This week I've picked four problems that touch on a variety of skills and topics.
- The first brings us to Syria where we're asked to determine how far we'd be able to see standing upon one of the modest mountains of Addalia. The result is neat and is a great rule of thumb you can take with you.
- The next problem asks us to find the moment that a cylinder rolling off a table begins to slip. Though this situation might not come up all the time, resolving is takes some very careful work balancing all the dynamical quantities.
- Next, we're introduced to the RNA/DNA hairpin, a common structural motif in molecular biology. At the cellular scale, life occurs under the full force of thermal fluctuations that biology must find ways to counter. When scientists manipulate DNA in order to do things they want, they can use temperature to their advantage and for instance cause double stranded DNA to separate into two strands. Here we consider a simple model of DNA hybridization and ask how many bonds it will keep on average when we raise the temperature very high.
- Finally, we have the simply stated problem of a small ball being driven into a wall by a heavy mass. Each time the ball and mass collide, the mass slows a bit and the ball accelerates. We are asked to propagate this system through 10 collisions and find the conditions such that the block will come to rest at that time. The solution is very beautiful and reveals a deep connection between the problem and the geometry of rotations.
I think this set in particular is quite rewarding.