How Can We Accurately Determine the Number of Small Marbles in Each Bag?

[nothing123]

We had to conduct a similar practice to Millikan's oil drop experiment in class.
This is the situation:

There are 10 bags each containing a different amount of small marbles of the same mass. There is also one big marble of different mass to the small marble added to each bag. Each bag is massed and that mass is given to us. Now through techniques similar to those of Millikan, we must:

1. Determine the mass of one SMALL marble.
2. Determine the number of SMALL marbles in each bag.

Our technique was simply to find the difference between all the bags and thus having 45 differences. This difference represents the net mass. That is, since all the bags contain one large marble, this difference eliminates the mass of the large marble as well as the bag since they were constants of each total mass. This leaves the net mass of just the small marbles in each bag. Moreover the smallest difference between the differences were found. If this difference was divisible into all the other differences as a whole number factor, it was concluded that this was the mass of a single small marble. Now how do we find the number of small marbles in each bag? We don't have either the mass of one big marble nor the mass of the bag. I know there are some serious flaws with this method so feel free suggesting a more sound, error-proof method that may involve mathematical equations and more physics concepts.

Thanks.

 

[nrqed]

We had to conduct a similar practice to Millikan's oil drop experiment in class.
This is the situation:

There are 10 bags each containing a different amount of small marbles of the same mass. There is also one big marble of different mass to the small marble added to each bag. Each bag is massed and that mass is given to us. Now through techniques similar to those of Millikan, we must:

1. Determine the mass of one SMALL marble.
2. Determine the number of SMALL marbles in each bag.

Our technique was simply to find the difference between all the bags and thus having 45 differences. This difference represents the gross mass. That is, since all the bags contain one large marble, this difference eliminates the mass of the large marble as well as the bag since they were constants of each total mass. This leaves the net mass of just the small marbles in each bag. Moreover the smallest difference between the differences were found. If this difference was divisible into all the other differences as a whole number factor, it was concluded that this was the mass of a single small marble.

You can't be completely certain of this, right? What if there are 10 little marbles in one bag, 12 in another, 16 in another, 18 in another, 22 etc...

 

[nothing123]

You can't be completely certain of this, right? What if there are 10 little marbles in one bag, 12 in another, 16 in another, 18 in another, 22 etc...

Yes, that is the basis of the experiment. Given varying values like Millikan and finding a common multiple among them. The teacher knows how many small marbles he put in each bag beforehand. As for our method, it's the best one we can think of so far. Keep finding differences until there is a small single value that practically divides into all the other differences (it is virtually impossible to have a common factor among 45 differences simply because of the measure of error in massing the bags).

 

[Hurkyl]

The problem I think you've identified (and nrqed pointed out) is that the number of small marbles in each bag can have a nontrivial common divisor.

So the question is how likely is that the greatest common divisor of the number of small marbles in each bag is greater than one?

If the number in each bag is "small", I think it's very unlikely. However, if the number is "big", I think it's very likely. (but the common divisor is likely to be big too)

 

[nothing123]

Based purely on inspection, it looked like there was at least 15 small marbles in the lightest bag ascending to almost 200 marbles in the most massive bag. Also note that the mass of each bag ranged from 25g - 450g leading us to believe that the mass of a single small marble would be around 0.5g and we did come up with a number very close to that.

 

[Hurkyl]

That's good then!

I wonder if this is a good application of the maxim "don't compute anything unless you already know the answer"?

 

[JenHeart]

The Decimal Dilemma

We are doing a similar experiment with 16 bags of marbles, where we have to find the mass of an individual marble. This number of bags would give us 120 differences. How can we find the lowest common denominator if it is a decimal?