Measuring Pi

Introduction Joaquin

In this experiment, we wanted to know if we can measure circular objects using basic measuring tools, and by using some sort of carbon trail to identify length to see how they relate with the value of pi. Some of us used powder to compensate for the lack of carbon paper, while some of us used measuring tape to measure the length of how far the object rolled. The point here was to learn about uncertainty and measurements to see how the lengths correlate with the value of pi. We expect to see the values create a slope on a graph that is or close to the value of pi.

Procedure : Ivan

INSERT SKETCH HERE

In the following section, we will cover our procedures and how we collected our data. Our goal was to measure the trace (distance traveled when spinning in a forward direction) of eight circular objects, two per person, in an effort to measure a value of PI.

To collect data for the Penny we marked the coin with a sharpie on the circumference, placed the mark against paper and marked the starting with a pencil. Upon readying the penny, we rolled the object and marked the end point with a black mark against the paper. The roll was in a forward motion, away from the origin. We wanted to account for the scenario where the trace was longer than the entirety of the measuring device. To account for this, we marked the point to which the device reaches with a pencil and moved the device to that point. We then added the measurements at the end of the full trace and established our uncertainties, which included the uncertainty of the measurements, the thickness of a sharpie line, thickness of pencil line, and the slippage on the roll. We repeated these set of a rules to find the trace of a quarter, using an engineering ruler as the measuring device of choice.

The next two objects we measured were one 2.5lb dumbbell and a nickel. We placed a garbage bag on a table and poured flour on top of the garbage bag to mimic the behavior of a carbon paper that would show the trace of the object. We marked a starting point at one end of the table and rolled the object until it stopped. We used a standard measuring tape to measure the trace, this device permitted us to measure the entire trace without needing to separate the measurements—unlike the use of a traditional engineering ruler for the quarter and penny. For the uncertainties of these two objects, we chose to account for coin spillage, the measuring device’s unit capacity (0.125 inches), and the act of stopping the circular object. The only difference in uncertainties for the two objects was due to the dumbbell having more mass than the nickel, allowing it to fall on its own.

For the next two circular objects, we used a disk ion battery and a half dollar coin. We used a similar method of pouring powder over a table to compensate for the carbon paper effect. The measuring device of choice was a digital caliper to measure the diameter of the object and a 16ft pullout measuring tape to measuring the trace of the objects. (INSERT SOMETHING ABOUT UNCERTAINTIES)

The final two objects we measured were the cap of a nuts container and a cylindrical cup (Starbucks cup). Flour was used to simulate the effects of carbon paper, similar to the previous objects. We marked the starting points with a sharpie, rolled the objects, and marked the ending points. The measuring device of choice was a ruler to measure the trace and measure the diameter for each object. The uncertainties that we accounted for these two objects included the thickness and smoothness of the flour, the instant of catching where the circular objects where they stopped, the slippery surface (granite kitchen island), and the ruler not being long enough.

An iPhone was used in the process of recording all eight of our objects being rolled. This is how we collected our data which will be presented in the following section.

Data :Yan

Yan:

	Cap of the Nuts Container Diameter - 12.5cm±.1cm - 0.8% Uncertainty	Starbucks Cup Diameter - 9.1cm±.1cm - 1.1% Uncertainty
Trail #	Length of Trace	# of Turns	Circumference	Deviation	Length of Trace	# of Turns	Circumference	Deviation
1	113.7±0.3cm	3	37.9±0.1cm	1.0 cm	74.1±0.5 cm	3	24.7±0.17 cm	-0.9 cm
2	109.5±0.3 cm	3	36.5±0.1cm	-0.4 cm	74.7±0.5 cm	3	24.9±0.17 cm	-0.7 cm
3	105.6±0.3 cm	3	35.2±0.1 cm	-1.7 cm	76.8±0.5 cm	3	25.6±0.17 cm	0 cm
4	113.1±0.3 cm	3	37.7±0.1 cm	0.8 cm	77.1±0.5 cm	3	25.7±0.17 cm	0.1 cm
5	110.1±0.3 cm	3	36.7±0.1 cm	-0.2 cm	75.6±0.5 cm	3	25.2±0.17 cm	-0.4 cm
6	110.7±0.3 cm	3	36.9±0.1 cm	0 cm	79.2±0.5 cm	3	26.4±0.17 cm	0.8 cm
7	112.5±0.3 cm	3	37.5±0.1 cm	0.6 cm	80.1±0.5 cm	3	26.7±0.17 cm	1.1 cm
			Std. Dev.	0.85cm			Std. Dev.	0.69cm
			Percent Dev.	2.31%			Percent Dev.	2.7%
	Mean Diameter	12.5cm±.1cm	Mean Circumference	36.9±0.1 cm	Mean Diameter	9.1cm±.1cm	Mean Circumference	25.6±0.17 cm

Uncertainty:

· Since I measured both of them on the top of the flour instead of carbon paper, the thickness and smoothness of the flour is one of the uncertainties.

· After roll a complete revolution, the instant of catching where it stops may be different each time

· I used the cap of the nut container and a cup, sometimes it is hard to see where it stops because the marking point on the object is facing to the ground

· I am measuring them on the island in the kitchen, the surface may be slippery after putting a layer of flour on the top. And this may cause the object to slide somehow instead of rolling.

· My ruler is not long enough. Every time when I try to continue with the previous measurement, I may not match exactly to where it was left.

· The reason I choose 0.3 cm as an uncertainty variable for the cap of the nut container is every time I measured the length of a full cycle, it always fell between that range. The reason is I marked the start point with the figure tip. And the thickness of the figure tip mark is around 0.3 cm.

· For the cup, I choose 0.5 cm as an uncertainty variable because on the top of the thickness of finger tip, the thickness of the label “Starbucks Coffee” is about 0.2 cm thick because I used that label as a beginning point to roll.

Daniel:

	Penny Diameter - 1.8cm±.1cm - 5.56% Uncertainty	Quarter Diameter - 2.35±.1cm - 4.26% Uncertainty
Trail #	Length of Trace	# of Turns	Circumference	Deviation	Length of Trace	# of Turns	Circumference	Deviation
1	17.65cm±0.4cm	3	5.88cm±0.13cm	-0.118cm	22.55cm±0.5cm	3	7.51cm±0..16cm	-0.08cm
2	18.0cm±0.5cm	3	6.0cm±0.16cm	+0.002cm	22.75cm±0.6cm	3	7.58cm±0.2cm	-0.01cm
3	12.0cm±0.4cm	2	6.0cm±0.13cm	+0.002cm	22.75cm±0.4cm	3	7.58cm±0.13cm	-0.01cm
4	12.15cm±0.5cm	2	6.08cm±0.16cm	+0.082cm	15.25cm±0.6cm	2	7.62cm±0.2cm	+0.03cm
5	18.0cm±0.4cm	3	6.0cm±0.13cm	+0.002cm	23.00cm±0.7cm	3	7.67cm±0.23cm	+0.08cm
6	17.95cm±0.5cm	3	5.98cm±0.16cm	-0.018cm	22.80cm±0.5cm	3	7.6cm±0.16cm	+0.01cm
7	18.15cm±0.6cm	3	6.05cm±0.2cm	+0.052cm	22.80cm±0.5cm	3	7.6cm±0.16cm	+0.01cm
			Std. Dev.	+0.058cm			Std. Dev.	+0.034cm
			Percent Dev.	+0.96%			Percent Dev.	+0.44%
	Mean Diameter	1.8cm±.1cm	Mean Circumference	6.0cm±.0057cm	Mean Diameter	2.35cm±.1cm	Mean Circumference	7.59cm±.004cm

Uncertainties:

· Penny: The uncertainty comes from coin slippage (I calculated this from estimating the rotational slippage). There was an additional 0.1cm from the ruler on each trail, and 0.2 cm comes from the thickness of the mark made on the penny.

· Quarter: The uncertainty comes from coin slippage (I calculated this from estimating the rotational slippage). There was an additional 0.1cm from the ruler on each trail, and 0.3 cm comes from the thickness of the mark made on the penny.

Ivan:

	Nickel Diameter - 2.1cm±.1cm - ___ Uncertainty	1.25 dumbbell weight Diameter – 12.7±.1cm - __ uncertainty
Trail #	Length of Trace	# of Turns	Circumference	Deviation	Length of Trace	# of Turns	Circumference	Deviation
1	19.71±0.4cm	3	6.57 cm±0.13cm	-0.03cm	80.1±0.5cm	3	40.01 cm±0.17cm	+0.31cm
2	19.86±0.4cm	3	6.62 cm±0.13cm	-0.02cm	81.6±0.5cm	3	39.51cm±0.17cm	-0.19cm
3	19.83±0.4cm	3	6.61 cm±0.13cm	+0.01cm	84.6±0.5cm	3	39.80 cm±0.17cm	+0.1cm
4	19.74±0.4cm	3	6.58 cm±0.13cm	-0.02m	78.3±0.5cm	3	39.92 cm±0.17cm	+0.22cm
5	19.89±0.4cm	3	6.63 cm±0.13cm	+0.03m	83.7±0.5cm	3	38.95 cm±0.17cm	-0.75cm
6	19.77±0.4cm	3	6.59 cm±0.13cm	-0.01cm	82.2±0.5cm	3	39.79 cm±0.17cm	+0.09cm
7	19.8±0.4cm	3	6.6 cm±0.13cm	0.00cm	84.9±0.5cm	3	39.88cm±0.17cm	+0.18cm
			Std. Dev.	0.02cm			Std. Dev.	0.33cm
			Percent Dev.	0.30%			Percent Dev.	0.84%
	Mean Diameter	2.1±0.1cm	Mean Circumference	6.6±0.13cm	Mean Diameter	12.7±015cm	Mean Circumference	39.7±0.17cm

Uncertainties:

For my uncertainties, I accounted for 0.2 cm for coin slippage, similar to Daniel. I used a ruler which accounts for 0.3 cm uncertainty, which goes down to 0.125 inches and I converted this to cm. There is a final uncertainty added of 0.3 cm which is accounting to stop the nickel. For the weight, I used the same logic except it was heavier and there was no slippage and I did not have to stop it because it was much heavier than the nickel and would fall much more easily after three turns.

Joaquin:

	Watch Ion Battery Diameter - 2.00 ±0.01 cm	Half Dollar Coin Diameter - 3.06 ±0.01 cm
Trail #	Length of Trace	# of Turns	Circumference	Deviation	Length of Trace	# of Turns	Circumference	Deviation
1	62.83 ±0.01 cm	10	6.28 ±0.001 cm	0.00cm	76.88 ±0.01 cm	8	9.61 ±0.001 cm	+0.06cm
2	25.28 cm ±0.01 cm	4	6.32 ±0.01 cm	+0.04cm	68.04 cm ±0.01 cm	7	9.72 ±0.01 cm	+0.17cm
3	37.56 cm ±0.01 cm	6	6.26 ±0.01 cm	-0.02cm	46.25 cm ±0.01 cm	5	9.25 ±0.01 cm	-0.3cm
4	18.81 cm ±0.01 cm	3	6.27 ±0.01 cm	-0.01cm	74.48 cm ±0.01 cm	8	9.31 ±0.01 cm	-0.24cm
5	25.2 cm ±0.01 cm	4	6.30 ±0.01 cm	+0.02cm	88.38 cm ±0.01 cm	9	9.82 ±0.01 cm	+0.27cm
6	37.74 cm ±0.01 cm	6	6.29 ±0.01 cm	+0.01cm	38.48 cm ±0.01 cm	4	9.62 ±0.01 cm	+0.07cm
7	18.75 cm ±0.01 cm	3	6.25 ±0.01 cm	-0.03cm	47.85 cn ±0.01 cm	5	9.57 ±0.01 cm	+0.02cm
			Std. Dev.	0.022cm			Std. Dev.	0.192cm
			Percent Dev.	.035%			Percent Dev.	2.01%
	Mean Diameter	2.00 ±0.01 cm	Mean Circumference	6.28 cm±0.01 cm	Mean Diameter	3.06 ±0.01 cm	Mean Circumference	9.55 cm±0.01 cm

Uncertainty:

Uncertainty was calculated by looking at the smallest interval of my measuring device and divided it by 10.

Analysis : Barnabe

Conclusion - dan

During this experiment we were successful in measuring pi via finding the circumference of a round object at comparing it to the measured diameter. The slopes of our graphs show a number averaging within a deviation of ____ to the actual value of pi. Taking in account the known and analysed uncertainties we were well within acceptable boundaries of producing a measured pi value instead of using a known table value. The methods of this experiment ranged from using powered tail traces to manually rolling and marking revolutions on a paper. This range of methods gave us more sets of data to reinforce the values we calculated.