ۥ-!@ <-t3*$*$nnnnnnJXn\ppppp"ppqLqqqqqqqqqqqqqqet4t/qknqq10.1: Analyzing the Motion
of a Pendulum
PROBLEM:
What is the relationship between the length and the frequency of a simple pendulum?
DIAGRAM:
**See back of title page**
APARATUS:
SYMBOL 183 \f "Symbol" \s 10 \h apparatus as illustrasted
SYMBOL 183 \f "Symbol" \s 10 \h stopwatch
SYMBOL 183 \f "Symbol" \s 10 \h protractor
METHOD:
1. A simple pendulum was attached to a rigid support, with the centre of the bob about 100 cm below the pivot point. The length of the pendulum from the pivot point to the centre of the bob was measured.
2. The length of the 100cm pendulum was decreased, in steps of approximately 20cm. The frequency of the pendulum was determined for each length. Observations were recorded in a chart, as seen in OBSERVATIONS.
3. A graph was plotted of the pendulums frequency (f) against its length (L).
4. The length of the pendulum was adjusted to the identical length used in the first observation. This time, the bob on the pendulum was released from aproximatley 10o so the pendulum would vibrate with a smaller amplitude. The frequency was compared with that obtained when the pendulum oscillated with a larger amplitude, in step 1.
5. The bob was replaced with one of a larger mass, making sure that the length of the pendulum remained the same. The frequency was determined and compared with that obtained in step 4.
OBSERVATIONS:
Mass (g)
Angle (o)
Length (cm)
Cycles
Time (s)
20g
20o
100cm
10
20.40.5
20g
20o
80cm
10
18.120.5
20g
20o
60cm
10
15.750.5
20g
20o
40cm
10
12.710.5
20g
20o
20cm
10
9.290.5
20g
10o
100cm
10
20.490.5
20g
20o
100cm
10
20.830.5
ANALYSIS:
Frequency(Hz)
=cycles/time
or
=1/T
Period(s)
=time/cycles
or
=1/f
0.49019
2.04
0.55187
1.812
0.63492
1.575
0.78678
1.271
1.0642
0.929
0.48804
2.049
0.48007
2.083
**See following page**
Sources of Error
The angle of release was changed slightly after measurement due to the unsteadiness of the hand holding it and the fact that the person letting go pulled it back slightly before release. This change in the angle would change the amplitude but it would not directly affect the results because only the number of cycles and the time were being measured and they depend only on the length of the pendulum. This error could be minimized by using a stationary device to hold the pendulum at the desired angle.
The length of the pendulum was not exact due to the gradations on the metre stick and human error. A change in this length would directly affect the results obtained for the length of the time that each cycle took. This would then in turn affect the calculations obtained for frequency and period. This error could be minimized by using a millimetre ruler, and by being more careful.
The string could not be exactly fixed to the rubber stopper. The weight of the bob caused the string to slide up and down through the hole in the stopper which changed the length of the pendulum several times during each experiment. A change in this length would directly affect the results obtained for the length of the time that each cycle took. This would then in turn affect the calculations obtained for frequency and period. This error could be minimized by using something other than a rubber stopper to attach the string to.
The starting and stopping times were not exactly matched to the real time due to the slow reaction time of the person and the difficulty that exists in determining the exact point in which to start and stop. These discrepancies in the length of time would affect the results that were calculated for frequency and period. This error could be minimized by using a computer with motion detectors to record the exact times at which the pendulum started its
first cycle and ended its tenth.
In addition to swinging from side to side, the mass of the pendulum was also spinning round and round. This spinning would slow the mass down which in turn would affect the time of each cycle, which would then change the calculations obtained for frequency and period. This error could be minimized by using a stationary mass and a better set-up.
Due to the amount of people in the room, air currents were created that may have slightly changed the direction of the swinging of the pendulum causing it to swing in more than one plane. This error could be minimized by performing the experiment in a vacuum.
The table that the pendulum was set up on was completely stable and due to the members of group sitting near it was moved slightly during each experiment. This would move the pendulum causing it to swing in more than one plane. This error could be minimized by setting up the experiment on a table secured to the floor that is not being used at the time.
The pendulum was not swinging is just one plane. There was lateral movement that would change the results. The time of each cycle would be increased slowly which would then change the calculations obtained for frequency and period. This error could be minimized by correcting all of the errors mentioned above.
All of these errors are minimal and can be ignored for one cycle, but they grow worse as time continues. Because the time of one cycle was measured by finding the average of 10 cycles, the actual time the first cycle took is less than the average time because these errors when compounded increase the time of each cycle.
This experiment could be improved by performing the experiment in a vacuum, on a secured table, with a proper set up and accurate instruments.
Questions
1. What effect does a change in the length have on the frequency and period of an oscillating pendulum?
As the length decreases, the frequency of an oscillating pendulum increases and the period decreases. As the length increases, the frequency of an oscillating pendulum decreases and the period increases.
2. For a pendulum with a fixed length, what is the effect on the frequency and period of an oscillating pendulum of:
a) a change in amplitude?
A change in amplitude has no effect on the frequency and period of an oscillating pendulum.
b) a change in the mass of the bob?
A change in the mass of the bob only affects the amplitude. Therefore a change in the mass will have no effect on the frequency and period of an oscillating pendulum.
CONCLUSIONS
The frequency and period of a pendulum are directly affected by the length. As the length decreases, the frequency increases and the period decreases. As the length increases, the frequency decreases and the period increases. For a pendulum of fixed length, a change in amplitude and a change in the mass of the bob have no effect on the frequency and period.
The results of the experiment were valid as the procedure was followed exactly and the sources of error were mentioned and accounted for.
APPARATUSillustratedcentercenterapproximatelymillimeter
EMBED MSGraph \s \* mergeformat
Pendulum Frequency vs Pendulum Length
EMBED MSGraph \s \* mergeformat
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Present at the Lab:
Andrew Likakis, Michael Hancock, Eva Chen
Absent from the Lab:
Christina Mitchel
Title Page: Andrew Likakis
Problem: Andrew Likakis
Diagram:Andrew Likakis
Aparatus: Andrew Likakis
Method: Andrew Likakis
Observations: Andrew Likakis
Analysis: Mike Hancock, Eva Chen
Conclusions: Christina Mitchel
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