# [Physics] Astronauts on a tether

forcesfree-body-diagramnewtonian-mechanicsstring

Recently, I went to a competition with a science test. One of the questions was as follows:

Two astronauts (A and B) are tethered together in space. Simultaneously, Astronaut A pulls on the tether with 20 lbs. of force, and Astronaut B pulls on the tether with 8 lbs. of force. What happens with to the astronauts?

A. Nothing.
B. They are pulled together with 28 lbs. of force.
C. They are pulled together with 12 lbs. of force.
D. (Can't remember what this answer was)
E. Cannot determine without the astronauts' masses.

Apparently, the correct answer is apparently C, but I kept getting 32 lbs. towards each other:

There are four forces to consider.

1 <--|--> 2   3 <--|--> 4
A-|-------------|-B


Forces 1 and 2 are resulted from A's tug and the opposite reaction to his tug; same for 3 and 4 but for B's tug. Therefore, we have the following forces:

20-->|<--20   8 -->|<-- 8
A-|-------------|-B

|____|_____________|____|
X        Y        Z


Segment $X$ goes to the right with 20 lbs. of force, segment $Y$ goes to the left with 12 lbs. of force, and segment $Z$ goes to the left with 8 lbs. of force. So we have:

20-->|    <-- 12   |<-- 8
A-|-------------|-B

|____|_____________|____|
X        Y        Z


So the force in segment $Y$ "overrides" the force in segment $Z$. So we have:

20-->|    <-- 12
A-|---------------B

|____|__________________|
X          YZ


I thus conclude that they are pulled together with 32 lbs. of force.

a. What did I do wrong in my solution?
b. How do I get 12 lbs.?