How long for a mass to slide down a spring-loaded trap door?
This is a problem I first proposed to a friend of mine in college. He claimed to be able to solve it by setting up a differential equation, but I've never been able to crack it.
Imagine a spring-loaded trap door, mouse-trap style, one for which $\tau = k\theta$: when the door is open, the restorative torque (trying to close the door) exerted by the hinge is proportional to the angle at which it's opened. We'll say the angle can't exceed $\pi$ so that the door can't open past the floor on which it's installed.
The trap door, which is rectangular and physically uniform, has length $\ell$ and mass $M$. It is opened "all the way" and held there, so that $\theta_i = \pi$. A small mass $m$ is placed at the end of the trap door.
With the mass placed at the end of the door, the system is then released so that the trap door starts to close and the mass starts to slide down the increasingly tilted door. 512
(The reason the mass $m$ is "small" is so that it won't hold the door open when it's released.) The coefficient of friction of the mass on the door is given by $\mu$. For simplicity's sake, assume that only kinetic friction acts the whole time, so there is no "lurch" when the object starts moving, and there is no need to distinguish between $\mu_k$ and $\mu_s$.
The question is this. How do we derive a function $L(t)$ which gives the length along the door which the mass has traveled after a given amount of time? I'd like to use this function to derive an expression for how long it takes the object to slide down the incline and be "dumped" into the opening beneath the trap door.
Work thus far:
* I've gone down the obvious roads of setting up the static situations. However, I can only come up with equations that involve $L(t)$ related to $\theta(t)$ in complicated equations, and turning it into a differential equation doesn't make it better; the fact that torque is involved just makes the whole thing blow up because of the product rule.
* I tried solving the simpler situations of a mass sliding down an incline that tilts at a constant rate, and that's achievable. The fact that $\theta(t)$ is so difficult to nail down in this case is the villain.
The real nut of this problem is that there is nothing constant to tie myself to. Even the moment of inertia of the mass rotating about the hinge changes, since its distance from the hinge changes with time. 766