A sheet of metal with a circular hole in the middle is heated uniformly. Does the size of the hole increase, decrease, or stay the same?
It increases. I'm not too happy with my explanations though.
Explanation 1:
As the neighbouring molecules in the sheet are being heated up, they start to occupy more volume of space, forcing an increase in the distance between them by a certain amount directly related to the temperature. Any pair of neighbouring molecules will produce the same result, regardless of where they are in the sheet. One can plot a path of pairs of molecules, each pair neighbouring the next, across the sheet to connect any two points therein. By adding the distance increase(either by plotting it all on a cartesian reference plane and adding each pair's contribution to the distance increase along bothe x and y axes, or by relying on simple vector addition techniques known from secondary-ish school mathematics) one can find out that regardless of the shape of the path plotted, and whether or not the path encompassess any holes of whatever shape, the distance between any arbitrarily chosen starting and ending points(molecules) in the path always increases, and in direct, linear proportion to its starting separation distance.
For example, imagine that this is a metal rod, with a gate-like bend in the middle:
__|-|__
It's easy to see that the gap at the bottom of the gate will expand accordingly with the rest of the rod's expansion. It can be extrapolated to any shape and size, as long as there is an unbroken path connecting all the molecules in the material, and that the material is otherwise rigid.
Explanation 2:
Imagine a one molecule-thick ring of metal, corresponding to the rim of the hole in the sheet. It's circumference is πd(d=diameter). It is made up of N molecules, with each molecule occupying L amount of space alongside the lenght of the rim. So, πd
0 = N*L. After heating the ring up enough, the molecules now occupy 2L space, so πd
1 = N*2L; After simple substitution, we find out that: d
1 = 2d
0, that is, the diameter must have increased, to accomodate the more voluminous molecules.
We can extrapolate this to any number of additional rings of material, each lying Δd farther than the previous one, where Δd is the distance to the next ring related to the size of a molecule (L) albeit measured in the radial direction:
π(d
0+Δd) = N
2*L;
πd
1 = N
2*2L
d
1 = 2(d
0+Δd)
to find out that each one of them increases it's after-heating-diameter by the same amount as the previous one plus 2*Δd, which corresponds to the increase in volume of a molecule measured radially, thus freeing space for the inner rings and in no way obstructing the expansion.
A rock that is significantly more dense than water is frozen in an ice cube such that the ice cube plus rock is significantly less dense than water. The ice cube is placed in a glass of water, and over time the ice melts. What happens to the water level in the glass? (you do not need to give an exact answer, a rough one will do)
This had been discussed earlier(see page three of this thread). I remember I had an argument with G-Flex about this.
edit: typos, so many typos. I was slightly tipsy when writing this, if this can somwehow absolve me.
Author
Topic: The questions, riddles and puzzles thread (Read 37371 times)