Hypothesis
I predict that the longer the string is, the longer the time period for 1 full oscillation is, and that the Time Period ² will be proportional to the string length which would mean we could calculate the time period of one oscillation with only needing to know the length of the string.
When a pendulum is swung, at point 1, the pendulum has the maximum potential gravitational energy. At point 2 this energy is converted to kinetic energy which is moving energy and at point 2, the kinetic energy is at its highest.
This means that the pendulum is moving at its fastest at point 2.
When the pendulum reaches point 3, it has risen and the kinetic energy is transferred back to gravitational potential energy.
After this, then there is the fact that air molecules in the air that cause friction to the pendulum and makes it lose energy so the gravitational energy that started from point 1 will now be less at point 3 because some energy is lost and eventually the pendulum stops.
The reason a pendulum moves at a slower time period the longer the string is, is because the longer the string, the pendulum will have even further it has to travel because there will be a wider arc. We can compare the length of the string to the radius of a circle. The longer the radius of a circle is, the bigger the arc is (circumference).
So if we take the formula for the circumference of a circle(2∏r) into consideration we can predict the time period for different lengths of string using the formula
T = 2∏√(l/g)
T being the time period for the pendulum to oscillate once, l being the length of the string and g being the gravitation acceleration at Earth which is a constant 9.81 m/s².
The graph is not good enough to be used because our hypothesis states that the time period should be proportional to the length so is we wanted to calculate the time period easily we would just need the length of the string.
I will now try to make a graph with t² against length
The is what I predict my experiment results will look and when I compare the 2 graphs I should get a very close match. If not then my Hypothesis in wrong.
Fair Test
To keep this a fair test, we must ensure that the string length is accurate, the angle of displacement is same through out and to make sure that reaction time make as minimal effect as possible by measuring the time for 10 oscillations instead of 1.
Safety
There should be no safety precautions as there is nothing heavy or explosive or in anyway harmful being used.
Method
Firstly the apparatus shown in the diagram is set up. The string is held in the clamp by adding 2 square wooden blocks between the string. From the start of the wooden blocks a measurement is taken of the string length.
We had 3 people in our group conducting the same experiment, one held the stop watch, one measured the length of the string and the other would measure the displacement of the pendulum and then let go of it to start the pendulum swinging. The Timer would also count the oscillations.
When the Displacement was measured (10 degrees) the pendulum would be let go off and after the 1st oscillation the timer would start the stop watch and count 10 oscillations before he stopped the timer.
This was repeated twice for every measurements from 5cm to 50cm with intervals every 5cms.
It was repeated twice so a more accurate time could be recorded and we would count 10 oscillations instead of 1 do decrease the reaction time effect.
The reason for the 10 degrees displacement was because we are also looking at how simple harmonic motion works. We can only look at this with a small displacement angle of the pendulum.
Results
Analysis
The table of results can be used to make a few graphs that will help visually show the trends and links in this experiment and can be used to give extra information.
From the data we can now compare the T² from the predicted graph and compare it to the data we go from the experiment.
This Graph proves that the formula T = 2∏√(l/g) is accurate and works and can be used to work out the time period for any string length of a pendulum swinging in a harmonic motion
This graph shows that the relationship between t² and the length of the string is proportionate, this means that the time period is determined on the string length of the pendulum. It shows us that t is not directly proportionate to length as also shown in the prediction graph.
The gradient is 1.1s²/30cm which is 0.34s²/cm. The reason for this connection is because the formula is not dependent on the 2∏ and g because we have shown that
t is proportional √L and so t² is proportional to L.
This proves my hypothesis is correct and that t² and L have a direct correlation and that string length will increase time period and the larger the length is the larger the arc of displacement.
Evaluation
Apart from the 3-4% error margin and the difficulty of obtaining an exact string length to the nearest centimetre, we had very accurate results.
Because of this our results where very uneven but they were not that bad and were still useful.
I think that the data shown in the results and analysis show that my hypothesis was correct.
