An Experiment Using a Pendulum to Find the Acceleration due to Gravity.

Aim:

The aim of my experiment is to find the acceleration due to gravity otherwise known as ‘g’. To do this I could do an experiment that involves a pendulum and the formula

which can be put into the equation of a straight line y=mx+c Another experiment I could undertake uses a trolley and ramp but a different formula involving mass, which again is put into the equation of a straight line.

I am going to pursue the pendulum idea, as it was the original experiment used by Sir Isaac Newton who’s value for the acceleration due to gravity still stands today (even with all our modern technology). The trolley and ramp idea seems insufficient as the trolley isn’t really in free fall and the friction from the ramp would surely affect my results.

I need to make a pendulum that:

Has minimal friction at its pivot point,
Can have its length easily changed and be accurately measured,

Will not swing in a circle,
Has a small set angle of swing (drop angle),
Non elastic stem

I propose two different ideas.

Meccano idea:

The rigid structure will stop the pendulum swinging in a circle.
The even spaces mean that the length can be easily changed and easily measured.
Using a wheel as a bob means that you can easily measure to the centre of it.
Oil can be used to decrease friction at the pivot
A protractor can be placed at the pivot and then the drop angle can be easily controlled.
A light gate could be used to calculate oscillations more accurately.

Double fishing line idea:

The two points of contact will stop it swinging in a circle

The line is almost massless which means it has little air resistance and has little negative affect on the experiment.
The line can be clamped at the pivot and therefore has minimal friction.
By measuring the line and marking divisions with a pen you can clamp the line at exactly the length you want, and easily slide it up and down to change it.
Again a protractor can be used to control drop angle.

Comparing the two ideas I find that:

They both stop the pendulum swinging in a circle. This is important because I need to measure as accurately as possible the time period of a certain amount of oscillations. If the pendulum is swinging in a circle then the measurements will be less accurate. It could also collide with something, which would disrupt the experiment.

They can both be measured accurately, although it is much easier to change the length of the meccano idea. The length is included in the formula, it will be one of the things used in calculating the acceleration due to gravity, and therefore needs to be measured as accurately as possible.

The fishing line idea due to its clean crisp nature will have less friction at its pivot than the meccano idea. I would have to use oil for the pivot point of the meccano but it still involves the rubbing of metal against metal without bearings.

They can both have their drop angle measured.

The meccano idea could involve a light gate because its square shape will mean the light can be cut precisely whereas the fishing line idea has a round bob with no definite cut of point and the fishing line is so thin it would not cut the light. As long as the cut of point is the same each time a light gate should be very accurate but it would take a lot of detailed co-ordination to achieve this. Also generally varying light levels occurring naturally in the room could affect the light gate. So as long as I work out my margin of error doing it manually would be just as accurate.

Rigid pendulums are used in clocks so they must be accurate as timekeepers. Yet an Internet site (http: kossi.physics.home.edu/Courses/p23a/Experiaments/Pendulum.html) about the experiment stated that it recommended the use of a massless, inextensible string. All experiments I have seen also use some sort of string rather than a rigid structure.

On this basis and previous reasoning I am going to use the fishing line idea.

Apparatus:

Fishing line
Clip board clips
Reasonably small cylindrical weight with attaching ring
Two points of bearing so that the position of the bob at rest can be accurately seen when oscillating.
Either a stand or clips from the ceiling or table.
A stopwatch
A protractor to measure the drop angle.
A meter ruler or tape measure to measure the length of the pendulum.

Fair test:

There are three variables that could affect the result of my experiment. They are the drop angle the mass of the bob and the length of the pendulum. The two former are not included in the formula so should not affect the outcome of the experiment, non-the less I will keep them as constant as possible throughout my investigation.

I will drop the pendulum from the same angle each time. This angle will be 10 degrees, anything more than that and the difference in amplitude of the oscillations will change more rapidly from the first to the last. This makes the time it takes for each different length of pendulum to complete the oscillations more variable.

I will not move my experiment as not to change its set up between tests because this could affect my results.

I will make sure that my pendulum stand is rigid so that it wont move and absorb some of the energy ...

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I will not move my experiment as not to change its set up between tests because this could affect my results.

I will make sure that my pendulum stand is rigid so that it wont move and absorb some of the energy from the swing.

Before I undergo the experiment I will use a computer program, which tests my reaction time, I can then work out the margin of error in my results. I will take the test 3 times and get an average I should be looking at something between 0.2 and 0.25 seconds.

I will complete my experiments in a draft less area, as friction from a stream of air particles will have an adverse affect on the swing of the pendulum.

Measurements:

I will take measurements using a stop clock for the time it takes to complete 30 oscillations. This is enough to make human error and reaction time fairly insignificant but not too much so that the pendulum will stop before completion of its oscillations. In a book ( ) it recommends 50 oscillations and previous results show a successful experiment using only 20 so I’m going for the middle ground. The stop clock will measure accurately to 1/100th of a second, my reaction time after calculation will be somewhere between 0.2 and 0.25 of a second.

I will measure the length of the pendulum but keep this as my controlled variable. I will measure from the pivot to the centre of the bob. The length measurements I will use will range from 10cm to 1m with divisions of 10cm. This means that from 0 there will be equal divisions, the graph will therefore look tidier and have a good range of results over an equal spread. I can measure with a tape measure precisely up to 0.1 of a centimetre (1.0mm).

I will repeat my experiments 3 times and take an average. I will do this to check reliability, a small range in results means they are reliable.

I will record my results in a table like this one: -

Detailed plan:

Find a suitable place to build the pendulum either from the ceiling, or on two stands between two tables to allow a meter of pendulum beneath them. A rigid structure is important otherwise energy is absorbed in the swaying of the stands.

Build the pendulum as shown in the picture on the previous page. By attaching the two clip board clips to something so that the distance between them does not change. Cut two pieces of fishing line longer than a meter and tie them both to a weight. Clip the line into the clipboard clips. Attach a protractor to one side so that the angle the pendulum is at can be seen from the other.
Use a smallish mass e.g. 50g so that the fishing line doesn’t slide through the clipboard clips.
Measure the length of the pendulum to be 10cm1.0mm
When constructed hold the pendulum back to 10 degrees as shown on the protractor.
Let go and at the same time start the stop clock.
Count 30 oscillations and stop the clock.
Note down the time on the chart in excel.
Repeat the experiment another two times and note the results down on the chart. An average will automatically be calculated.
Repeat with the next length e.g. 20cm1.0mm
And then carry on until all lengths have been done 3 times.
The results will automatically come up on a scatter graph in excel and should show a strait line.

Safety:

I will make sure any stands are fixed securely so that they won’t fall on top of anyone.

I will make sure the weight is secured tightly so that it won’t fall of on to anyone.

I will make sure everyone is clear of the pendulum so that it won’t swing into anyone.

I won’t swing the pendulum past 10 degrees, as this is unsafe and may hit some one. Prediction:

I predict my results will show that T2 is directly proportional to ; the gradient of the line will be around 4.00 and therefore a value for ‘g’ can be calculated at 9.8ms-2 (2sf)

Hypothesis:

T2 will be directly proportional to because as the length of the pendulum increases so does its displacement therefore so dose the time it takes.

I predict this because Isaac Newton whom the story goes, had an apple fall on his head, recorded from theory that all objects had a gravitational pull or gravitational field strength due to the fact that masses attract. He successfully calculated using the pendulum experiment that the acceleration due to gravity was 9.81ms-2. The reason my results will not necessarily come up with this exact figure is because there will be a degree of uncertainty. This will be due to the accuracy of my measuring ability, which will be controlled by the equipment I use and in some case my reaction time.

Another factor that plays a role in calculating ‘g’ is where you are on the earth; In some places you weigh less than in others. This is due to things like the density of the rock that you are standing on. Igneous rock on continental plates, which is denser than others types will make ‘g’ larger where as sedimentary rock on oceanic plates which is less dense than other types will cause a smaller value of ‘g’. If this is so then doing the experiment out at sea or elevated from the ground on a high-rise building will also have a different value for ‘g’. You also have to take into account gravitational pull from the sun and moon or even other smaller bodies of mass like say the walls in the room the experiment is undertaken. These will all pull the pendulum an opposite way to the effect from the earths gravitational field strength. Generally the affect is so small the apparatus I will be using won’t pick up any difference in results. And I don’t have enough funds to go travelling the world.

Implementing

Changes I made to my plan while implementing where:

Instead of using a protractor to measure a 10 degree drop angle, I drew this angle on to a piece of card so that I could see it clearly and therefore measure more accurately the angle at which I was dropping the pendulum each time.

Because my pendulum involved two points of contact the fishing line could not just be measured from the clip to the bob to gain a value for length, as this would have been incorrect. Instead I had to measure the distance from the ground to the bottom of the clip (155.5cm) and make sure this value was the same on both sides by adjusting the clamps. I would then measure from the ground to the centre of the bob and take it away from 155.5cm to gain the value for length.

Although I said I would not move the pendulum structure as not to affect the results, due to the time it took to build my structure I could not finish my experiment in the first sitting. I had to move it to keep it safe. I took measurements so that I could put it back in almost the same place for the next sitting.

Observing and recording

Here are the results that I collected from my experiments:

While implementing my experiment I added two more columns to my chart. The standard deviation column was added so that I could see weather my result taking was reliable and then make a decision on whether to repeat them or not. The increment column was added so that I could see how close to a straight line my results were heading. Both of these additions allowed me to see while doing the experiment whether I was doing it right and therefore whether I had to change anything.

The results in green where results that I repeated, and the results in red are ones which I have earmarked for closer evaluation due to the large standard deviation from them. The green results for 0.1m were all repeated because the first time I collected them I counted every swing as an oscillation, therefore only timing it for half the oscillations it should have completed. It there had a much lower result for time period than is shown in green (in fact only half that). I could have doubled the result but that would have been bad practise, wind resistance may have slowed down that second half of oscillations, which I would not have observed.

For the time period of the oscillations I wrote down the results to two decimal places because I wanted to have the level of precision given by the stop clock. Because of this in all other calculations originating from the time period of the oscillations I also used two decimal places.

Interpretation and Evaluation

From the chart I made a graph on excel which plotted time squared (T²/s²) against length (/m). I expected to see a straight line with strong positive correlation through (0,0) because it is directly proportional.

This graph shows that time squared is directly proportional to the length of the pendulum. This is because of simple harmonic motion caused by the pull of gravity. If the graph was just time against length it would be a curve. I can now work out the acceleration due to gravity but first I have to put the equation that relates T2 to length into the formula for a straight line.

y=mx+c

I then used the gradient, 4.03 to eventually find a value for ‘g’.

Theory shows that the value for c (the intercept) should be 0 because the line is directly proportional. Although my result for c is a very tiny way off 0 it is too small to be used in the calculation. The reason it does not follow theory is because of an error in the measuring of the length. There are many reasons why that may be. One reason is that because of the long-winded way I had to measure the length of the pendulum there was much room for small errors, like whether the ground was all level. Another reason could be that because my bob had a small extension above its main body its centre of gravity would not have been exactly at its main body centre, this would mean a slight error in measuring the length each time. I could work out the bobs centre of gravity and if I had more time that’s what I would do to extend this experiment. But now I am quite happy with knowing that my value for ‘g’ is not affected by this small error. If I were to do the experiment again I would take more time and caution when measuring and maybe find a better more accurate way to do it. I would use a bob that could be described as a point weight or work out the centre of gravity for an irregular weight.

Degree of uncertainty:

I am going to round my value of gravity to 2 significant figures. This gives me a value for ‘g’ at 9.8ms-2 and a percentage error of compared to the accepted value of 9.81ms-2. To find out where this percentage error came from I have to trace back and work it out from the limitations of my measuring equipment.

Actual error for time is 0.01s for the limitation of the stop clock + 0.14s for the variance in my reaction time. If it takes you 0.2 seconds to react to the dropping of the pendulum and then 0.2 seconds to react to it finishing its final oscillation then the two cancel each other out. The problem is you don’t have the same reaction time each occasion you stop or start the clock. To find out my minimum and maximum reaction time I used a computer program and found that my fastest result was 0.19s and my slowest was 0.26s a 0.07 second difference. This means that there is a 0.14 second uncertainty, 0.07 at the start and 0.07 at the end.
To find the percentage uncertainty for time I divided the actual error by the average time and then multiplied this by 100.
To find the percentage error for time squared I multiplied the percentage error for time by two.
To find the actual error for time squared I divided the T2 by 100 and then multiplied this by the percentage error for T2.
The actual error for length is due to the limitation of the measuring equipment and means that I can only measure accurately up to one millimetre.

To find out the percentage error of my result for ‘g’ I need to now draw another graph with error bars, then from this find the maximum and minimum possible values of the gradient. After that I will use the formula to work out a value for the acceleration due to gravity using the lowest gradient and then the same with the highest gradient. From that I can then work out a percentage error.

From the graph you can see that my two values for ‘g’ were 9.99ms-2 and 9.62ms-2 this gives me an actual error of 0.2ms-2 rounded to one significant figure and a percentage error of 2%.

This shows that the true value for ‘g’ lies within the percentage error of mine 9.8ms-22%. These errors came from the accuracy of the equipment that I was using or the accuracy of my ability to read of the results from the equipment. If I wanted to eradicate any hint of error caused by reaction time in any future experiments I could use a light gate to record the time taken for oscillations. This is a device that when the light is disrupted records a result on a computer. I had previously disregarded this idea because it was not suited to my experiment but I’m sure that in future I could adapt it.

Correlation:

I know that my gradient produced an accurate value for ‘g’ taking into account the percentage error, but what about each individual result for time. How accurate are they? How strong is there correlation? And what could I have done to make them more accurate? I researched a formula that could be used to calculate correlation called the: -

Product Moment Correlation Coefficient:

With the help of a graphic calculator I worked this out to be 0.99997431 (≈ 1) in a scale where 1 is absolute positive correlation and 0 is no correlation it is quite clear that these results bear a very strong positive correlation.

Although the correlation is strong there is a reasonable amount of standard deviation between my 3 repeats of each length of pendulum. From my table of results you can see that generally the longer the pendulum the greater the standard deviation, especially on the last length highlighted in red. I suggest that the longer the pendulum swings for the more friction air resistance and any other external effects affect it. The reason that the last result was so far different was, I think, because the table that my experiment sat was knocked during the taking of one result. This would have meant some of the energy from the swing would have been absorbed in the swaying of the table and structure.

You can also see from the increment that there wasn’t a perfect step up in results this could be due to the unevenness of the floor, which I was measuring the length from. The way I was measuring the angle left a small margin for error each time I dropped the pendulum. The fishing line could have slipped a bit on some results although it was held quite firmly it was held by the spring tension of a metal clip board clip rather then say a fixed clamp. Due to the fact that the experiment took me a while to set up (mainly because of trying to solve my angle measuring problem) I didn’t get to finish all the result taking in one session. Although I tried my best using measurements to set it up exactly the same in the next session the results may have been affected. Though looking at my chart there is no uniform alteration in increment or standard deviation, which would support this. To improve my experiment for next time I would pay more attention to detail when measuring the angle of swing and do all the result taking in one session.

One other factor that may have caused a small anomaly in my value for the acceleration due to gravity is as I explained in my plan the difference in the Earths gravitational field strength. Because the different density of the earth at different points it dose not have a uniform gravitational field strength and the place where I conducted my experiment may have a different value for ‘g’ than the place Newton conducted his. (I must note that this anomaly will only be tiny but very interesting if I wanted to extend my experiment any further)

Conclusion:

I conclude that my result for the acceleration due to gravity of 9.8ms-2 reflects an accurate attempt at supporting the value discovered by Sir Isaac Newton. The 2% uncertainty that I gained from the limitations of my measuring equipment due to their accuracy show that Newton’s value lies within the boundaries of mine. If I were to do the experiment again and follow all the modifications that I have stated then I am sure that I could if not only repeat the level of accuracy shown by my result of ‘g’ to 2 significant figures maybe even find it to 3 (9.81ms-2).

Furthering my investigation:

To further my investigation I could find out the effect on time period by changing mass (although I know from Newton who stated that the time period is independent of mass or swing length, the fact that they are not in the formula supports this).

I could complete the experiment in different parts of the world where I know the density of rock beneath me is different to see if I could gain different results for ‘g’. My experiment would have to have been refined to great perfection though so as to notice any change.

I could investigate the simple pendulum as a parametric oscillator by changing either its length or acceleration due to gravity during oscillations as to keep it swinging at a constant rate.