LENGTH
It is evident from the results that I have achieved, and the graphs that I have drawn, that there is a very strong relationship between the period of one swing, and the length of the string. This relationship shows to be a positive one. i.e. As the length of the string increases, the period increases, for we find that at 0.1m (the smallest length of string used) the period is 0.65s and the period increases as the length increases, until the period reaches its maximum of 1.99s when the length is 1m (the longest length of string). This proves my first prediction right, which states that as the length increases, the period increase. This is justified in allot of detail on page x. Basically, the reason for the period to increase as the length increases, because an increase in length, leads to an increase in the arc (the distance the suspended object has to travel through). When the distance increases, the time taken for the bob will undoubtedly increase, leading to an increase in the period.
My second hypotheses for this experiment, refers to a proportional relationship between period squared and length. So not only am I looking for a general relationship between period and length, I am looking for a proportional relationship between them. A proportional relationship means that in doubling or tripling one factor (such as period squared) the other factor triples as well. I will test if this proportionality is true for period squared and length:
0.1m = 0.42s
The figures above show that when the length of string is 0.1m, the period is 0.42s. When I double 0.1m, my period should also double. My results show:
0.2m = 0.81s
The above shows that when I doubled the length of string to 0.2m, the period increased to 0.81s. In my opinion, this value satisfies the proportional relationship as it is almost double 0.42s. Hence, I can say that my experiment showed that T2 ? l and that it satisfies the theoretical formula T2 = 4? l / g.
In looking at the graphs, we can very clearly see that the accuracy of the experiment shows to be very high. This can be said, as the actual line is very close to the linear line. My experiment however, is not 100% correct as my actual line is not on line with the linear line. I will discuss this in more detail in the evaluation.
2- VARYING THE ANGLE
When looking back at the data recorded on page x, and looking at the graph plotted on page x for varying the angle, a general comment can be said about the relationship between amplitude of angle, and period. It is, as the angle increases, the period increases. I say general relationship, because there are some values which show that the angle increases, but the period stays the same. For example, at 50 degrees the period is 0.95s. When the angle is increased by 10 degrees to 60 degrees the period remained the same at 0.95s.
Eventhough though there are anomalous results, there is a general relationship between angle and period which must not be ignored. The fact that this positive relationship exists, proves my prediction wrong, which stated that an increase in angle will not have an effect on the period. The truth of the matter is, that an increase in the angle did have an effect on the period. It must be stated however that the difference between the smallest angle which was 10 degrees, and the largest angle which was 80 degrees (70 degrees in difference), was only one second.
When the amplitude of the angle increased, the bob attached to the string was lifted higher. The greater the angle, the higher the bob. This meant that the bob was going to have greater GPE. This is because gravitational potential energy is given the formula:
Gravitational Potential Energy = Mass * gravitational acceleration * Height
When height increases GPE increases as Mass & gravitational acceleration remain constant. An increase in the GPE of the bob, will lead to an increase in the KE of the bob. This is because a pendulum can be seen as a device whose energy is continually changing. When the pendulum swings to & fro, its energy changes from gravitational potential energy to kinetic energy. KE is given the formula:
Kinetic Energy = 0.5 * Mass * Velocity squared
An increase in Kinetic energy, means an increase in velocity because mass remains constant. Therefore with a greater angle, the bob travels faster and in theory should have a shorter period as velocity is inversely proportional to time. However, this is not the case, as increasing the angle increases the arc and hence the period increases as distance is proportional to time. Our results show that as the angle increases, the period increases. This must mean that the increase in velocity in larger angles could not compensate for the greater distance that come as a result of larger angles, and hence the period increases, as the angle increases.
Notice that on the graph on page x, there are 2 lines on the graph. The actual line, which was achieved from the results, and a linear line which was derived from the formula T = 2?? l / g (1 + 1/4 Sin2? ..... ). When using the linear line, we can see that in theory, the period should increase as the angle increases. There is another purpose however, for this linear line. It is, to compare it to the actual line, so a comment could be made about the accuracy and reliability of my results. It is evident from the graph that my results are very inaccurate, as the actual line is very far from the linear line. I will discuss this in more detail in the evaluation.
- Angle (?)
2- rank (of column 1)
3- Period (T) (s)
4- rank (of column 3)
5-difference between columns 2 & 4 (d)
6-difference squared (d2)
0
8
0.9
6
2
4
20
7
0.9
6
30
6
0.91
5
40
5
0.93
4
50
4
0.95
3
60
3
0.95
3
0
0
70
2
0.98
2
0
0
80
0
0
?d2 =8 _______________________________________________________________________
I will insert 8 into the formula. This formula should give me a value between 1 & -1. If it is a positive number, then there is a positive relationship between Angle and period, and if it is a negative number, then there is an inverse relationship between angle and period. A value of +1 or -1 indicates a perfect relationship. A value between + or - 0.7 & + or - 1 indicates a strong relationship. Anything below 0.7 or above - 0.7 indicates a weak relationship, getting weaker as you become closer to 0. The formula is:
R = 1 - (6 + ?d2) / (n3 - n) R = Spearman's Rank, n = number of values (8 in this case).
By substituting my values, I get:
R = 1 - (6 + 8) / (512 - 8) = 0.97
The value of 0.97 which I have achieved, shows that the relationship is positive, this agrees with my graph, and it also shows that there is an extremely strong relationship. This means that in the majority of cases, as the angle increases, the period also increases. This may sound as a good thing, however in looking at the linear graph, we find that in all cases the period should increase as the angle increases and therefore the ideal value I should achieve from this statistical test is 1 and not 0.97. This provides me with further material proof that there are some anomalous values in my results.
EVALUATION
- VARYING THE LENGTH:
This experiment, in which I varied the length of string in the simple pendulum, was successful as it allowed me to meet the demands of my aim and prove both my predictions right and hence arrive at a valid conclusion. In looking at the graphs on page x and x which present data for this experiment, we find that the experiment is of a very high degree of accuracy. This can be seen as the actual line is very close to the linear line in both the graphs. The fact that this experiment shows to be of a very high degree of accuracy, reflects one thing: it is that very few experimental practical difficulties were experienced during the experiment and hence very few difficulties interfered with our result. Since the result is not 100 % accurate means that some practical difficulties were experienced when carrying out this experiment. Before I speak of the practical difficulties that encountered us, I will like to present a graph which will provide a clear view of how accurate my experiment was. In it, the reading with the greatest and the smallest errors can be identified. I will do this, by rearranging the formula T = 2?? l / g,
2 2
to g = 4 ? l / T . As is known, gravitational acceleration has a fixed value of 9.81 m/s. By using the formula and inserting the periods and lengths for each period, I can compare the value I achieved for g, with the theoretical value for g. See graph below:
From the graph, we can straight away indicate the most and least accurate results. We can do this by comparing the values to the standard fixed line which I have drawn for gravitational acceleration which is equal to 9.81. The value which is furthest away from the standard line is the least accurate, whereas the value which is the closest to the standard line is the most accurate. We see from the graph, that the most accurate result is found when the length of string was 0.4m. Here we find that the gravitational acceleration works out to be 9.79 in comparison to the standard 9.81 hence having a percentage error of only 0.2 %. The most inaccurate readings, were found at the lengths of at 0.5m, 0.8m and 1m. where the difference between the values I achieved for gravitational acceleration were 0.16 above and below the standard value. This works out to a percentage error of 1.6%. Although this is the greatest % error, when looking at it practically, we find that it is very small. I calculated my average % error in this experiment, to be 0.5%, which in my opinion is very good.
Although my percentage errors for this experiment are very low, I must clarify the practical difficulties that have led to errors existing. They are:
- In releasing the bob, I exerted a force, which I tried to avoid, which in turn interfered with the oscillation of the pendulum.
2- Since I was carrying out this experiment alone, I could not release the bob and instantaneously start the stopwatch. Therefore there was a small lag time between releasing the bob and starting the stopwatch. This in turn has affected the length of the period.
3- I am likely to have experienced human error in adjusting the length of the string. I must emphasise however that I was very careful to measure the length from the centre of mass of the bob.
4- As the human eye was my only tool in this experiment, I could not judge precisely when the oscillation was complete and hence mistakes were likely to be made.
In the readings with the greatest % errors, the practical difficulties which I mentioned above had a greater impact on these lengths, than they did on the lengths where there was a smaller % error.
As the experiment was carried out under limitations of time, I did not have the time to overcome these difficulties. However, now that I have completed this experiment, I can mention some of the solutions that could improve the efficiency of the experiment, by overcoming some of the practical difficulties. The solutions are:
- Work with a partner when carrying out this experiment, so one could release the bob while at the same time, the other one starts the stopwatch.
2- Film the experiment on a video camera. In the film show the stopwatch so that I could determine the exact second in which the oscillation completes.
3- Spend more time in measuring the length, and use a tape measure to measure the length rather than a metre rule.
To further my experiment, I will like to carry out the same method on a greater number of lengths, so than I could be certain that my conclusion is valid. I was capable however on experimenting on 1 more length of which was 22.45m. I went to the science museum, and there a 22.45m simple pendulum was set up. I timed how long it took 10 complete oscillations to occur and found it to be 95.5s. I then divided this value by 10, to work out T,which was 9.55s.Having done that, I inserted it into the formula g = 4?2 l /T2 , to see how accurate my reading was. I found that this gave me a result of 9.72, which when compared to the standard 9.81, has a % error of only 0.9%.
2- VARYING THE ANGLE:
It is difficult to judge the success of this experiment. It was successful in allowing me to investigate the effect on varying the angle on the period of a simple pendulum and hence allowing me to arrive at a conclusion regarding this. However, this success is brought down when we come to consider the accuracy of this experiment. The statistical test alone shows how unreliable my results are, as they did not score the maximum of 1 on the test. For such an experiment as mine, 1 should be the normal and not an achievement as all 1 is saying that as the angle increases the period increases and nothing more than that. The accuracy of my experiment was reduced more when I considered the big distances between the actual line I achieved from my results, and the linear line a derived from a mathematical formula. Saying that my experiment is inaccurate implies that my practical difficulties affected it greatly. I will graphically display my inaccuracies for this experiment, the same way in which I have done when evaluating the former experiment.
It is evident from this graph that the range for the errors in this experiment are very large. For we find a couple of results which are very accurate, and we also find results which are very inaccurate. When we speak about the accurate results of this experiment, we must make specific reference to the result achieved at 80 degrees. The graph shows that this result is 100% accurate and has 0% error as it is inline with the standard line of 9.81. The reading at 10 degrees is also very accurate. This reading has a percentage error of only 0.1%. The most inaccurate result was found at the angle of 60 degrees, which worked out to have a gravitational acceleration of 10.39. When comparing this to the standard 9.81, we find that there is the maximum % error of 5.9%. I calculated the average percentage error for this experiment to be 2.5%.
The practical difficulties I experienced in this experiment, were the same as the one experienced for the former experiment. However, there were more practical difficculties that I encountered, and this is why my average % error for this experiment were larger than the former experiment. They were:
- It was very difficult to line up the pendulum with the angle required. I found it difficcult to get it exactly at the angle I want. Therefore my angles were adjusted to the nearest degree or two.
2- Notice that my actual line on the graph was always less than the linear line. This is because the pendulum could not be kept constant at the angle I wanted, and instead decreased by the time 10 complete oscillations occurred. It decreased because of air resistance acting on it.
Again, I was under constraints of time when I carried out this experiment and now that I have completed it, I can provide possible solutions to the two above mentioned problems:
- For the first problem, I can produce an enlarged protractor. This will make it easier for me to line up my pendulum with the angle I require.
2- For the second problem, I suggest that this experiment be carried out in a vaccum, where air resistance will have no influence on this system.
To further this experiment, I will like to investigate the effect of other factors on the period of the simple pendulum. The factors which I wil like to experiment on are: 1- Mass of bob 2- Surface area of bob.
This is as far as the simple pendulum is cocerned. However, I will also like to learn more about other types of pendulums, and factors which affect their oscillation and period such as the Bifilar pendulum which is used to record the irregalar rotation of the earth and the Foucault pendulum which is used to demenstrate the rotation of the earth.