Find the Mass of the Candy Knowing the Total Mass of the Bags

[mrhood]

[Mentors' note: There's no template because this thread was moved from General Physics. The specification of the problem seems adequate without the template]

Hi everyone,
I am working through stuff based on the Millikan Oil Drop experiment. There is a question that asks to determine the mass of a piece of candy in a paper bag. I know the mass of 4 bags with candy and one bag being empty.
Empty Bag: 10g
Bag 1: 38g
Bag 2: 90g
Bag 3: 62g
Bag 4: 30g
I would like some guidance on where to start the process work into this question.
Thanks!

 

[phinds]

Check out Diophantine equations

 

[Nugatory]

A good first step might be think about the amount of candy in each bag.
There are some assumptions that you'll have to make to solve this problem. What did Millikan assume about electrons going into his experiment?

 

[haruspex]

It might help to consider this...
Suppose you have found a mass m that satisfies the given information. What other masses would then also satisfy it? What does that tell you about an assumption you need to make regarding m?

In this candy question, you can assume all the measurements are exact. In Millikan's experiment you would need to allow for error ranges, making it rather trickier.

 

[mrhood]

I am trying to read into the assumptions Millikan made prior to the experiment regarding electrons, but can't find it. As far as Masses that work; 1g, 2g, and 4g. As far as finding the greatest mass the single candy could be, the greatest divisor would have to be contained in the smallest total mass value of candy. In this case 20g (after taking 10g off of the 30g), would be the smallest value.

This would result in a maximum per candy of 4g but I still don't understand what assumptions I should be making to rule out values, now that I know it has to be one of the above stated masses.

 

[jbriggs444]

It is a pretty small data set from which to draw reliable conclusions. And it is questionable practice to be making probability judgements after the experimental results have already been obtained but...

Suppose that the candy size was 1g. How likely would it be to find that all four of a set of randomly selected samples had a number of pieces that was a multiple of four? If 2g, how likely that all four had an even number of pieces?

 

[mrhood]

Neither of those are very likely, especially comparing to the probability of it being 4g. If 4g, then there are both even and odd amounts of candy in different bags , and it contains no higher divisors. I just dislike using probability to answer a 'determine' question, also without using assumptions.

 

[haruspex]

I am trying to read into the assumptions Millikan made prior to the experiment regarding electrons, but can't find it. As far as Masses that work; 1g, 2g, and 4g. As far as finding the greatest mass the single candy could be, the greatest divisor would have to be contained in the smallest total mass value of candy. In this case 20g (after taking 10g off of the 30g), would be the smallest value.

This would result in a maximum per candy of 4g but I still don't understand what assumptions I should be making to rule out values, now that I know it has to be one of the above stated masses.

There is no reason it should be a whole number of grams. Could be 4g/n for any positive integer n.

Neither of those are very likely, especially comparing to the probability of it being 4g. If 4g, then there are both even and odd amounts of candy in different bags , and it contains no higher divisors. I just dislike using probability to answer a 'determine' question, also without using assumptions.

As I noted in post #4, in the real world you would have to allow for inexact measurements. This makes a probabilistic approach the only option.
You could set the measure of error as the sum of squares of differences from the nearest whole multiple of the candidate candy mass, but you would have to add a cost function to bias against small masses. Otherwise, you would end up choosing an infinitesimal mass as the best fit.
So here's an interesting question: what is a suitable cost function?

 

[mrhood]

Using the empty bag of 10g, I can make this cost function:

total mass = 10g + (x)m
so,
38 = 10g + (x)m
90 = 10g + (x)m
62 = 10g + (x)m
30 = 10g + (x)m

where m is the mass , 4g/n, for any positive integer n. And x is the number of additional units.

 

[haruspex]

Using the empty bag of 10g, I can make this cost function:

total mass = 10g + (x)m
so,
38 = 10g + (x)m
90 = 10g + (x)m
62 = 10g + (x)m
30 = 10g + (x)m

where m is the mass , 4g/n, for any positive integer n. And x is the number of additional units.

No, you misunderstand what I mean by a cost function here.
Suppose in general you have measurements x₁,..,x_r which you wish to explain as multiples of x.
Define the error function as E(x)=∑_imin_ni{(x_i-n_ix)²} but add some cost function c(x) which tends to infinity as x tends to zero.
The task is then to minimise E(x)+c(x) wrt x.
What would work well as a cost function? 1/x would make no sense dimensionally. Maybe (var({x_i})/x)²?

In the present case, x=4g gives E(x)=0. But what if we had measured bag 2 as 91g? Would it make more sense to choose x=4g or x=1g?

Edit:
On further thoughts, a multiplicative cost function might make more sense, but merely 1/x² wouldn't cut it. That wouid still lead to infinitesimal x.
Judging from Hurwitz' theorem, might need 1/x⁴.

 

[mrhood]

I'm not going to lie, I have no idea what any of this means. All I know is that I was asked to find a method to figure out the single candy mass of it. Doing Calculus next semester so don't have too much into mathematics.

 

[haruspex]

I'm not going to lie, I have no idea what any of this means. All I know is that I was asked to find a method to figure out the single candy mass of it. Doing Calculus next semester so don't have too much into mathematics.

Fair enough. I was just explaining why it is appropriate to use probability in making deductions from such an experiment.