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1Maurice Yap 6946 – AS Physics Coursework – Unit G493(b) Quality of Measurement

MEASURING ‘G’, THE GRAVITATIONAL ACCELERATION CONSTANT USING AN OBJECT IN FREE-FALL

My aim is to measure the gravity of Earth as accurately as possible by

measuring the acceleration of a free-falling ball bearing after it is rolled

off the edge of a table.

INITIAL PLANI had an initial brainstorm of ideas as to which method to use in order to find ‘g’ most accurately. They included:

Rolling a trolley down a rampThrowing a tennis ball vertically upwards into the airDropping an object from a set height

After having carefully considering each one, my chosen method is to roll a marble or dense ball bearing off the edge of a table into a plastic tray. I chose this method because a falling dense marble or ball bearing would be affected less by air resistance because of its heavy weight; it is easy to precisely measure the height of a table (and therefore the distance which the object is travelling, ‘s’); and also, rolling it off the edge of the table ensures that the vertical component of its initial velocity is exactly zero.

METHOD

Figure 1: an illustration showing my planned setup.

First, I will roll a ball bearing along and off the edge of a flat table. Then, using a stopwatch, I will start timing from when the ball leaves the edge of the table until when it hits the bottom of the plastic tray, at which point, I will hear an audible sound caused by the metal ball hitting the plastic.

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2Maurice Yap 6946 – AS Physics Coursework – Unit G493(b) Quality of MeasurementI will change the height between the surface of the table and the bottom of the tray by adjusting the boss and clamp, and using a metal tape measure, with millimetre intervals, to measure. Starting from the lowest possible height, I will move the tray up, and therefore decrease the height, by 10cm intervals each time. In order to improve the reliability and repeatability and also to identify anomalous results which I would need to disregard.

CALCULATION‘g’ is the acceleration caused by gravity, therefore in the context of my experiment, is the magnitude of the ball bearing’s acceleration towards the plastic tray. The following formula is used to determine the displacement of the ball bearing, given the time, initial velocity and acceleration:

(SxS y)=(uxu y) t+ 12 (axay ) t2

Because the horizontal (x) components can be totally disregarded because they don’t affect and are not relevant to my result for ‘g’, the top row of each matrix above can be ignored. Also, the initial vertical velocity is zero, meaning that the entire uy t term can be removed as a term of zero would be added, which has no effect to the result. This resulting formula can therefore be rearranged fora y, vertical acceleration (caused by the Earth’s gravity), as the subject, because it becomes the only unknown (time, t, is the dependent variable which will be measured and displacement, s, is the independent variable which I will change):

a y=2 s yt2

Because, is I have already explained, the horizontal components are irrelevant, it does not matter how fast I push the ball bearing off the table (horizontal velocity), but I must make sure that the table is flat, to make sure there is no vertical velocity before the ball rolls off the table’s surface.

EQUIPMENT LISTBall bearingPlastic trayTape measureStopwatchRetort stand with boss head and clamp (x2)

RISK ASSESSMENTThere are two main hazards which I have identified.

The dense metal ball bearing is a dangerous “heavy object” in my view. Therefore:

I will not wear sandals or other footwear with open toes. I will make sure those working around me also have adequate footwear.I will make sure that the plastic tray is sufficiently deep so that the ball bearing does not bounce out and injure myself or someone else.I will check the condition of the retort stand, boss head and clamp to make sure they are in good working order and not likely to break or fail.I will check that these apparatuses are securely attached to the tray and each other.I will warn others working around me in the classroom of this source of danger.

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3Maurice Yap 6946 – AS Physics Coursework – Unit G493(b) Quality of MeasurementThe ball bearing will be accelerating towards the ground, in other words, falling. This moving object is also a hazard, so for that reason:

I will allow for more space than is required for the ball bearing to fall safely without obstructions.I will not conduct the experiment at eye level. This includes other people around me; therefore, I will need to check to make sure that no one close to me, for whatever reason, has their head at a height below that of the table i.e. if they are kneeling or sitting on the floor.I will keep my desk and the floor around the apparatus completely clear of the fast moving ball bearing. I will remove all bags, coats, stationary and other objects.I will mark out a safety zone around my setup to warn others of the danger.

In addition to this, I will not run or play practical jokes on others while in the classroom where this and other experiments are taking place.

RESULTSThis table shows my raw results. Results which I have judged to be anomalous have been included but are crossed out.

Table 1: raw results of experiment including uncertainty values, all numerical values are to two significant figures.

Height, 's' /m (± 1mm)

Time, 't' taken for ball bearing to hit bottom of the tray /s

Mean time, ‘t’ /s

Value of ‘t 2’ /s2

Uncertainty in value of t 2 /s2

Value of 2 s y /m (± 2mm)#1 #2 #3 #4 #5 +ve - ve

0.20 0.35 0.55 0.22 0.37 0.38 0.33 0.11 0.04 0.06 0.400.30 0.28 0.20 0.67 0.21 0.15 0.21 0.04 0.03 0.02 0.600.40 0.30 0.28 0.21 0.21 0.35 0.27 0.07 0.05 0.03 0.800.50 0.32 0.39 0.37 0.39 0.31 0.36 0.13 0.03 0.03 1.00.60 0.63 0.46 0.42 0.52 0.51 0.51 0.26 0.14 0.08 1.20.70 0.57 0.67 0.51 0.89 0.64 0.60 0.36 0.09 0.10 1.40.80 0.64 0.59 0.50 0.56 0.69 0.60 0.36 0.12 0.11 1.60.90 0.63 0.60 0.62 0.64 0.61 0.62 0.38 0.03 0.02 1.8

The uncertainties have been calculated by taking the highest and lowest values for t, then calculating t 2 for these two values. These are the positive and negative uncertainties respectively. I have also calculated the uncertainty for the height value (2S) to be ± 2mm, because the scale of the tape measure I was using was 1mm, so my height measurement was precise to 1mm either side.

The following shows these results plotted on a graph, including uncertainty bars in both the x- and y- directions.

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4Maurice Yap 6946 – AS Physics Coursework – Unit G493(b) Quality of Measurement

Figure 2: scatter graph showing the results of my second experiment

From my previous formula, the acceleration is calculated by dividing double the displacement (height) in metres by the time squared. I have therefore chosen to plot these on the y- and x- axes respectively. This allows me to calculate and estimate for the size of the acceleration by

simply finding the gradient of the line of best fit, which is∆ y∆ x

. From this, my estimate for ‘g’ is

3.2±3.3ms-2 to two significant figures. The uncertainty value was calculated using the maximum and minimum gradients which could be generated by these (dashed lines on my graph).

ANALYSISAs can be seen from my graph, the line of best fit does not appear to cross the origin if extended, as should be the case. Instead, it appears to intercept my y-axis for height. This suggests a systematic error because the projection suggests that to produce a result where the height of the drop would be greater than zero in order to record a time of zero.

I think that this could have been caused by the reason that the tape measure I was using was a flexible metal one. This means that if it was bent while I was measuring the height, I would have recorded a value greater than the true vertical height between the table’s surface and the bottom of the tray. Also, I did not make much of an effort to ensure that the tape measure was in a perfectly vertically upright direction, therefore it could have been slightly angled,

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5Maurice Yap 6946 – AS Physics Coursework – Unit G493(b) Quality of Measurementagain, causing me to read and record a height which was more than the correct value. This means that I can make the assumption that the heights recorded were more than their true values, causing all my results and hence the line of best fit, which should go through the origin, to shift up in the positive y-direction.

Another factor which compromises the accuracy of my results is the human error element due to my human reaction time when operating the stopwatch. This added a significant amount onto each of my results for time, hence shifted all results to the right in the x-direction. Evidently from the graph, this error has had a smaller impact than the previous systematic error.

My estimate of 3.22ms-2 is a very significant amount less than the 9.81ms-2 value for gravitational acceleration which has been widely accepted to be correct (CGPM, 1901).

I think that this has been caused mainly by human error in the timing of each drop. My average reaction time, which I have recently measured to be around 0.35 seconds, is quite a long length of time compared to the times in the data I collected. Also contributing massively to this large error is the fact that I was watching, and therefore pre-empting the moment at which the ball would hit the tray and produce a sound. This could have caused me to press the button on the stopwatch too soon. When the same problems repeated for the point at which the ball leaves the table causes the extent of this error to be doubled, it suggests that this is probably the reason why my results are so far away from the correct value and are pretty inaccurate.

In principle, another factor which would have affected my results was the factor of the speed of sound. My ears were probably roughly a metre and half away from the tray, where the sound was produced. Using the speed of sound as 343.6ms-1 (Collaboration for NondestructiveTesting, 2008),

speed=distancetime

∴time=distancespeed

→time= 1.5m

343.6ms−1=0.004 seconds

This length of time is insignificant to the number of significant figures (two) which I have used in my results. In practice, it was probably unimportant, if not totally irrelevant, because I was using my eyes to watch more than my ears to listen.

IMPROVED EXPERIMENT METHODTo get rid of the problem of an incorrect measurement for the height, I decided to use a straight, stiff and rigid wooden metre rule alongside a spirit level along with a set square for good measure. This ensured that I was measuring the full distance, since the wooden metre rule was straight and not easily bendable. The spirit level ensured that the ruler was exactly vertically upwards in direction, perpendicular to the Earth’s flat surface. I will also use it to make sure the table and tray are exactly parallel to the Earth’s surface.

In order to eliminate the human error caused by reaction times, instead of using a human operated stopwatch, I will instead use a phone with a microphone to measure the time taken for the ball bearing to hit the base of the plastic tray. To do this, I will place the device on the table surface and start recording. Then, as the ball rolls along the table, vibrations will be picked up by the microphone. As soon as the ball leaves the table surface, this will no longer be present. When the ball hits the bottom of the tray, a strong audio signal will be produced

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6Maurice Yap 6946 – AS Physics Coursework – Unit G493(b) Quality of Measurementand received by the microphone. The time which I will record will be the length of time between the start of the absence of the vibrations caused by the rolling ball bearing, and the distinct sound of the ball bearing hitting the bottom of the tray. I will obtain this through analysis of the digital audio signal using computer software, such as Adobe Audition. This screenshot and annotation shows one of these audio samples. A problem I encountered with this method was the loud background noise of my classmates conducting their experiments. This made it slightly difficult to distinguish the vibrations of the table through the audio signal’s amplitude alone. I solved this problem by equalisation of the audio, where I reduced the volume of the higher frequencies in order to emphasize the lower frequencies of the vibrating table surface. This solved the problem completely.

Figure 3: screenshot from Adobe Audition of one of the samples which I analysed.

My phone’s microphone records at a sample rate of 44,100 Hz, therefore, it would give a precision of about 22.7μs. It is therefore necessary, in my opinion, to correct each result for the speed of sound. I will place the microphone end of my phone as close to the edge of the table as possible, in order to make the time difference for when the ball leaves the table negligible. I will then adjust each result by taking away the time taken for sound to cover the height set (assuming the speed of sound in the room to be 343.6ms-1 (Collaboration for Nondestructive Testing, 2008)).

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7Maurice Yap 6946 – AS Physics Coursework – Unit G493(b) Quality of Measurement

Figure 4: illustration of my improved method

RESULTSI conducted this experiment while recording using my phone’s microphone. I later analysed the sound files to compile a set of results:

Table 2: raw results from second experiment

Height, 's' /m (± 1mm)

Time, 't' taken for ball bearing to hit bottom of the tray /s

Adjustment for sound /ms

Mean time /s (adjusted)

Value of ‘t2’ /s2

(adjusted)

Uncertainty in value of ‘t2’ /µs2

Value of ‘2S’ /m (± 1mm)#1 #2 #3 #4 #5 +ve -ve

0.20.2032

0.2024

0.2023

0.2012

0.2023

-0.582 0.2017 0.0407 372 435 0.4

0.30.2482

0.2483

0.2482

0.2483

0.2483

-0.873 0.2474 0.0612 20 30 0.6

0.40.2865

0.2867

0.2867

0.2866

0.2867

-1.16 0.2855 0.0815 34 80 0.8

0.50.3205

0.3205

0.3204

0.3206

0.3205

-1.46 0.3190 0.1018 64 64 1.0

0.60.3516

0.3518

0.3519

0.3519

0.3519

-1.75 0.3501 0.1226 56 154 1.2

0.70.3798

0.3793

0.3794

0.3796

0.3796

-2.04 0.3775 0.1425 196 181 1.4

0.80.4060

0.4059

0.4060

0.4060

0.4059

-2.33 0.4036 0.1629 32 48 1.6

0.90.4311

0.4309

0.4311

0.4311

0.4313

-2.62 0.4285 0.1836 171 171 1.8

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8Maurice Yap 6946 – AS Physics Coursework – Unit G493(b) Quality of Measurement

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Value of 't2' /s2

Valu

e o

f '2

S' /m

Figure 5: scatter graph showing the results of my second experiment

I recorded results to the nearest 0.1ms, because this was the precision of my chosen method and of my microphone’s audio format. The adjustment for the speed of sound was calculated using the formula:

speed=distancetime

∴time=distancespeed

→time taken for sound¿ travel=height343.6

In order to affect the final result by as little as possible, I decided not to round this result at all. This adjustment was taken away from the mean time (and hence, t2) and for the times used to calculate the uncertainty. I calculated the uncertainty in exactly the same way as my first experiment.

I used spread sheet software to calculate all my results. Although those presented in the table have some degree of rounding, I did not round any numbers before the final results in order to affect this by as little as possible by inaccuracies caused by rounding.

The gradient of the line of best fit, and therefore my estimate for the value of ‘g’ is 9.81ms -2

±0.02ms-2 to two significant figures (rounded from 9.8085, which was the value of the gradient produced by my spread sheet software). The uncertainty in this was calculated using the respective uncertainty values for each height. They produced a result of 0.0019ms -2. I have not included this on the graph.

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9Maurice Yap 6946 – AS Physics Coursework – Unit G493(b) Quality of MeasurementMy graph does include uncertainty bars, but because my raw results for each height were extremely consistent, they are not very visible at the scale I have chosen in the horizontal, x, direction because the uncertainty was so tiny. The uncertainty bars in the y-direction for height are of the same size as those used in my first experiment, because the metre rule which I used to measure the height had a precision, again, of 1mm, so I judged my own precision to be a millimetre either side (2mm for ‘2S’).

I did not record any anomalous results in my improved experiment.

ERRORSThe following table and graph compares the errors of each of my heights for each experiment. The ‘correct’ value of ‘g’ is taken to be 9.80665ms-2 (CGPM, 1901).

Height /m

#1 %age error

#1 g value /ms-2

#2 %age error

#2 g value /ms-2

0.2 62.5 3.67 0.26 9.83

0.3 38.7 13.6 0.03 9.80

0.4 11.9 11.0 0.10 9.82

0.5 19.5 7.89 0.18 9.82

0.6 52.6 4.65 0.15 9.79

0.7 60.0 3.92 0.18 9.82

0.8 54.1 4.50 0.14 9.82

0.9 52.3 4.68 0.03 9.80

I have decided to present my data on a line graph with a logarithmic scale for the percentage error axis so that it is easier to compare the errors between my two experiments. It would be difficult using a linear scale because the distance between them would be too great to visually compare.

Firstly, it shows the massive magnitude of the difference between each experiment. My second experiment has errors roughly of the order 0.1%, however, my first experiment has results all of which are between 100 and 1000 times less accurate than this, of the order 10%. This shows the big improvement which my improved method had on the results, compared to my initial experiment.

My first experiment’s errors don’t show a clear pattern or correlation between error magnitude and height. I would have expected there to have been a negative correlation because a longer time would have ‘diluted’ the error of my bad reaction time. However, the systematic errors were so bad, and my reaction times were so inconsistently bad that there is no obvious correlation. The results are all very erroneous and are not really comparable to each other to generate a conclusion.

Figure 6 (right): line graph comparing the errors of my experiments. NB: the y-axis has a base 10 logarithmic scale.

Table 3 (below): estimates for 'g' for each height and their respective errors for each experiment

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10Maurice Yap 6946 – AS Physics Coursework – Unit G493(b) Quality of MeasurementThe errors for my second experiment don’t show a correlation with height either. I think that this is because any error there was, was caused by a systematic error, as opposed to a measurement error; more specifically, the height of drop: there was some physical movement in the apparatus, especially the two clamps holding my tray. This may not have necessarily moved it closer to the ground, but could have changed the angle of the tray slightly, making the angle of the tray a new unwanted variable which I did not attempt to control. Also, although I used a spirit level in two directions to make sure my tray was perfectly flat, I didn’t do this between tests for the same height. There was also some slight angle range for the spirit level to measure flat. I do not think that there was a significant error, if any error at all, in the process used to measure the time. When analysing sound samples, I zoomed in to the highest possible resolution, to the point where I could distinguish between each individual sample to ensure maximum precision.

A systematic error which could have occurred was that the ball bearing could have hit the side of the tray instead of the bottom, giving a measurement of less time than would be correct, as the ball bearing would have travelled less distance. This fortunately did not happen on any occasions.

OVERALL ERRORSMy final result for ‘g’ for my first experiment was 3.2ms-2. This is an error from the correct amount of about 67%. This was, as I have already explained, due to lots of errors in my first experiment, which affected all the results by a lot, the biggest factor being human reaction time.

My second experiment produced a value of ‘g’ of 9.8085ms-2, which is about 0.02% error. The big improvement and reduction in error was probably down to the removal of many of those factors which caused the error in the first experiment. It was also down to the excellent accuracy and precision of the equipment I decided to use to take measurements, which was a lot better than myself using a stopwatch.

REFERENCE LISTCGPM. (1901). Comptes Rendus de la 3e., (p. 70).

Collaboration for Nondestructive Testing. (2008, January 31). Temperature and the Speed of Sound. Retrieved February 3, 2014, from NDT Resource Center: http://www.ndt-ed.org/EducationResources/HighSchool/Sound/tempandspeed.htm