Precision: could be divided into two major part: 1) resolution 2) sensitivity
(1) (Resolution): The ability of a measurement to be consistently reproduced
(2) (Sensitivity): the number of significant digits to which a value has been reliably measured. If on several tests the temperature sensor matches the actual temperature while the actual temperature is held constant, then the temperature sensor is precise. By the second definition, the number 3.1415 is more precise than the number 3.14.
Look at the examples below:
- Examples from my results contain the results for 10cm length in which as we can see the average of my results is close to that calculated from expression but the three results are far from each other(precise but not accurate)
- And in the length of 55cm my results are precise but not accurate. And the rest of my results are both accurate and precise and in the part evaluation I will consider them in details and I will figure out why these invalid results are obtained but for now we know enough that why we need to draw error boxes in our graph.
Conclusion about length:
Graph: Now by the aid of a graph I can consider the relationship between length and square of period (because only the graph of squared period against length will give me a straight line) and then if they lie in a straight line then I can say that my experiment does approve the Galileo’s formula. But as we saw in earlier part there might be some errors that we need to take them into account while drawing our graph and to achieve it we need to draw error boxes.
How to draw error box: to draw error box I first need to see what the range off errors in my data for the variables in the x-axis and y axis are. And in doing so we need to find the sensitivity of our instruments and the errors involved at human action then draw the error boxes.
My variable in x-axis is length (independent variable) and the sensitivity of the meter was ±0.05 cm so now we can draw the error bar for length so the length of error bar is 0.1 cm.
My variable in y-axis is period and its uncertainty has to do with two factors.
1. The sensitivity of the stop clock which is ±0.005 s.
2. The errors involved at human action (starting or stopping the clock at the right time) which is going to give me errors as big as I think ±0.01 so the total uncertainty is ±0.005±0.01= ±0.015 but the graph is T² and not only T so we need to multiply the error by 2 so 2 ×±0.015=±0.03 so the length of error bar will be 0.06.
conclusions: as we can see from the graph above the pink line is the theoretical results which are all in a straight line and the blue line which is the line of best fit of my experimental results and as we can see because the line of best fit can be drawn through my error boxes it means that the square of time IS proportional to length and now how do we know if my results are consistent with the Galileo’s equation as well .well as the pink line lies in the area of my error boxes too so it means that my values could have been those.
Mass: according to Galileo’s equation the period is independent to mass of the object but I want to experimentally prove it and to do so I choice 6 different masses with the same length of 55 cm and I started with 50 gram and each time added a weight of mass 50 gram to the pendulum and I got the following results.
The results above shows I have quite good results and I just need to draw them on a scatter graph to investigate the effect of mass on the period and as we can see according to Galileo’s formula mass has no effect on the period and lets see if I can prove that with those results obtained earlier.
Conclusion for mass: This graph might be a bit tricky because the practical values look a bit far from each other but if you look at the scale of the time you will realise that each horizontal line division in graph is 0.002 of a second which is an extremely small fraction of time and if we compare the systematic error which is ±0.03 s with the percentage error of our results which is (1.478 ←1.484→ 1.492 which in average is ±0.007 or 0.7%) it means that if we draw error boxes for the graph the theoretical value is in the range of error box too. from these information I can ascertain that mass has no effect on the period.
Angle: according to Galileo’s theory the angle should not effect the period of a pendulum and now I will practically prove it so in doing so I kept the length constant at 55 cm which is proven to affect the period of a pendulum. And I got the following results:
conclusion for angle: in this case I think there is no use of drawing any graph because my data are considerably out of range and there is no way to draw a straight line trough them even with the aid of error boxes. But as we can see there is a pattern of increasing in the period as we increase the angle which shouldn’t be so according to Galileo’s equation but the reason I think we have an increase is because of the air resistance which is a random error and I think happens because when we increase the angle the height of the pendulum is increased too so it gains potential energy so when it reaches the vertical position its speed is greater than it is for smaller angles so because of the high speed viscous drag is great so it slows it down again and as it should cover a longer distance with the same speed the period of the pendulum increases as we increase the angle. And to see how much viscosity can affect my results I modified my experiment so that I can do the experiment in water (which is more viscous than air).
Modification: I did the same experiment but I filled a sink with water and I put the pendulum in water so that it can swing in water but I tried to not to let the string to be in water too because the resistance opposing the motion should be only considered on the pendulum and not the string. And I did the same experiment for length and following results were obtained:
Now as we calculated the error bars and hence draw an error box for our length scatter graph earlier we can use the same values here s my graph will look like the one below:
Conclusion for viscosity :As we can see from the graph we can draw a straight line through our data (if the error boxes are drawn) which again means that even in water the square of the period is proportional to the length of the thread but if we look at the graph the line drawn through the theoretical values(Galileo’s formula) is steeper than the line for the experimental value and hence what we can conclude from that is that the period for a pendulum with the same length in water is shorter than that of air (or vacuum) but why and does it help me to evaluate my results for different angle and explain why I got those anomalous results for different angles(because according to Galileo’s formula changing the angle has no effect on the period) :
- As we increases the viscosity the pendulum being released from a angle will loose more of its kinetic energy to over come viscous drag and upthrust force so when the pendulum is perpendicular to horizontal it has less kinetic energy than it has In air so the pendulum can not again swing up as much as it did in air so it travel a shorter distance and in each swing the distance to travel for one period gets very relatively shorter so the pendulum travel one swing very quickly.
- Second factor is the speed of the pendulum because the pendulum moves very slowly in water which gives an increase to the period but I think the first effect well outweighs the second effect so the period gets shorter so the gradient of the line for period of pendulum in water is less.
Evaluation for my angle results: as we saw the viscosity does affect our results but why the periods for my angle alternation got more as I increased the angle I think is that as we increased the angle the distance for pendulum the complete got more but because the speed of the pendulum gets fast as well these two factors are cancelled out but when we do he experiment in a non vacuum space the viscosity of air decreases the speed of the pendulum and that is why we got bigger results as we increased the angle of release.
Evaluation: throughout the experiment there are some factors that make my experiment inaccurate and invalid from which I can name of: the sensitivity of ruler or the sensitivity of stop clock or how accurate and on the spot I start the clock and I stop it. Some of these errors could be minimized but there are some that we have no control over like the air resistance or be very specific viscosity. So errors could be divided into two major categories:
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Random errors: are those errors that happen and we can’t control it but we can minimise for example:
- Air current: as we saw air current can effect the period of the pendulum by slowing it down when the pendulum is going towards the position where kinetic energy is maximum i.e. when pendulum is perpendicular to the floor and it can affect it when the pendulum is gaining potential energy and losing kinetic energy. We can’t eliminate this air current unless we do it in the vacuum but we can at least make sure that there is no wind that can give pendulum some extra speed or slow down.
- preventing any circular motion of the pendulum (before I talk about this part I should say that these error could be considered as both random error and systematic error): when we are swinging the pendulum there might be some circular motion instead of an exact linear, and to prevent this of happening we have to make sure that we are swinging the ball as straight as possible.
- The friction between string and where it is hanged to is unavoidable and it could give us some error as well.
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it is impossible that human brain can process so accurate that can stop the clock when the pendulum is at its pick so I thought of a new way of doing the timing but I dint have time to implement it but I will just illustrate what that was: if I had the opportunity to do the experiment again I will put a device like in the opposite figure under the pendulum and then I will release the pendulum for example from right had side and as soon as the pendulum is vertical or the needle is exactly behind the ball I would start the clock and one period is when the ball goes to left then right and then again is back to vertical position I think this is much easier for us human to measure.
Actually random errors affect the precision of the experiment because random errors are quoted in the form of a fractional or in a percentage for example if we have three results for the length 25 cm such as: 10.06s, 10.02s and 10.00s we can find the fractional errors by averaging the three results (which is 10.03) then see what is the range of changes and that is ±0.03 (the precision) and the percentage error is ±0.30% which I think is a really small error in physical context because sometimes in physics we can have errors as big as 100 % in astronomical researches. So in general a precise value is achieved when random errors are small.
Systematic errors: Systematic error can be caused by an imperfection in the equipment being used or from mistakes the individual makes while taking the measurement. E.g. A balance incorrectly calibrated would result in a systematic error. An accurate value is achieved when systematic errors are small.
My most favourable results were the length of pendulum results. These were very good as they had a high degree of accuracy. The graph that I produced that compares my results to the theoretical results proves this. The difference between them was minimal. This shows that my methods were highly accurate in achieving those reults. However, there could have been a degree of chance in getting the timings so accurate. I did not expect these results to be so accurate.
I did not like my angle of release results or my mass results. These were not as expected, but this may have been down to timing inaccuracies or in the case of angle the viscosity. The way in which the angle of release experiments changed were justifyable because I could hypothesise as to how the results came to do that by doing an extra experiment (modification) . In the end, I believe that I came up with a fairly good theory for the way in which the results behaved. I do not believe that it is possible to expalin the mass experiments’ results. These were anomalous because of the way that they changed so much with no particular pattern. I suspect that as the other experiments were faily accurate in their timing, it could be due to the method of the experiment itself - perhaps it was not a fair test. My end conclusion did not correspond with the actual results but did correspond with my hypothesis. Therefore, I have no evidence to back up my conclusion, even though I believe it is the right conclusion.