Experiment to investigate the factors affecting the swing of a pendulum.
Title: Experiment to investigate the factors affecting the swing of a pendulum.
Aim: To investigate how the period of oscillation of a pendulum is affected by the length of the string and the mass of the weight attached to the string.
Introduction:
A simple pendulum consists of a weight hanging at the end of a string. When a force is applied to the pendulum so that it is displaced by a small angle, the pendulum will swing to and fro at a constant rate under the effect of gravity. The path travelled by the weight is the arc of the pendulum and the period of the oscillation is the time taken for the weight to pass to and fro once over this arc.
In this experiment I will be investigating how the period of oscillation of the pendulum is affected by:
. The mass of the weight attached to the string
2. The length of the string.
In this experiment, I will also determine the value of the acceleration due to gravity using the relationship between T; the period of the oscillation and L; the length of the string.
Hypothesis:
. I predict that the period of oscillation will increase with increasing length of string. This is because a longer string will travel a grater distance than a shorter string given the same angle of displacement.
In the diagrams above, it can be seen that the longer string has a longer distance to move than the shorter string, hence its period of oscillation will be greater than that of the shorter string.
2. I predict that the period of oscillation will be independent of mass provided the angle of oscillation and the length of the string are constant. This is because the pendulum will always travel the same distance no matter the mass of the weight attached to it.
Apparatus:
. Retort stand and clamp,
2. G-clamp,
3. Metre rule,
4. 80cm string,
5. Stop watch.
6. Wooden blocks
7. Masses
8. Hook to attach masses to
Variables:
This experiment will be divided into two sections. The first part of the experiment will be investigating how period of oscillation changes with constant length of string and angle of displacement. The independent variables are the length of the string and angle of displacement while the dependent variable is the time taken for a specific number of oscillations.
The second part of the experiment will be investigating how the period of oscillation changes with increase in the mass of the weight attached to the string. The independent variable will be the length of the string and the angle of displacement while the dependent variable is the weight of the mass attached to the string.
Procedure:
In the first part of the experiment, the clamp was attached to the retort stand. The G-clamp was used to fasten the retort stand to the table so as not to allow the retort stand to topple over. ...
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The second part of the experiment will be investigating how the period of oscillation changes with increase in the mass of the weight attached to the string. The independent variable will be the length of the string and the angle of displacement while the dependent variable is the weight of the mass attached to the string.
Procedure:
In the first part of the experiment, the clamp was attached to the retort stand. The G-clamp was used to fasten the retort stand to the table so as not to allow the retort stand to topple over.
Using the metre rule, 80cm of string was measured and cut. One end of the string was attached to the hook on which the masses will be placed. The other end of the string was suspended in-between two wooden corks and held tightly with the clamp of the retort stand. The string was released slightly and with the aid of the metre rule, 10cm of the string was measured so that the length of the spring suspended from the retort stand was only 10cm.
The string was then clamped tightly again in-between the wooden blocks. The other part of the string was wound round the clamp, away from the 10cm suspended, so that it does not affect the oscillation of the pendulum.
50g mass was placed on the hook attached to the pendulum and pendulum was displaced through a small displacement and the time taken, t1 for 10 oscillations was measured using the stopwatch and recorded. The pendulum of length 10cm was displaced again through a small angle and the time t2 for 10 oscillations was measured and recorded.
The above procedure was repeated increasing the lengths of the string to 20cm, 30cm, 40cm, 50cm and 60cm. Two time readings were taken for each length of string. The mass attached to the string was kept constant throughout the experiment.
Using the same set-up as above, in the second part of the experiment, 40cm of the string was allowed to suspend freely from the retort stand. 50g mass was placed on the hook and the pendulum was displaced through a small angle. The time taken for 10 oscillations was measured using the stopwatch and recorded.
The procedure was repeated using masses 100g, 150g, 200g, 250g and 300g. Two time readings were measured and recorded using each mass. The length of the string was kept constant at 40cm during the experiment.
Results:
Experiment 1:
Length, L/cm
Time, t1/s
Time, t2/s
0
20
30
40
50
60
8.28
0.43
1.97
3.75
5.00
6.31
8.12
0.34
1.97
3.88
4.94
6.31
Experiment 2:
Mass, m/g
Time, t1/s
Time, t2/s
50
00
50
200
250
300
3.76
3.88
3.88
3.78
4.00
3.84
3.80
3.78
3.84
3.82
3.78
3.94
Analysis of results:
Experiment 1:
Length, L/cm
Time, t1/s
Time, t2/s
Time, t/s
Period, T(t/10)/s
0
20
30
40
50
60
8.28
0.43
1.97
3.75
5.00
6.31
8.12
0.34
1.97
3.88
4.94
6.31
8.20
0.39
1.97
3.82
4.97
6.31
0.82
.04
.20
.38
.50
.63
In the above table, time t/s was calculated by finding the average of the two time readings.
The graph of length, L/cm against period, T/s was plotted and the following was noticed:
. The graph does not start at the origin
2. There is a direct relationship between length of string and the period of oscillation. This thus shows that the period of oscillation increases with an increase in the length of the string.
Experiment 2:
Mass, m/g
Time, t1/s
Time, t2/s
Time, t/s
Period, T(t/10)/s
50
00
50
200
250
300
3.76
3.88
3.88
3.78
4.00
3.84
3.80
3.78
3.84
3.82
3.78
3.94
3.78
3.83
3.86
3.80
3.89
3.89
.38
.38
.39
.38
.39
.39
From the above results, we can see that the period of oscillation is approximately constant irrespective of the mass of the weight attached to the string.
From the above results and from the graph drawn, it can be seen that my hypothesis has been proven right. Therefore for a pendulum, I have come to a conclusion that the period of oscillation is dependent on the length of the string, but independent on the mass of the weight attached to the string.
From the theoretical part of physics, I know that:
T = 2? ?L/g-------------- (1)
Where;
T = period of oscillation
L = length of string
g = acceleration due to gravity
From equation (1), squaring both sides of the equation we get:
T2 = (2?)2*(?L/g)2
T2 = 4?2 * L/g -------------- (2)
Therefore, to further my investigation I calculated the value of the acceleration due to gravity by plotting a graph of the square of the period, T2 length, L. I used the results from my first experiment.
Length, L/cm
Period, T(t/10)/s
T2/s2
0
20
30
40
50
60
0.82
.04
.20
.38
.50
.63
0.67
.08
.44
.90
2.25
2.66
From the graph,
Gradient, m = A - B
C - D
Substituting the values we have:
m = 2.42 - 0.90 = 1.52
54 - 16 38
m = 0.04 s2/cm = 4s2/m
Re-arranging equation (2), we have:
T2 = 4?2 * L -------------- (3)
g
We can see that equation (3) follows the form of the equation:
y = mx + c -------------- (4)
Combining equations (3) and (4), we can see that :
y = T2
x = L
m = 4?2 -------------- (5)
g
Therefore substituting for the value of m in equation (5), we have:
4 = 4?2
g
4g = 4?2
g = ?2 = 3.1422
g = 9.87ms-1
From my experiment, the value of the acceleration due to gravity calculated was 9.87ms-1.
Conclusion and evaluation:
From the above results, it can be seen that my hypothesis has been proven correct. I can therefore conclude that the period of oscillation of a pendulum is dependent on the length of the string and independent on the mass of the weight attached to the string.
During the experiment, I ensured that the windows of the laboratory were shut throughout the experiment to prevent errors in the oscillation of the pendulum as a result of air resistance. I also ensured that the pendulum was displaced through a very small angle to ensure that only one the effect of one factor was tested each time. This is because if the angle is being increased or decreased, the period of oscillation of the pendulum may be affected. It was quite difficult keeping the angle constant because there was no means of measuring the angle scientifically.
In the second part of the experiment, I expected the period of oscillation to be the same irrespective of the mass attached to the string. This was not quite so because the period of oscillation was not exactly constant, it was only approximately constant. If the experiment were to be repeated, it will be better to measure the time for 50 oscillations instead of 10 used in my experiment. This is to ensure that the error that arises through being unable to start or stop the stopwatch when the pendulum is exactly in the intended position is greatly reduced.
I expected the graphs of the period of oscillation against length and that of the square of the period of oscillation against length to start at the origin, but they did not. This may be due to experimental errors, for example, in measuring the time of oscillation or in the calibration of the metre rule used in measuring the length of the string used. The error was recurrent and therefore, a systematic error because although the graphs did not start at the origin, the graphs still show a direct proportionality between the period of oscillation and the length of the string.
The value of the acceleration due to gravity calculated was 9.87ms-1 as opposed to 9.81ms-1, which is the well-known constant, used in physics today. The value I calculated was not quite right and this may be due to the experimental errors that have been carried over.
If this experiment was to be carried out again and all the errors are taken into account and prevented, I think the experimental results will be more reliable. The effect of the angle through which the pendulum is displaced may also be investigated in further experiments.
References:
A- Level Physics, 4th Edition by Roger Muncaster - page 90-92
Advancing Physics A2 - page 15-17
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