Physics and Astronomy Labs/Radioactive decay with dice

Lede

This graph shows (in pink) the amount of a radioactive sample that remains after several half-lives have passed. After one half-life, half the sample is left; after two half-lives, one half of the remainder (or one quarter) is left; and after three half-lives, one half of that (or one eighth) is left. Note that, in reality, the decay of radioactive elements in a rock sample would not cause any visible change in the appearance of the rock; the splashes of color are shown here for conceptual purposes only.

This will be the Lede. It should mention the application to radioactive dating in OpenStax Astronomy. Remove the header Lede when this has been written.

Images

201 dice were rolled and all the "ones" were removed and counted at each throw. The process was simulated five times for comparison with the actual experiment.

Dice half-life decay

Excel Spreadsheet

Column roll denotes the number of rolls. We rolled 201 dice 18 times, removing the "one" each time. Column exp denotes the number of dice removed on each roll in our experiment. Columns sim1 through sim5 represent five simulations. Column theor represents the theoretical value one is most likely to obtain: the first value is 201 divided by 6, and each consecutive value is 5/6 times smaller, representing exponential decay.

Column SEexp is the square of the error associated with the experiment. For the first role, this is obtained as follows:

$SEexp(1)=(33-33.5)^{2}=0.25$

Columns SE1 through SE5 represent the squared errors associated with the 5 simulations.

These squared errors are summed at the last row. The values of SoS (Sum of Squares) represent a measure of how well the simulation or the experiment matched the theoretical ideal. If the experimental SoS sufficiently exceeded any of the simulated SoS values, we might wish to question the experimental method.

roll	exp	sim1	sim2	sim3	sim4	sim5	theo	SEexp	SE1	SE2	SE3	SE4	SE5
1	33	33	27	29	35	37	33.50	0.25	0.25	42.25	20.25	2.25	12.25
2	25	34	25	28	30	23	27.92	8.51	37.01	8.51	0.01	4.34	24.17
3	21	18	22	27	23	23	23.26	5.13	27.71	1.60	13.96	0.07	0.07
4	18	20	26	24	17	15	19.39	1.92	0.38	43.74	21.28	5.70	19.24
5	17	18	14	21	13	14	16.16	0.71	3.40	4.65	23.47	9.96	4.65
6	14	13	20	13	15	12	13.46	0.29	0.21	42.73	0.21	2.36	2.14
7	11	13	9	15	14	8	11.22	0.05	3.17	4.92	14.30	7.73	10.36
8	13	6	9	12	9	10	9.35	13.33	11.22	0.12	7.03	0.12	0.42
9	12	9	9	6	6	16	7.79	17.72	1.46	1.46	3.21	3.21	67.39
10	9	5	3	3	5	4	6.49	6.29	2.23	12.20	12.20	2.23	6.21
11	5	6	4	4	11	7	5.41	0.17	0.35	1.99	1.99	31.24	2.53
12	4	4	6	6	1	6	4.51	0.26	0.26	2.22	2.22	12.31	2.22
13	2	4	3	5	2	3	3.76	3.09	0.06	0.57	1.54	3.09	0.57
14	1	4	3	1	4	4	3.13	4.54	0.76	0.02	4.54	0.76	0.76
15	2	5	6	3	2	3	2.61	0.37	5.72	11.50	0.15	0.37	0.15
16	1	3	1	1	3	3	2.17	1.38	0.68	1.38	1.38	0.68	0.68
17	1	2	2	0	0	2	1.81	0.66	0.04	0.04	3.28	3.28	0.04
18	0	0	3	1	0	1	1.51	2.28	2.28	2.22	0.26	2.28	0.26
sums	201							67	97	182	131	92	154

Matlab codes

To know if the random error is consistent with the laws of probability for such a decay process, we use matlab codes. These codes simulate any number of labs for an arbitrary number of dice.

Simulates five virtual labs

this code simulates but we graphed in Excel

clear all;close all;clc; Nworms = 5; Nstart = 201; Nstop = 18; data = zeros(Nstop,Nworms); for nworm=1:Nworms ncurrent=Nstart; % initiate throws for count = 1 : Nstop %iterates throws by all the students n2remove=0; for diceCount = 1:ncurrent if rand < 1/6 n2remove=n2remove+1; end % ends if end % finish thowing all the dice ncurrent=ncurrent-n2remove; %remove some dice data(count,nworm)=n2remove; %record number left end end

Simulates and also graphs

this code graphs in matlab

clear all;close all;clc; Nworms = 5; Nstart = 201; Y = [42 32 29 25 11 10 14 7 3 6 3 3 2 1 3 1 1] sizeArray =size(Y); Nstop = sizeArray(2); data = zeros(Nstop,Nworms); for nworm=1:Nworms ncurrent=Nstart; % initiate throws for count = 1 : Nstop %iterates throws by all the students n2remove=0; for diceCount = 1:ncurrent if rand < 1/6 n2remove=n2remove+1; end % ends if end % finish thowing all the dice ncurrent=ncurrent-n2remove; %remove some dice data(count,nworm)=n2remove; %record number left end end for wormcount = 1:Nworms x=[1:1:Nstop]; y=data(:,wormcount); plot(x,y,'r'); hold on; plot(x,Y,'o'); end

Collects data from a data file

This code allows the user to enter classroom data into an Excel file that can be opened by matlab.

matlab code that reads an excel file

clear all;close all;clc; Nworms = 5; Nstart = 201; Y = xlsread('engineersData.xlsx'); %It is necessary to save an Excel file with this name and extension, and %then place into "Current folder". This was done on a click and drag the %only time I have ever tried it. % The data in that file were: % 42 32 29 25 11 10 14 7 3 6 3 3 2 1 3 1 1 sizeArray =size(Y); Nstop = sizeArray(2); data = zeros(Nstop,Nworms); for nworm=1:Nworms ncurrent=Nstart; % initiate throws for count = 1 : Nstop %iterates throws by all the students n2remove=0; for diceCount = 1:ncurrent if rand < 1/6 n2remove=n2remove+1; end % ends if end % finish thowing all the dice ncurrent=ncurrent-n2remove; %remove some dice data(count,nworm)=n2remove; %record number left end end for wormcount = 1:Nworms x=[1:1:Nstop]; y=data(:,wormcount); plot(x,y,'r'); hold on; plot(x,Y,'o'); end

Future project(s)

Matlab codes have been used to make spagetti plots. In the future, it would be nice to also use matlab to perform the sum of squares analysis.This can be of ultimate usefulness because of radioactivity of some elements under the hood.

HTW fall 19

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