Expected Chord Length Through a Disk (Random Radius Method)

A chord is drawn through a unit disk as follows: pick a point uniformly at random inside the disk, then draw the chord that passes through that point and is perpendicular to the radius at that point. What is the expected length of the chord? Note: this is one of the three classical sampling methods in Bertrand's paradox -- each method gives a different answer. Be explicit about which method you are using.