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Gravitational acceleration -
This will be the main force acting upon the pendulum when it is moved away from its stationary point. The gravitational acceleration on earth is constant at about 9.8 N/kg.
The gravitational force will be responsible for supplying the pendulum with energy (from potential to kinetic) to swing back and forth.
This is because it contributes to the potential energy the pendulum will posses as shown in the formula m.g.h.
If this same experiment were being conducted on the moon, the results would be quite different, as gravitational acceleration is only 1/6th of what it is on Earth.
Apparatus:
Clamp stand
Clamp
Boss
2 squares of wood
Pendulum
Stopwatch
Protractor
Ruler
Diagram:
Hypothesis:
The rate of pendulum oscillation depends vastly on the forces acting upon it.
A pendulum is a small mass hanging from a piece of string.
When the pendulum is hanging straight down it is a state of equilibrium where all the forces acting upon it cut each other out.
When the pendulum is displaced then the gravity, its weight and the tension on the string are the forces acting upon it.
For small displacements the net force is proportional to how far the pendulum has moved.
Therefore this is an example of a simple harmonic oscillator.
Below is a diagram to show which forces are acting upon the pendulum and therefore its direction and force.
As we will be using a simple pendulum and a small displacement every time, this experiment will therefore be an example of simple harmonic motion.
For simple harmonic motion the period (time taken for one complete swing) of the pendulum is independent from the mass and amplitude of the swing.
Therefore the following formula can be used to predict the results.
T = 2π √(L/g)
T is the time period for the pendulum to complete one swing.
L is the length of string that the pendulum is attached to, in meters. g represents the gravitational acceleration at earth, which remains constant at 9.8 m/s2 .
2π is in the formula because as the tensional force is acting on the string it is moving in a circular motion, but for only a small amplitude, therefore parts of the circumference formula are included in the equation.
This formula can give us a graph of predicted results, as L is the only variable.
It seems odd that one can predict the results by not even accounting for the mass or amplitude.
This is because simple harmonic motion is independent of mass and amplitude but dependant on the length of string, the gravity and the tensile force (which causes its direction hence the 2 π in the equation).
This is because no matter how much it weighs or how large the amplitude is every time period for one length of string is the same
This is because; say you have a large amplitude at the beginning of the test but a smaller one near the end due to air resistance and friction slowing it down.
Each period would be the same because if the pendulum had a large amplitude it would also be travelling proportionately faster, whereas when it has little energy the pendulum would move slower through the small amplitude.
The 2π is essential to the equation because it corresponds to the distance the pendulum must travel. Imagine the point from where the string is attached to is a centre point of a circle.
The distance the pendulum would therefore be the circumference i.e. 2πr (or in this case 2πL).
But as the pendulum doesn’t travel the whole of the circle the equation uses the 2π and manipulates the length together with the gravitational acceleration to give the formula.
This also leads to the reason why the longer the string the longer the period.
Again imagine that the pendulum is part of a circle, and an arc of the circumference is the distance it travels.
To work out the distance of the arc you use 2πr.θ/360 where θ is the angle from the centre point.
Take for instance the angle that the pendulum is released from is 10° to the normal.
Therefore the total angle would be 20° because when the pendulum is released, it will pass the normal and go to about 10° to the other side.
If we were to calculate the distance the pendulum has to travel considering the length of the string is 20cm this is how we would do it:
D= 2π. 20. 10/360
= 2π. 20/36
= 3.5cm
This same formula also displays a direct link between the time period for an oscillation and the length of string.
T = 2π √(L /g)
2π will always be a constant because it is just a value of a specific number.
“g” is also a constant because it represents the gravitational force of the earth, so unless one was to do the experiment elsewhere the value would always be the same (9.8m/s2).
If you take the two constants out of the equation because their only job is to make the “numbers add up”, you are left with:
t α √L
What this does show is that there should be a directly proportionate correlation between t and √L
When the pendulum is pulled to its starting point, it gains potential energy.
This is because it has moved to a higher position and so gravity is giving it extra energy.
If the length of string is about 20cm and a 20g pendulum is moved to 10° to one side we can calculate the potential energy and its velocity as follows:
Adj = Cos10 × 20
=19.7
h =20 – 19.7
= 0.3cm
PE =m.g.h
= 0.02kg × 10× 0.003m
= 0.0006J
m.g.h = 0.5 mv2
v 2= 0.0006J /0.01kg
v = 0.24m/s*
*Assuming 100% efficiency
The pendulum will obviously not swing forever.
This is due to two reasons, firstly as the pendulum oscillates it collides with air particles, losing energy – this is air resistance/friction.
The second reason is that when the pendulum is released it has potential energy, which is quickly converted to kinetic energy, this makes the pendulum move to the other side where it again gains potential energy and then it happens all over again.
Every time there is an energy transfer the pendulum loses heat energy.
As a result of these two reasons the pendulum slows and comes to a stop.
Fair test:
In order to ensure that the test is fair we must keep all the factors apart from string length constant.
This means that there can be no or minimal change in gravitational acceleration, degree of displacement, mass air resistance and tensile force.
The tests should be conducted as accurately as possible. In order to make sure the reaction times for the stopwatch have a minimal effect the period for10 oscillations should be measured.
Method:
A clamp stand will be erected and using a clamp and boss the pendulum will be held up in between two pieces of wood.
The length of string will be measured using a ruler.
The pendulum (20g) will be positioned to ten degrees from its stationary point using a protractor and will be released.
As the pendulum will be released the stopwatch will be started and stopped after the pendulum after the pendulum has returned to its starting point for the tenth time.
This method will be repeated for each different length of string.
The pendulum will be left to oscillate 10 times before stopping the watch.
This is because the reactions of the student using the stopwatch are about 0.25s for stopping and starting. So if one oscillation took 1 second the student would have around 1.5s on the stopwatch.
This would mean there would be an error of 33%, which is very high.
However, if the experiment were conducted for ten oscillations, the same margin of error would result in only a 3% error, which is minimal.
It is important that the angle is about 10 degrees.
This is because this experiment also looks at how simple harmonic motion works.
Simple harmonic motion can only take place when there is a small displacement of the pendulum.
In order to obtain the most accurate results and define trends in this investigation we must take as many readings as possible over a large range.
As we have about 1½ hours to conduct the investigation, we will be able to manage about 10 readings of different string lengths.
We will conduct the experiment for varying lengths of string from 5cm to 50cm at intervals of every 5cm.
This allows us to get a good range and the intervals are close enough together to give an obvious trend.
Problems:
As we did the experiment, unfortunately things didn’t go as they were planned.
The main problem was that we were unable to securely hold the pendulum at the intended lengths, as the string would not correctly wrap around the wood to give us exactly the right string length.
Although our results were therefore a little uneven and untidy the results were still valid as we took them at an average interval of 3-4 cm ranging from 8cm to 52cm and in total taking 12 readings.
These results can still be plotted on a graph for analysis and therefore can still correctly show any valid trend.
Safety:
Considering this experiment ensures that pendulums have very little energy, the procedure is very safe as the pendulums won’t hit anybody and even if they do it will be harmless.
Results:
Analysis:
The table of results can be used to produce several graphs to visually display various trends and links in this investigation and can be manipulated to give extra information.
The first step is to manipulate the data.
The second step is to construct graphs using this data.
The graph above shows how our results compared with the theoretical or predicted results. It shows that the results are very close to the predicted results and that they both have a very similar shape. This proves that the equation T = 2π √(l /g) is correct at determining results for the time period of one oscillation for a pendulum in simple harmonic motion. This is because all the relevant and depending forces are used in the formula whereas the independent ones such as mass and amplitude are left out.
On average the actual results are about 3-4% wrong compared to the theoretical results, the error margin can easily be explained. When testing the hypothesis we used a stopwatch to time each period over 10 oscillations. If we had a reaction time of about 0.3s each way, if the period was 10s long then there would be an error of about a 6%. Other than that measuring the amount of degrees was a little tricky while positioning the pendulum.
The graph above shows that there is a directly proportionate relationship between t2 and length of string. This means that the time period is dependent on length. It displays the fact that t may not be directly and solely proportionate to l but its square is, therefore increasing the length will increase the time period in correspondence with its square. The gradient works out as 1.1s2/30cm i.e. 0.34s2 /cm. The reason for this connection is that in the formula even the other dependant parts i.e. 2pi and g have been taken out of the equation leaving only t = sqrt l which is equivalent to t2 = l. This therefore proves the hypothesis correct i.e. there is a direct correlation between t2 and l. L will naturally increase t because the larger the l the larger the arc of displacement or amplitude because not only will it have more of a fractional circumference but more potential energy as the height difference will increase in proportion.
This chart shows that there is a directly proportionate correlation between the length of string and the potential energy that the pendulum gains. It proves the last point in the previous paragraph “l will naturally increase t because the larger the l the larger the arc of displacement or amplitude because not only does it have more of a fractional circumference but more potential energy as the height difference will increase in proportion.” However this will not effect the rate of oscillation much, as bigger amplitude also means bigger velocity and so the rate will stay pretty much the same.
Evaluation:
The experiment was far from perfect, mostly because we were unable to obtain the planned readings, as we could not get the string length to exactly what we wanted. As a result the results were uneven. What we could have done was to set the apparatus as below:
The accuracy of our results could have been enhanced if we had repeated experiments and taken the average of each reading. The 5% stopwatch error margin could not be avoided and positioning the pendulum was a little tricky to start with when measuring the degrees.
Apart from that our results were very accurate with only a 3-4% error margin. Also the results clearly showed the directly proportional relationships without any anomalous results. I believe there is strong evidence in this investigation to support the conclusions and the hypothesis.