I'm given a position vector $\overrightarrow r$ which depends on $\theta=\theta(t)$. I found the velocity vector by simply taking the derivative. But then I noticed that the problem also mentions that the particle is subjected to a gravitational acceleration in the $-z$ direction.
Does a position vector already account for this gravity? My guess is yes. If I were to simply put in a value of time, I would find out exactly where the particle is, regardless of anything else.
A small bead of mass m is constrained to move on a helix: $\overrightarrow r(θ) = (R \cos(θ), R \sin(θ), q θ)$ where $R$ and $q$ are constants, and $θ=θ(t)$ describes the position of the bead along the helix at time $t$. The bead is also subjected to a gravitational acceleration $g$ downward ($-z$ direction). Find the following quantities in terms of $θ$ and $dθ/dt$ :
c) potential energy U