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

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Introduction

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 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: -

 Length of pendulum /m Time for 30 oscillations (s) Average period of time T/s Time for one length squared T2/s2 EXP 1 EXP 2 EXP 3 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

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 10cm 1.0mm
• 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. 20cm 1.0mm
• 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.

Middle

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.

Conclusion

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.

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