# Rotational motion along a hinge

When a mass moving with some velocity collides with the door and sticks to it after collision, the door gets a torque and starts rotating, keeping the angular momentum constant only along the hinge. What I'm not getting is what kind of forces are acting through the hinge. Are they normal reaction forces? Are they external? Can I say the hinge force is internal if I consider the hinge, door, and the mass to be a system? Can the angular momentum be conserved if the hinge, door, and the mass are considered as a system instead of taking only the door and the particle as the system?

what kind of forces are acting through the hinge. Are they normal reaction forces

Yes. Normal forces hold up the door and hold it in as well, so it doesn't fall off.

Are they external? Can I say the hinge force is internal if I consider the hinge, door, and the mass to be a system?

Yes, then those normal forces i just mentioned that come from the hinge are internal. But the wall holds on to the hinge, so you still have external forces there. You will never get rid of them unless you include the whole earth in the system.

Can the angular momentum be conserved if the hinge, door, and the mass are considered as a system instead of taking only the door and the particle as the system?

Yes, but it doesn't matter. The external forces at the hinge are there and you can't get of them. But the lucky thing about angular momentum is that it is based on torques - which are based on distances from the rotational centre.

So, if you choose the rotation axis to be that hinge, then even though those external hinge normal forces are still present, they cause no torques. Their distances to that rotation centre are zero. Angular momentum conservation then is still usable, because no external torques are acting, even though some external forces are present.

So, in general:

• Linear momentum conservation is usable when there are no external forces, and
• angular momentum conservation is usable when there are no external torques.

1) The hinge applies a centripetal force, which keeps the door from flying off (laterally).

2) By normal reaction force do mean Newton's Law: equal and opposite force? The door pulls on the hinge and the hinge pulls on the door. Equal and opposite.

Conservation laws depend on what you call your system. If your system is everything, then energy, momentum and angular momentum are conserved.

If your system "lives" in a external gravitational field then:

i) Energy is conserved (because the system is time independent), and energy gets exchanged between kinetic and potential energy.

ii) Momentum is not conserved (because it's NOT translation invariant): Potential energy is a function of height. Particles with zero net force (that is, in free fall), do not have constant vertical momentum. Their horizontal momentum is conserved, because the system is invariant under horizontal translations.

iii) Angular momentum is not conserved (because space is not isotropic): there is a preferred direction (up/down). Example: a precessing top.