Circle chord required, but an odd twist?

Question

Circle, 12 feet diameter.
Pins sticking out of the circle, 0.3 inches.
Imagine the the chord line as secure base for the pins (Therefore pins are longer towards the middle but still only protrude 0.3").
The pins can vary to a maximum of 3.5 degrees off tangent.
What is the widest chord possible given the above?
If all the pins were tangental, the pins could not be withdrawn inwards from the circle. This is the key part, the pins which are attached to the chord must be able to move inwards, hence using the 3.5 degree allowance to keep the pins at 90 degrees to chord.

Apols.....Pins are 50 thou 0.050" diameter.

Answer 1

Sorry, but I do not understand your key part about pins must be able to move inward, hence keeping pins perpendicular to the chord. If the pins are always perpendicular to the chord, then they are all parallel to each other. Also not sure what you mean by off tangent. What tangent?

A diagram is needed here to netter explain the problem.