2009-07-20

Marble Problem 3

Sture has finally got tired of playing with marbles, but Bengt continues with the fervor of a true scientist. He is still building pyramids, of four marbles each. But the floor can be a little slippery sometimes, and then the pyramids won't stick together. Bengt intends to do something about that. One of his ideas is to give the marbles electrical charge, in order to make them stick to each other.
The marbles have diameter 1 cm and mass 1 g. They are not conductive, so they do not lose their charge when touching each other or the floor.

  1. An easier way to get marbles to stick together is to use glue. Suppose each marble exerts a glue-force on each marble that it touches. How big must the glue-force be to hold the pyramids up, if there is no friction at all? (That is, neither between a marble and the floor, nor between a marble and another marble.)
  2. Forget the glue, and let's investigate the friction a little more. What happens if there is a large friction between a marble and the floor, but no friction between a marble and another marble?
  3. And what if it's the other way around - large friction between a marble and another marble, but no friction between a marble and the floor?
  4. Suppose the friction against the floor is very large. How big must the the marble-marble friction coefficient be to keep the pyramid up?
  5. Suppose both the frictions are the same. How big must the friction coefficient be to keep the pyramid up?
  6. Bengt also wants to build the ball-shape mentioned in Marble Problem 1: One central marble, surrounded by as many other marbles as possible, all touching the central marble. With no friction, what glue-force is needed to keep this arrangement standing?
  7. Without glue, could enough friction keep it up? Assume very large friction against the floor. How big must the the marble-marble friction coefficient be?
  8. Now suppose we want to do something even more extreme: Let there be a force between two marbles, which is just enough to overcome gravity, so that Bengt can hold one marble and let the other hang from it. Clearly this is possible with a glue force. But could this force be described by a friction coefficient, if it is extremely large?
  9. Let's try with those electric charges. Two marbles have equal and opposite charge, enough to hang one marble from the other. How big must the charge be?
  10. What if you have three marbles hanging like a chain, the one in the middle being positive, and the others having equal but negative charge?
  11. What if you have n marbles in a chain, with equal and alternating charges?
  12. Returning to the case with three marbles - is there a better way to distribute the charges? Find the setup such that the sum of the absolute values of the charges is as small as possible.
  13. And for n.
  14. We can also use charges to get the pyramid to stay up. If the top marble has positive charge, and the bottom ones have negative charge of the same magnitude, will that help the pyramid stay together, or will it actually force it apart?
  15. Suppose instead that the charge of the top one can be larger than the others. What should the charges be to get the pyramid to stay up? Assume there is large floor friction but no marble-marble friction, and minimise the largest charge magnitude.
  16. Even more freedom: Arrange the charges, magnitude and polarity, any way you like, and again minimise the largest magnitude.
  17. And what if there is no friction at all?
  18. What about the ball shape? If there is more than one possible shape, which one is better? Assume that there is large floor friction but no marble-marble friction. Also assume for convenience that the middle marble is positively charged. To get the best result (that is, minimise the largest magnitude), which marbles should be positive and which should be negative?
  19. And what would those charges be?
  20. And what if there is no friction at all?
  21. There is yet another, even more unrealistic way to get the marbles to stick together. What if they were really heavy, so that gravity kept them together? Consider the case of one marble hanging from another. If they have the same mass, how heavy must they be?
  22. And what about the pyramid? Assuming large floor friction but no marble-marble friction, and same mass for all marbles.
  23. And what if there is no friction at all?
  24. For completeness: The ball shape. Large floor friction but no marble-marble friction.
  25. And what if there is no friction at all?
  26. Bengt decides to finish off the alphabet by putting u and i together. Take one of the marbles from question u, charge it up, and hang it from a normal 1 g marble like in question i. How big would the charge have to be? And how big, roughly, is the theoretically biggest positive charge 1 g of normal material can have (that is, with every electron removed)?