One Full Oscillation Of A Pendulum Essay, Research Paper
An investigation on the factors affecting the period of one complete oscillation of a simple pendulum In this investigation I aim to discover and investigate the factors which affect the time for one complete oscillation of a simple pendulum. It is important to understand what a pendulum is. A simple pendulum is a weight or mass suspended from a fixed point and allowed to swing freely. An oscillation is one cycle of the pendulums motion e.g. From position a to b and back to a. The period of oscillation is the time required for the pendulum to complete one cycle of its motion. This is determined by measuring the time required for the pendulum to reoccupy a given position.
I am going to do a simple preliminary experiment to investigate which of the factors I test have an effect on the time for one complete oscillation. The factors basic variable factors I can test are:
? Length (the distance between the point of suspension and the mass)
? Mass (the weight in g of the item suspended from the fixed point)
? Angle (the angle between the point of equilibrium and the maximum point the pendulum reaches)
*The point of equilibrium is the point at which kinetic energy (KE) is the only force making the mass move and not gravitational potential energy (GPE).
I will test the extremes of these factors as I can assume that if they have any effect on the period of oscillation it will become obvious. To make sure my results are reliable and to allow for any anomalies I will repeat the experiment 4 times for each extreme. I will also keep all the other factors constant so if the results change for the different extremes I can be sure which factor is causing this change, as all the others will remain constant. To keep the results as accurate as possible I will measure the period of 10 oscillations and only use one decimal place to allow for my reaction time. Results Angle (º) Time Taken (sec) for 10 oscillations
90º 13·4 13·4 13·8 13·7 Average:13·6
45º 13·2 13·2 13·1 12·9 13·1
Length: 0·3m, Mass: 20g Mass (g) Time Taken (sec) for 10 oscillations
400g 11·1 11·3 11·3 11·4 Average:11·3
100g 11·6 11·1 11·0 11·2 11·2
Length: 0·2cm, Angle: 45º
Length (cm) Time Taken (sec) for 10 oscillations
0·25m 10·4 10·5 10·5 10·3 Average:10·4
0·65m 16·8 16·0 16·5 15·9 Average:16·3
Mass: 50g, Angle: 40º
I can see from the results that there is one clear factor, length. For Angle and mass the period for 10 oscillations is roughly the same for both of the extremes. 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 GPE 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 KE which can be calculated using 1/2 x mass (kg) x velocity2 (m/s2) and GPE. At the point of equilibrium the pendulum just uses KE and then it returns to KE and GPE and finally when the pendulum reaches maximum rise it is just GPE and this continues. From this I can deduce that KE = GPE. 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. From this I can now explain why the amplitude and the mass have no effect on the period of oscillation. As the amplitude is increased so too is the GPE because the height is increased which affects the GPE and therefore the KE must also increase by the same amount. The pendulum then oscillates faster because height or distance is involved in v2 in the KE 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 GPE and the KE 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.
GPE = KE
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.
The distance between a and b is greater in the first pendulum. However the pendulum has gained no amplitude so therefore no additional GPE or KE 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. Apparatus:
? Clamp stand
Method: The apparatus was set up as shown above. The amplitude was always 45ºand the mass 10g. I held the string taut and started the stopwatch when I released the pendulum; I then stopped the stopwatch after the tenth oscillation. I used a range of 10cm to 100cm to use a suitable range of measurements. I also repeated each length 4 times to make the average gained more reliable and to allow for any anomalies. To make the results more accurate I also counted 10 oscillations meaning if you divide the period by ten your reaction time, which affects the length of the period, is reduced by 9/10. To make sure this was as fair a test as possible I:
? Tried to create as little friction as possible where the string is attached to the clamp.
? Let go with out adding any extra forces
? Kept the string taut
? Made sure the mass and angle remain the same in case they have a small effect on the period.
? Keep the whole experiment in the same place so that the gravitational field strength does not change
To make this a safe experiment:
? No weight above 400g
? No angles above 90º
? The clamp stand is secure Results:
A Table to show the periodic time for 10 oscillations for various lengths Periodic time (seconds)
0·1 6·1 6·1 6·1 6·0 Average periodic time: 6·08
0·2 8·6 8·7 8·6 8·7 Average periodic time: 8·65
0·3 10·2 10·3 10·3 10·3 Average periodic time: 10·28
0·4 12·6 12·4 12·6 12·3 Average periodic time: 12·48
0·5 14·0 14·1 13·9 13·8 Average periodic time: 13·95
0·6 15·2 15·3 15·3 15·3 Average periodic time: 15·28
0·7 16·4 16·4 16·5 16·6 Average periodic time: 16·48
0·8 17·8 17·6 17·5 17·7 Average periodic time: 17·65
0·9 18·8 18·5 18·4 18·8 Average periodic time: 18·63
1·0 19·6 19·3 19·3 19·3 Average periodic time: 19·38
Angle: 45°, Mass:10g
To find the average period of one oscillation I must divide my average for 10 oscillations by 10.
A table to show the period of one oscillation for various lengths
Length Period Period according to formula
0·1 0·61 0·63
0·2 0·87 0·89
0·3 1·03 1·09
0·4 1·25 1·26
0·5 1·40 1·40
0·6 1·53 1·54
0·7 1·65 1·66
0·8 1·77 1·78
0·9 1·86 1·88
1·0 1·93 1·99 I am now going to create a table by rearranging the formula I found so length is directly proportional to the period of oscillation squared. A table to show the period squared for various lengths Length Period squared Period squared according to formula
0·1 0·37 0·40
0·2 0·76 0·79
0·3 1·06 1·19
0·4 1·56 1·59
0·5 1·96 1·96
0·6 2·34 2·37
0·7 2·72 2·76
0·8 3·13 3·17
0·9 3·46 3·53
1·0 3·72 3·96
The above two tables are to two decimal places so that the data is easier to draw a graph from. I did not include the period or the period squared on my graphs because the results are too close together but the tables indicate how near my results were to what they should be in theory.
From my results I have found out that the period squared is as predicted directly proportional to the length of the pendulum because my graph is a straight line and goes through 0. Also if you take the length at 0·1m the period squared is 0·37 and then if you take the length at 0·5m the period squared is 1·96m. The point of this is to show that that both the period and the length go up by nearly exactly the same proportion because 0·5/0·1=5 and 1·96/0·37=5·3. The graph with period plotted against length also provides the useful information that period and length have a relationship, which involves the indice 2 . I have noticed the pattern that if you divide the period squared of the pendulum by the length of the pendulum you get roughly the same figure each time and that the ratio between length and the period squared is roughly 1:38.
I can draw a conclusion from my evidence that the formula:
This is correct because if you rearrange it to form T2= 2pl this fits perfectly
with the graph and my results. If you remove the constants from the formula you are left with a direct link from T2 to l. The curve also reinforces the original formula by showing that as l increases by 0·1 the period increases by a much larger percentage. As the length increases the period goes up in smaller and smaller amounts, which again agrees with the formula. These results totally support my original prediction and they also support the scientific theory. The shape of the graph immediately shows this. The results obtained show that my experiment was successful for investigating how length effects the period of an oscillation because they are the same and agree with what I predicted would happen. The procedure used was not too bad because my results are very similar to what they would be under perfect circumstances. My results are reasonably accurate as they fulfil what I thought and said would happen. However there are a few minor anomalies which can be seen in the graphs and in the tables. They have a larger gap from what they should have been according to the formula than usual. At 1m there is an anomaly which is a few fractions of a second away from the line of best fit.
Most of the procedure was suitable because it gave a useful and relevant outcome but it could have been improved in a number of ways. The reliability of the evidence could be increased by making the angle more precise, making sure the string is taut when the pendulum is released and making the string the exact length it should to be. The anomalous results I have may be down to a number of reasons but could mainly be blamed on my releasing the pendulum and providing it with an external force, which would affect the period. My timing of the stopping and starting of the stopwatch could be inaccurate. The overall results may be a 1/100 of a second out because I used the gravitational field strength of 10 when the actual field strength may be different. If the above improvements were added in, the results would be more accurate and reliable. To further this work, I would repeat each length providing more accurate averages. I would provide additional evidence for my conclusion by increasing the range of lengths and decreasing the intervals between the lengths to five centimetres. These additions would extend my investigation further.