Clamp, string, measuring equipment, stopwatch, person who does all the measuring and timing, weights, position where knot is tied, angle where it is released (amplitude).
To make sure our results are accurate we need to keep everything but the variable constant. Below are some simple guidelines to ensure that our testing is fair.
Note: Although during my research I ascertained that the mass of the bob does not effect the period of the pendulum, I should still keep this constant, as I should only have one variable in my experiment.
Note: The friction on the string caused by the air will affect the results. Ideally, this experiment would be conducted in a vacuum. However, we have no equipment in school that we could use to achieve this. So, therefore we will have to conduct the experiment with the knowing that all are results will be slightly affected by the surrounding air atoms.
Safety
There are many accidents that could happen if this experiment was not carried out safely; below I have outlined a few simple guidelines to prevent such accidents occurring.
Apparatus
For our experiment we needed:
- A length of String
- A bob
- Clamp and retort stand
- A heavy mass
- A large protractor
- A Stopwatch
- A meter ruler
- Scissors
Method
- Measure the string to the desired length
- Place string on centre of bob
- We then pulled the string back to 15 degrees
- We then released the bob and started the stopwatch at the same time.
- We let the bob swing backwards and forwards 10 times
- We then stopped the pendulum swinging and recorded the times.
- We repeated the experiment with the same length 3 times
- We then repeated steps 1-6 for string lengths 10cm, 20cm 30cm, 40cm, 50cm
Diagram
Errors in Measuring/Judgment: In many cases we found that when we repeated the experiments, we found that the time or the amplitude was different. This was because every time we did it there was a margin or error. Nobody in the world could ever measure it and get it right with your bare hands. We therefore took the results by averaging the result from three repeated tests so that we won’t get one very strange result from one particular area. We could not measure very accurately either. Many times when a person measures it again, we found that it was often different by 1-3mm. So we will be comparing all results by showing the true time mathematically by the sum: T= 2π √(l/g).
Preliminary Work
To confirm that the theory’s are correct we performed some preliminary experiments using different variables.
With a small bob at 29 cm’s it come to 1.189 seconds for one swing. With a heavier bob it came to 1.207 seconds, so the weight doesn’t affect it, as proved in the theory. The difference between the previously mentioned results was because of unavoidable human error.
However with the small bob with a short string it took 0.929 seconds compared to the long string which took 1.207 seconds. So obviously the length of the string affects the time. The smaller the string the bob is attached to the smaller the time it takes for a swing.
We also investigated whether the angle the ball is dropped from affects the time. With a big angle it took 12.85 secs so their was no big difference.
My prediction, based on the preliminary work is that the smaller the string the bob is attached to the smaller the time it takes for one swing. In contrast, the larger the string is, the longer the bob takes for one oscillation.
Were using a retort stand and clamp to swing the pendulum from. We will measure time for 10 ,20 ,30 ,40 , 50, 60, 70, 80, 90 7 100 cm’s length strings.
We will get 3 measurements and then average the results.
For each result we will let the pendulum swing for 10 periods and then average to eliminate human error as much as possible.
The angle will be same that we drop it from, also the weight of the bob will be the same. Were using a protractor to keep the angle the same.
We will put weights on the stand to make the results accurate.
We will not be going over 15 for the angle.
Results
Analysis
By looking at my results, I can immediately tell that the longer the length of string the longer it takes the pendulum to complete one period. I have drawn a graph, which shows the period for each oscillation.
The graph shows that the length of the string the bob is attached to is proportional to the time taken for 1 period. The larger the string, the longer for one period and in contrast, the smaller the string the smaller the time taken for 1 period.
The graph looks like the length of string is directly proportional to the time taken for one period (a straight line). However, I noticed a slight, but not perfect curve in my graph. This means that they may not be directly proportional. I have illustrated the difference between the results in a table and a graph. It doesn’t give a perfect line on a negative gradient. It is somewhat random. However the gap between the first plot and the last plot is large and something to be considered when coming to a conclusion. This difference is most probably due to human error when timing and measuring angles, or just the retort stand rocking slightly affecting the results.
I can see from the results that there is one clear factor, length and swing size. For mass the period for 10 oscillations does nothing to change the factor. The variation between the averages is small enough for me to conclude that these factors have a minimal effect if any on the period of an oscillation. From the information from this preliminary experiment I can now go onto investigate how precisely length effects the oscillation period of a pendulum. I have also learnt from this preliminary it is necessary for the clamp stand to be held firmly in place so it does not rock.
Scientific Theory
As a pendulum is released it falls using gravitational potential energy which can be calculated using mass (kg) x gravitational field strength (which on earth is 10 N/Kg) x height (m). As soon as the pendulum moves this becomes kinetic energy, which can be calculated using 1/2 x mass (kg) x velocity2 (m/s2) and gravitational potential energy. At the point of equilibrium the pendulum just uses kinetic energy and then it returns to kinetic energy and gravitational potential energy and finally when the pendulum reaches maximum rise it is just gravitational potential energy and this continues. From this I can deduce that kinetic energy = gravitational potential energy. If these were the only forces acting on the pendulum it would go on swinging forever but the energy is gradually converted to heat energy by friction with the air (drag) and with the point the mass is hung from. The amplitude of the oscillation therefore decreases until eventually the pendulum comes to a rest at the point of equilibrium.
As the amplitude is increased so too is the gravitational potential energy because the height is increased which affects the gravitational potential energy and therefore the kinetic energy must also increase by the same amount. The pendulum then oscillates faster because height or distance is involved in v2 in the kinetic energy formula. However the pendulum has a larger distance to cover so they balance each other out and the period remains the same. The period is also the same if the amplitude is reduced.
For the mass as it is increased this affects both the gravitational potential energy and the kinetic energy as they both contain mass in their formulas but velocity is not affected. The formulas below show that mass can be cancelled out so it does not affect the velocity at all.
gravitational potential energy = kinetic energy
mgh = 1/2mv2
Length affects the period of a pendulum and I have found a formula to prove this and I will now attempt to explain it. The formula is:
T=period of one oscillation (seconds)
p=pi or p
l=length of pendulum (cm)
g=gravitational field strength (10m/s on earth)
This shows that the gravitational field strength and length both have an effect on the period. However although the 'g' on earth varies slightly depending on where you are as the experiments are all being done in the same place this will have no effect as a variable. Length is now the only variable. This means that T2 is directly proportional to length.
T2= 2plg
The distance between a and b is greater in the first pendulum. However the pendulum has gained no amplitude so therefore no additional gravitational potential energy or kinetic energy so it will still travel at the same speed. The first pendulum therefore has a greater distance to travel and at the same speed so it will have a greater period.
I can predict from this scientific knowledge that the period squared will be directly proportional to the length.
I can now justify my prediction and say that I was correct. I predicted that the longer the length of the string, the longer it will take for the pendulum to complete one period. This prediction was based on the assumption that this equation was correct.
The length of the string = L
G = Gravitational field strength
T = time for one period
Ignore the figures that are not input factors and you get this.
T=L/G
This means that there are 2 factors that will affect the length for one period of a pendulum. These are the gravitational field energy and the length of the string the bob is attached to.
From this equation I can say that the length of string, and the time for one period are directly proportional.
Evaluation
The evidence obtained from my experiment supported my prediction that as
the length of the pendulum increases, the period increases. This is also shown
in Graph, as the graph displays a smooth curve with a positive gradient. I discovered that T was directly proportional to L, providing all other values remain constant. This was shown by a straight line on a positive gradient. These results were
encouraging and led me to believe that my proposed method was sufficient for
the experiment.
Some of the results were not accurate, as they did not match the results
produced by the formula. This could have been due to human error. However,
the majority of my results were no more than a decimal place away from the
formula results and, therefore, quite reliable. Had there been any anomalous
results, I would have repeated my readings.
Factors which may have affected the accuracy of my results include:
- Error in measurement of angle of altitude. This angle proved difficult to
measure and it was hard to get the exact same angle for each result. To
improve the accuracy of this measurement, I could have attached the
protractor to the clamp stand so that it was in a fixed position.
- Error in measurement of string. To improve the accuracy of this, I
could have marked off the points with a pen to ensure they were as
accurately measured as possible.
- Human reaction time. Depending on human reaction time, the
measurement period time could have been measured inaccurately, due
to slow reactions when setting the stopwatch etc. This could have been
improved by involving another person to aid me with my experiment,
for a quicker reaction time.
The procedure was relatively reliable, excluding human error, and so I can
conclude that my evidence is sufficient to support a firm conclusion that:
That as the length increases, so does the period.
If I were to extend my investigation, I would investigate to provide additional
evidence to back up my conclusion, for example, changing the mass or angle
of altitude. The results gained would hopefully aid me further in supporting
my Scientific Theory.
I could also investigate the effects of what changing the gravitational field energy would do. I could investigate further whether the formula I found is correct by having the GPE as a factor.
I have heard that the angle the bob is released at affects the results after 15degrees. I could investigate this, and if what I have heard is true, then it would render the equation to be false.
