2D platform anchored in center, rotate only

Are there any joints or special physic objects I can use to create a platform that acts like it has a peg through the center, so it is only allowed to rotate on the Z axis, but cannot move (location)?

I tried just not allowing the position to change with this script:

Vector3 StartPos;

void Start () {
StartPos = transform.position;

void Update () {
transform.position = StartPos;

However, that throws off the physics – the way it interacts with the rest of the world and other objects. For example, if another object lands in the center of this object, it will slowly “pass through” the object, which just looks wrong/impossible. I imagine it is because the physics engine is actually trying to move my “fixed” object down, due to the weight of the other object landing on it… but then I’m forcing it back up, but in a way the physics engine doesn’t recalculate for.

Hopefully I’m just missing something simple.

Sounds like a HingeJoint2D to me.