A circular hoop of radius $r$ is stood vertically on a flat surface and held in place. A smooth wire is tightly stretched between the upmost point of the hoop and a point on the hoop at height $h$. A bead of mass $m$ is held at rest at the top of this wire, upon which it is threaded. When the bead is released, how long does it take to reach the bottom of the wire? Ignore air resistance. Give your answer in terms of $g$, $h$, $m$, and $r$.
A picture of the arrangement is shown below:
Here's a picture of my working:
I'll summarise the steps for clarity. First, I calculated the distance which the bead will travel in terms of $h$ and $r$, using some Pythagoras. Then, I calculated $cos(θ)$ in terms of $h$ and $r$ so that I could then resolve the acceleration $g$ into the component parallel to the wire, $gcos(θ)$. Finally, I applied the SUVAT equation $s=ut+\frac{1}{2}at^2$ and did some rearrangement to get an expression for $t$.
However, this answer is incorrect but I'm not sure what I've done wrong.
You're lucky I like investigating strange statements given without context such as "this answer is incorrect." I eventually found that particular problem on the website you mentioned and played with it long enough to find the issue.
In short, your answer is correct...
... but it's not simplified enough for that website to accept it. Specifically,
$$\frac{8r^2-4hr}{2gr-gh} = \frac{4r(2r-h)}{g(2r-h)} = \frac{4r}{g}$$ as long as $2r \neq h$ (if $h=2r$, then the question itself has problems, so I think it's safe to ignore that case)
That is, your answer is equivalent to $\sqrt{4r/g}$, which is accepted.
Is anything surprising about that answer? Where is the $h$?
That said, you do have one error in your work, even though it didn't end up mattering. You have assumed, in your diagram, that $h \leq r$. To be complete, you'd need to also show it works for $r < h < 2r$.
And as a follow-up question: Suppose we have infinitely many smooth wires intersecting at a point and making a plane which is not orthogonal to the gravity vector. Each wire has its own bead, all of which start at the intersection point.
All the beads are released at the same time and allowed to travel for $t$ seconds. Describe the shape formed by the beads at this time.