Control type problem from mechanics

[wrobel]

I saw this problem several years ago in a Russian journal devoted to physics olympiads. This problem was discussed there under some certain simplifying assumptions which I do not remember. I tried to solve it without simplifications but laziness and large formulas stopped me. Nevertheless, I find this problem interesting and solvable and perhaps some of PF participants will be interested in it.



A rod $AB$ of mass $M$ and length $\ell$ can slide without friction along the axes $OX$ and $OY$ by its hinges $B$ and $A$ respectively. A bug $C$ of mass $m$ runs downwards along the rod. Can the bug's run be such that the rod does not move?

My suggestion is as follows.
Let us regard the function $u(t)=|AC|$ as a given function. Then we have a Lagrangian system with one degree of freedom. For a generalized coordinate take an angle $\varphi=OAB$.
Then write down the Lagrange equation and substitute there $\varphi=\mathrm{const}$. We will have an equation for the function $u(t)$.

 

[Baluncore]

A bug C of mass m runs downwards along the rod.

"Downwards" suggests that the rod is being accelerated down the y-axis and along the x-axis by gravity.

Can the bug's run be such that the rod does not move?

Yes, but only for a finite time.

While the force along the rod, due to gravity, is equal to the force applied by the running bug, the rod will remain in equilibrium. The bug will accelerate down the stationary rod, until it reaches B, on the x-axis, then shortly after, it may be crushed by the rod after the rod begins to slide.

The question becomes, can the rod ever catch up with the accelerated bug?

 

[Lnewqban]

By accelerating its small mass (m) along the bar, the muscular force of the bug will need to induce a vertical component equal to (M+m)g.
That would be easier achieved at positions of the bar closer to vertical.