There are two forces acting on the pendulum bob. Gravitational force tries to pull the bob downwards but this is resisted by the tension in the string. When the pendulum is in the middle of its swing, these two forces are only balanced because they are in opposite directions only at that moment of time. For the rest of the time they are unbalanced which makes the bob swing back and forth. The affect of gravity on the period of a pendulum is related to geographical location. At different places, the gravitational force is different.
A small amount of energy is lost due to the air resistance, which makes each swing slightly smaller than the previous one.
Maximum displacement of the bob from its rest position is called Angular Amplitude (θ) of the pendulum or the angle between the extreme and the rest position of the string. The motion of the bob is simple harmonic if the oscillation are of a small amplitude, i.e θ does not exceed 10°. Period (T) is therefore independent of the oscillations and at a given place where gravity (g) is constant, it depends only on the length of the string.
Period of a simple pendulum is given by;
Where : T = Period (sec), l = Length (cm), g = gravity (m/s/s)
Prediction
Qualitative
I think that if the length of the pendulum is decreased the period will decrease as well because as the length decreases the arc gets steeper (as shown in the diagram) and steeper arc makes the acceleration bigger. A bigger acceleration means a shorter time (T) for each swing. The arc has a steepest gradient at the top and is flat when it reaches the middle. This tells us that the acceleration of the bob will decrease from maximum at the top of the swing to zero at the centre.
The diagram shows two arcs of 2 pendulums (X & Y). The 'Y' pendulum is
longer than 'X'. 'X' has always steeper angle
than 'Y' and 'X' is always above 'Y'.
The 'X' pendulum has the most gravitational potential energy at the top of the swing because it is higher. This means that its kinetic energy and hence speed through the centre will also be greater than for the 'Y' pendulum. A greater speed means shorter periods. So if the length of the string is decreased the period will decrease as well & if the length of the string is increased the period will increase as well.
Quantative
Looking at the equation I can predict that if I increase the length of the string, the period will increase as well because if one side an equation increases, the other side should increase as well. I think the time (T) taken for one complete oscillation will depend on the length (l) of the pendulum.
I think that according to my equation, if I draw a graph of length (l) against period (T) it will not be straight line and the length (l) will not be proportional to the period (T) but If I square root the length (l) and then plot it against period (T), the √l will be proportional to the period (T) thus I will get a straight line graph OR if I simplify the equation by squaring both sides (as shown below) and plot a graph of period-squared (T ) against length (l) , it will be straight line graph and the length (l) will be proportional to the Period-squared (T ).
T = 2π √l/g
Simplified by squaring both sides:
I have used A level books and books from library to explain the formula I was using and to help me rearrange it. The information helped me to plan my experiment and was also useful when I was explaining my prediction.
Apparatus
i) Plumb bob
ii) Metre Ruler
iii) String
iv) Clamp Stand
v) Split Cork
vi) Stop Watch
vii) Angle Measurer
Method
·Attach the string to the Plumb bob.
·Place the string between the split cork and place the cork in the jaws of Clamp stand.
·Measure the string to the length of 100cm using a metre ruler (From the point of suspension to the centre of Plumb bob).
·Set the bob oscillating through a small amplitude (approximately 10°), simultaneously start the stop watch and begin counting through the equilibrium position of the bob.
·Find the time for 10 complete oscillations and write down the results.
·Carry out the same method for two more times with the same length of the string and find the average (for accuracy).
·Repeat the same experiment decreasing the length of the string, 20cm each time. Last experiment will be at the length of 20cm.
Fair test
·There must be only 1 Independent Variable (Length of the string).
·Make sure the amplitude don't go up more than 10°.
·Use the same apparatus.
·Repeat each experiment 3 times and than find out average, which will give more accurate results.
Analysis
The result of my experiment is shown below;
With my these results I will draw two graphs: One period (T) against length (l) and one period (T) against square root of the length (√l).
My results show that with increasing the length of the string, the period increases as well. My results show that my prediction was right that “if the length of the pendulum is decreased the period will decrease as well”. From my results I have found out that when I square rooted the length and drew a graph it was proportional to time taken for 1 complete oscillation.
The graph 2 obtained from the results is a straight line. It seems likely from looking at my graph that the same trend would continue if the graph was made longer. On the other hand it also seems that shorter length would follow the same pattern although it will be more difficult to take the measurements as the length gets shorter.
My results show that my prediction was right; the time taken for one complete oscillation depends on the length of the pendulum: Longer the length, longer the period & Shorter the length, shorter the period.
Now I will find out the connection between period and straight line.
The equation for a straight line through the origin is;
Gradient 'm' measured from the graph is;
Gradient =Vertical/Horizontal = T/√l = 1/5 = 0.2
If T is the time for one swing in seconds, and l is the length in centimetres, the equation for the line can be written as;
Now I will test, wether my equation work for all lengths:
Example; If T = 1.25 (sec), find l .
T = 0.2√l
1.25 = 0.2√l
1.25/0.2 = √l
√l = 6.3 (cm) (1dp).
The example proves that my equation work for all the lengths.
Evaluation
I think that most of my results were accurate, and method I used to obtain my
results was good enough to provide me with valid results. Although my experiment was designed well but there were some problems:
Measuring the length
There were some problems in measuring the length of the string. One difficult part of measuring the length was deciding where the centre of the bob is. The uncertainty in determining this measurement is probably about 1 mm. Measuring the length beyond about 1 metre is more difficult than short lengths because the measurement has to be done in two parts using metre rulers. I didn't have this problem because last length in my experiment was 1 m.
Measuring the time
Although the stop-watch measures to one hundredth of a second but the overall accuracy of the time measurements are not that good. I think the human reaction time to start and stop the watch roughly cancels each other out as the same event is being observed, and reacted to in the same way, each time. Errors are produced by any variability in the reaction time of the individual which could be affected by many things.
No significant problems or difficulties were encountered during the experiment. The accuracy and reliability of the results and conclusions are very good. Within the accuracy of the method used, and for the range of values investigated, it is clear that the time for a complete oscillation of the pendulum is proportional to the square root of the length.
Although my experiment was good but some improvements could be made to get more accurate results. A longer ruler or piece of wood, could be placed level with the point of suspension, and a set square could be placed along the flat side, just touching the bottom of the pendulum. This distance could then be measured more accurately than trying to guess where the middle of the bob is. The diameter of the bob could be accurately measured with some vernier callipers so that the true length of the pendulum could then be calculated.
The counting could become more accurate if it is possible to have some sort of electronic detection system that could automatically count and time the swings. Something like a light gate as part of a computer based logging system might work. An alternative might be a very high speed digital video camera that could accurately record the position of the bob and the elapsed time.
Further work can be done on the factors, affecting the period of a pendulum. for example, changing bobs of different masses or trying out the experiment at different gravity levels to see if it affected the acceleration of the pendulum. There is an alternative way that a pendulum can swing. Instead of swinging backwards and forwards in a single plane it is possible to make the pendulum swing in a horizontal circular path. It would be interesting to investigate how the time for each revolution of this 'conical pendulum' changes with the length and to compare this with the ordinary simple pendulum.
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