We know the position as a function of time for an object moving in uniform circular motion is given by ~r = rcos (bt)ˆi + rsin (bt)ˆj where r and b are positive constants. we can turn this interesting 2-d motion into an interesting 3-d motion: a helical/spiral trajectory.

a) make a simple modification to the equation provided above such that the
object’s trajectory traces out a helix of fixed radius that is normal/perpendicular
to the xy−plane. justify your modification in a sentence or so.

b) based on other problems you’ve worked on this set you should be able
to modify the equation so the the helix is no longer of fixed radius, but gets
smaller and smaller or larger and larger. write down a relationship that
does this, justifying anything you add to your answer from a).

c) what would happen if we instead have ~r = rcos (at)ˆi +rsin (bt)ˆj, where
a is a new constant?