Imagine an object defined in cylindrical coordinates whose surface is defined as $r(\theta)$. The "cylinder" is infinite (for purposes of this of problem), so the mass per cross sectional area is a constant $\sigma$. The radius continuously increases from $\theta = 0$, where the radius is $r = r_0 > 0$ to $\theta = 2\pi$ (an involute $r(\theta) = r_0\sqrt{1+\theta^2}$ is an example of a function that meets these requirements). Placing a small object at $\theta = 2\pi$, imagine the object slides down along the surface from $2\pi$ to $0$, under influence of gravity alone. What function $r(\theta)$ would cause the acceleration of the small object to be uniform (constant) from $\theta = 0$ to $\theta = 2\pi$?
- The simple form would be to simply assume gravity pulls towards $(0,0)$, regardless of $r(\theta)$.
- The more complex form would be to use Newton's law of universal gravitation. $F = \frac{Gm_1m_2}{r^2}$.
- An even more complex form would have $\theta$ extend past $2\pi$, to any arbitrary value (i.e. a car on top of a mountain sliding down into a tunnel around and around and around the world) - defined by the road function ($r_1(\theta)$) and a top of tunnel function ($r_2(\theta)$). In this form, the minimum r should be 0, where the two functions would intersect.
I've self nerd sniped on this and need additional help.