Not entirely clear how a hollow square can be a cylinder so I've done this as per your diagram.
All are 3D pieces and all are 'Rigid Bodies'.
The T-piece is suspended from a "Rgid Body Constraint" set to "FIXED". That's positioned well above.
I see now I'm back online that your T-piece isn't moving, but hinged. That is just a matter of lowering the constraint Empty down close to the T. You could also try with the constraint set to "Hinge".
The red outer container (Cylinder/box/whatever) is a passive rigid body.
The brown 'Pusher' is also passive but ticked as 'Animated' and keyframed to move left to right.
Addendum 8th March, 2021
A 2nd Blend file has been added below, this one without the Rigid Body Constraint. That allows the T-piece to sit on the box/cylinder and react to surfaces it comes into contact with. In an extreme case where the pusher moves excessively, it will, as in the real world, actually force the T up far enough to get underneath it.
With a normal stroke and without the RB Constraint to hinge the T piece, it is able to react with collision surfaces in the manner you're seeking and still in similar fashion to the movie clip above.
You'll notice the T-piece's physics 'Shape' has been changed to mesh and it's Collision Margin changed to .01 .
1st Blend file -
2nd Blend file -