The clamp stand is connected to the boss clamp, which will provide the pivot for the pendulum and to ensure that it is high enough not to be obstructed by the desk. The string is tied securely to the boss clamp and the other end is connected to the 100g mass. The protractor is taped to the top of the boss clamp to accurately measure the angle of release. The G clamp holds the stand to ensure that the pendulum does not cause the stand to move and affect the results. The stopwatch is used to produce period measurements of high accuracy.
Diagram of Apparatus
Firstly, I will investigate how the amplitude affects the period of oscillation. I will keep all other variables constant, except for the angle of release. The length will be at 62cm, measured from the pivot to the middle of the 100g mass. g is constant close to the earth.
I will measure the period of oscillation, using the protractor, by raising the mass at increasing angles, 10 degrees each time, and recording the time it takes for the pendulum to complete ten oscillations. I will use angles between 10 – 60 degrees, which will give me a good range of results. I will time ten oscillations and then divide this by ten to find the average period of one oscillation. This will minimize the reaction time error from starting and stopping the stopwatch, as it will be spread over ten oscillations. After ten oscillations the air and pivot resistance would decrease the amplitude to much and affect the results. The experiment will be repeated 3 times to produce an average, which will reduce anomalies and other errors. As g is the constant that I cannot change I will use it to confirm my results. I will find the value of g for each angle to see how close it is to 9.81ms-2.
Safety
There are several safety issues in this experiment:
- The pendulum ‘bob’ may hit someone if the amplitude is too large. Therefore I will control the amplitude by using suitable angles.
- The clamp stand may fall off the desk and injure someone if it is not secured. Therefore the G-clamp will secure it in place.
Fairness
- All variables, apart from the one being tested, should remain constant in this experiment.
- The clamp stand should be secured because rocking will distort the results. The clamp stand is secured in place by a G-clamp.
- The angle should be measured accurately using the protractor.
Amplitude Results
I have taken the results to 2 decimal places because this was the limit of accuracy on the stopwatch. These results show that over a 50º increase in angle of release, the period of oscillation increases by 0.06. This is an unreliable measurement because the stopwatch does not have this degree of accuracy. In my prediction I stated that the period of oscillation ‘will not be affected by the angle of release or the mass of the ‘bob’ at low angles’. This was backed-up by research on the internet(3) This appears to be correct. At low angles my results were very close together. As the angle increased the period did increase slightly. I believe my result for 20º is an anomaly because it is lower than the result at 10º. None of my research suggests that the period would decrease as the angle increases. The 2nd and 3rd attempts do not show this decrease in period at 20º. So, this must be an anomaly.
A graph (I) of increasing angle of release shows the increase, as well as the anomalies.
If I find the value of g for each average period I will find how accurate they are. By using the equation:
Mass
For investigating the effect that mass has on the period of oscillation I am going to us a 60g mass made-up of 10g masses. This is so that I can change the mass of the mass of the pendulum easily without disturbing the setup. I will gradually increase the mass of the ‘bob’ by adding the 10g masses onto it. The mass is the only variable that I will be changing, so I will drop the pendulum at a 45º angle, measured by the protractor, for every attempt. The length of the pendulum will be 56cm. I chose 45º because it is halfway between the point of equilibrium and its perpendicular. This is to ensure that all other variables will remain constant throughout.
Mass Results
These results show that increasing the mass of the pendulum does not affect the period of oscillation significantly. The average period of oscillation has increased by 0.03s when the mass of the pendulum is 6 times larger. I have already shown how mathematically the mass does not affect the period. However, the air resistance acting on the mass will increase when the mass increases because of the larger surface area. The result of this is that we notice a slight increase in the period of oscillation with greater mass. In a vacuum this situation will not occur, because there is no opposing air resistance.
In my prediction I stated that the mass will not affect the period of oscillation. The results back this up and so does the graph (II) of mass v. period.
The value of g for each mass from the equation:
Length
From my prediction, the length of a pendulum will affect period of oscillation. This is because a greater length of pendulum, with the same angle, will cause the pendulum to travel in a greater arc at the same speed. Therefore, the time to complete one oscillation will be longer.
The length of the pendulum to start with will be 70cm and then cut down to 20cm in 10cm increments. This will allow me to show the relationship between period and length. Below 20cm the reaction error when timing the period of oscillation is too great to obtain a reliable average. Therefore, I have only recorded results up to 20cm. A 100g mass is attached and the angle of release will be 45º.
Length Results
The results and the graph (III) show that there is a clear increase in the period of oscillation as the length of the pendulum increases.
The period increase is 0.81s within 50cm increase in length. This suggests that my prediction was correct in saying that ‘the period of oscillation of a simple pendulum will increase as the length of the pendulum is increased’.
The values of g from the pendulum equation are.
All of the results for angle, mass, and length confirm my prediction ‘that the period of oscillation of a simple pendulum will increase as the length of the pendulum is increased. It will not be affected by the angle of release at low angles or the mass’. As g is a constant on earth, length was the only variable that I could prove affected the period of oscillation significantly. Therefore, I will carryout further experiments to investigate to effect of length on period of oscillation.
Length (further investigation)
The equipment used in this investigation will be similar to the set-up in the preliminary length experiment. However, I will be taking measurements for length in smaller divisions of 5cm instead of 10cm. This will give me results of greater accuracy because I will be able to distinguish any anomalies, and the graph will be smoother.
In my preliminary experiment I found that a source of error was the pivot friction. This made the pendulum slow down within the 10 oscillations and affect my results. I therefore adapted the pivot by attaching it to the pivot using masking tape, instead of tying it, to reduce this error.
Prediction
Increase in period
0.92s – 1.15s = 0.23s
1.15s – 1.31s = 0.16s
1.31s – 1.43s = 0.12s
1.43s – 1.59s = 0.16s
1.59s – 1.73s = 0.14s
------
Average = 0.162s
Increase in
Period
From the preliminary length results the average increase in period is 0.162s every 10cm. Therefore the average increase in period is 0.016s for every 1cm of length increase. I have shown this to 3 decimal places because of the small amount of increase between each centimetre. This relationship between period and length is supported by my earlier prediction and background knowledge.
I predict that as I increase the length of a pendulum, the period of oscillation will increase in proportion to the length. I predict an increase in period of 0.016s per centimetre increase.
Length Results
These results show an increase in period as the length of the pendulum increases. A table of increase will show how accurate my results were to my prediction.
The graph(IV) shows that the relationship between the period and the length is not linear. This shows that the rate of increase of time decreases as the length increases. This can also be shown by the equation of period:
The relationship between L and T is not linear because of the square-root.
g and π are constants in this equation therefore:
A graph(V) of T2 v L will produce a straight line.
I plotted these points on the graph and was able to produce a straight line. This tells me that there is direct proportionality between T2 and L, which was expected from rearranging the period equation. I noticed the same anomalies on this graph as on the length vs. period graph which would also be expected.
The gradient of this graph is 4.
So,
k = 4π2 / g
The gradient of the graph period squared Vs length is equal to k. therefore substituting 4 into the equation gives:
4π2 / 4 = g = 9.87ms-2
This value of g is very close to the constant of g at the earth’s surface. This proves that the results I have accurately show the increase in period with length, because the value of g does not change.
Percentage error = 9.81ms-2 – experimental value x 100
9.81ms-2
Percentage error = - 0.61%
This is a very low error value for the result and shows the accuracy of this experiment.
The equation of the line in graph (V) is:
T = 2√ L
This is derived from the period equation.
- At 65cm the value of T from this equation is 1.61 seconds.
- The value from my results show 1.62 seconds so this is a highly accurate equation for the line.
In the lower length pendulums the difference between the actual value of period I obtained and the expected value of T is high. In greater values of length, the difference between the actual and expected values of T is lower. At 15cm the difference is 0.12s, and at 65cm the difference is 0s. This shows that with longer lengths the period is greater and therefore the systematic reaction error is reduced.
Conclusion
I have investigated the factors that effect the period of oscillation of a simple pendulum, and have determined that mass does not affect the period of oscillation. I have also shown this mathematically and graphically. I have also shown that at low values the period is not affected by the angle of release. This is because sinθ ≈ θ at low angles and in the equation for T, θ is cancelled. Length was the only variable, in this experiment, that affected the period of oscillation. I found that the length of the pendulum was proportional to the period-squared, and have produced a graph with a line of best fit to show this.
I determined in my experiment that as the length of the pendulum was increased by a centimetre, the period increased by 0.016s. I found this to be inaccurate by 0.0016s, which is fairly accurate. There were several anomalies on this graph but they were all in the area of the graph where the period is shortest and the systematic and human error is greatest. As the length of the pendulum increases my results became more accurate showing that this error was reduced over time, and became less significant. I calculated the gradient of the graph (V), which was equivalent to the value 4π2 / g. This enabled me to show that the average value of g from my results was 9.87, a result which had a - 0.61% error from the real value of g.
In conclusion, my results prove that the only variable that affects the period of oscillation significantly is the length of the pendulum, which I predicted at the beginning of the experiment.
Evaluation
‘I predict that as I increase the length of a pendulum, the period of oscillation will increase in proportion to the length. I predict an increase in period of 0.016s per centimetre increase.’
The evidence I have gathered from this experiment has supported the prediction I made. The graphs I have produced show that the period increases as the length increases, and the graph of period squared shows that there is a T2 is proportional to L, as a straight line-of-best fit was produced. From the gradient of this line I was able to prove that my results were very accurate because the value of g from this graph is very close to 9.81ms-2. This showed me that my method was correct, because I satisfied my prediction.
However, in this experiment I produced several anomalies in the data. I believe that this is due to the human error in stopping and starting the stopwatch. This is because these anomalies appeared when the period was small and it was harder to stop the watch at the end of the oscillations accurately. When the period was increased this error is lowered and the values are more accurate. If there had been more anomalies that had affected the result I would have had to repeat the experiment to try and reduce these errors.
Several factors in the experimental technique which could have affected the quality of the results:
- I used a metre rule to measure the lengths. This was difficult to control and measure with at the same time when the string was suspended. I would use a different method of measuring if I were to repeat the experiment.
- There was error in the angle measuring as the protractor was taped onto the clamp that could move during measuring. I would secure it in place if I were to repeat the experiment.
- My human error in measuring the time affected the results significantly. However, this is a systematic error which is taken into account in the results.
There was also a mathematical error with the angle of release. The small angle approximation of sinθ ≈ θ is accurate below about 10 degrees (0.1745 radians).
Sin 0.1745 = 0.1736
However, at an angle of 45 degrees (0.7854 radians), which is where I was releases the pendulum:
Sin 0.7854 = 0.7071
There is obviously an amount of error built into my results because there is a 10% error in the value of sinθ and θ. If I were to repeat the experiment I would eliminate this error by releasing the pendulum at 10 degrees or less. The angle would then have no effect on the period.
If I were to repeat the experiment again I would reinvestigate effect on period at smaller lengths because my results had greatest inaccuracy at lower values. However, there are limitations to my equipment because the period would be too fast to measure at small lengths. I would have to use a more accurate instrument, such as a light gate, to measure the period. This would enable the use of a computer to log the data, and interpret the statistics more effectively. I would also try greater lengths to confirm that the trend continues. However, this could become impractical at great lengths, and the increase in the mass of the string would need a bigger mass or lighter string.
I would also like make gravitational field strength a variable, to investigate its affect on period. However, it is not practical with the equipment available, to vary the gravitational field strength significantly.
Graphs
( I )
(II)
(III)
(IV)
(V)
