Transient Experiments

Measuring the angular velocity of  a ceiling fan through probability .


What i am about to describe is a kind of  Monte Carlo simulation  , Monte Carlo simulation is a method which solves a problem through random number generation and measuring what conditions those generated number fulfill to reach the solution/answer  instead of solving it analytically . For example  lets solve this question through a Monte Carlo simulation .

Q . Three friends are playing a game of dice where each of them throws a die and  guesses a number between [3,18]  before the throw .  A person wins when his guess is equal to the sum of the numbers on 3 dice , and if no one is the winner we throw again .
What number will you guess ?

Let's do a Monte Carlo simulation ie we actually generate 3 random numbers between 1 and 6 and sum them to see which number we get , we do this 10 million times to see which sum is the most likely to appear
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import random
arr=[0]*19
for i in range(10000000):
    a=random.randint(1,6)
    b=random.randint(1,6)
    c=random.randint(1,6)
    arr[a+b+c]+=1
print("SUM  OCCURENCE")
for i in range(3,19): print(i ," = " ,arr[i])


Result :


SUM  OCCURENCE
3  =  46394
4  =  138715
5  =  278132
6  =  462384
7  =  694493
8  =  971408
9  =  1158053
10  =  1249968
11  =  1250244
12  =  1157254
13  =  971908
14  =  694466
15  =  462672
16  =  278549
17  =  138588
18  =  46772


Where you can clearly see that sum 10 and 11 are most likely and after that 9 and 12 are second most likely
so you should bet on 10 or 11 and if those are already taken you should go for 9 and 12 .


This problem is easy it can be solved analytically because it is easy to see that there are 27 combination which produce sums 10 and 11  whereas there are 25 combination that produce 9 and 12 and this goes on decreasing ans we move on to 8 and 13 and so on so 10 or 11 is your best bet .

But analytical solution is not always easy for example Monte Carlo  provides an easy method to approximate Pi  [1][2] whereas the mathematical way to derive it is slightly tricky .

So I was thinking to apply this simulation somewhere in real life to get some real result (useless they may be) , I love physics so i wanted to combine the two .

How to measure the speed of the ceiling fan through this method ? or it can be restated as how can you measure the speed of a fan by throwing something ( Be cautious )  near the edge of the fan ?
You can easily calculate the probability of something hitting the edge of a stationary fan (randomly)
for ease of calculation lets say radius of the fan is 'R'  and fan has 3 wings lets say each , edge of which subtend a 20 degree length on the center .

If we assume the fan to be stopped (at random places ) then the probability of hitting it will be (20*3)/360
but if the fan is moving say with speed of ω and object remains in the plane of the fan (the vertical position in which object can hit the fan ) for time T seconds so ω = [ change in θ ] or dθ  divided by time[ T ] now the fan will behave as if each of its wings are = 20 + dθ instead of being just 20 hence making a total wing area of 3*(20+dθ)

probability of hitting the fan becomes





where dθ=ω*T  so probability of hitting becomes

we can calculate p(hitting) by the actual throwing  where ω becomes





P(Hitting) is simply the number of hits divided by number of throws . You have to approximate T to get a satisfactory answer and your throws have to be consistent .










[1] http://niallohiggins.com/2007/07/05/monte-carlo-simulation-in-python-1/


2 comments:

  1. An ingenious solution , indeed, but the final expression depends heavily on T. T as far as I know would be the duration of length for which your projectile is in a "Hittable" state so to speak, which would be the Final velocity of projectile [upon reaching the fan] / component of length [of projectile] perpendicular to the fan surface.

    T = V(f) / L (perpendicular).

    It would be terribly difficult to keep these factors constant. Yet the fact that this is a thought experiment negates all practical ill-regard towards it.

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    1. Thanks Neeraj ,i am also trying to solve this problem , if it were something other than a fan we could have dropped the object from 1 meter height and approximated the final velocity that way, but if it has to be thrown above its a different problem . Consider a spring loaded mechanism you can approximate the initial velocity of a throw by this spring mechanism (v = Sqrt(2*g*H) ) this can be used for our throws now !!! Though i am still trying to build this

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