Given a continuous, differentiable convex curve such as a parabola, hyperbola, an ellipse, cycloid, catenary, semicircle, etc (with the domains accordingly restricted),
What is the time taken for a point particle to descend down the curve?
Note: this is a problem that I came up with myself and am wondering about, so it is possible that it is not solvable with the given information. If that is the case, please comment what other details I need to specify to make this solvable.
Assumptions:
- There is static friction
- It is a particle, not a ball, so assume no rolling and hence no rolling resistance/slipping
- Acceleration due to gravity is simply $-9.8ms^{-2}$
- Air resistance is negligible
- Please use any value for the coefficient of friction, or simply leave it as $\mu$, as well as any other quantities that I did not specify here.
Example/explanation
If we place a blue point particle on the curves below (parabola and ellipse),
It may look something like this:
The ball will descend the ramp, then climb to the right side of the curve, then go down, climb the left side... etc until it stops at the minimum point.
Things I tried
I tried using the concept of energy (gravitational potential/kinetic) and work done by friction, but to no avail.
For the case of the parabola, I parametrised it as $x=t, y=t^2$ and then parametrised it with respect to arc length by taking $r' = i + 2tj$ so $|r'|=\sqrt{1+4t^2}$ and $s(t)$ is just given by the integral of this, which Wolfram gives as $$1/4\Big(2x\sqrt{4t^2+1}+sinh^{-1}{(2x)}\Big)$$
However, $x$ here is a function of time i.e. $x(t)$. I wasn't sure how to find this, and even if I did, I wasn't sure how to proceed.