# Astronauts on a tether

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.?

• For some reason thinking of actual astronauts really confused me here. So here is what helped me: Imagine two rocket-powered space ships connected by a tether: if both are pulling the tether with the same force by boosting with their engines, which way are they facing and which way are they boosting? Are they going to move anywhere? Now if one is tugging the other, and second is not doing anything, what would happen? What would be the total force? Commented May 16, 2016 at 1:53
• @IlyaLapan The thing is that the astronauts cannot eject anything to move (e.g. rocket fuel), which is why the astronaut question is different. Commented May 16, 2016 at 1:54
• @AMACB Nevermind, that is not the same thing. You are absolutely correct. Commented May 16, 2016 at 2:02
• I think this question is poorly phrased. Commented May 20, 2016 at 20:39

Let me start out by saying this: subtracting the forces is certainly wrong. I have no idea why they would get the answer 20-8=12, other than wanting to subtract 20 and 8 and looking for an excuse (not exactly likely). A quick though experiment confirms what I say. If you extend their logic, then if both astronauts pulled at 20 lbs of force, then they wouldn't move at all, which is certainly wrong.

I propose looking from the point of view of a point in the center of the rope. From the perspective of that point, astronaut 1 pulls accelerates himself with 20 lbs of force towards the point. From the perspective of the point, astronaut 2 accelerates himself with 8 lbs of force towards the point. Therefore, they head towards each other with 28lbs of force.

Please explain down votes, so I can edit the post to fit the criticisms.

• I don't get why this answer was downvoted either. I agree that this seems strange that if both of astronauts were pulling towards each other with the same force, then by the same logic they should not move at all. Commented May 16, 2016 at 2:07
• You can't add two forces applied to two different objects as if they were applied to the same object. Commented May 16, 2016 at 21:10
• You don't have to. You simply have to shift the frame of reference to the center of the rope. By the way, I am making the assumption that the rope does not become slack as they approach each other, meaning they must carefully feed it behind them, or something of the sort. Commented May 17, 2016 at 0:32
• And I am not really adding them as if it is one object; I am adding them to find the force bringing them towards each other. Commented May 17, 2016 at 0:33

If Astronaut A were pulling with 20 lb of force then Astronaut B should be feeling 20 lb of force if he were just holding the line. If he is only "pulling" or "feeling" with 8 lb of force ,then he's actually letting the line slip through his gloves while using sliding friction to maintain 8 lb. of force. An insidiously tricky question if you ask me (and I'm not sure this is makes physical sense.) The tether should have a single vector of "force" or tension transmitted or shared in both directions.

• the astronaughts are likely harnessed in, so disregard what you address in you answer. Think of it like this: each astronaut is pulled by whatever force the other one pulls with by his harness, not his glove. Then, he adds on to the acceleration by pulling a bit with his arm. Commented May 16, 2016 at 0:47
• @DWin Yeah, what DevilApple said is what I mean. Commented May 16, 2016 at 1:52
• And also, the tether cannot extend past a certain point, but can retract (e.g. fold on itself). If this were a metal rod, then it would be a single vector of force, but it's not. Commented May 16, 2016 at 1:57
• Well, we do agree this question is probably nonsense. But I think his answer is wrong, ( but I'm not the downvoter, I just don't think it makes sense to say "they head towards each other with 28 lbs of force" .)
– DWin
Commented May 16, 2016 at 1:58
• @DWin Yes, it's completely true that my units are totally screwed up. You can't really "head towards each other" with force. Wrong unit. But what I mean is that the force pulling the towards each other must be the cumulative force of the two pulling. Commented May 17, 2016 at 0:38

Consider the frame of reference of the rope:

Astronaut A pulls on the rope with $20 \text{ lb-f}$, so by Newton's Third Law, the rope pulls on Astronaut A with the same force. There are no other forces acting on Astronaut A. Therefore, Astronaut A will accelerate along the rope with an acceleration appropriate to this force and his mass.

Similarly, Astronaut B pulls on the rope with $8 \text{ lb-f}$, so by Newton's Third Law, the rope pulls on Astronaut B with the same force. There are no other forces acting on Astronaut B. Therefore, Astronaut B will accelerate along the rope with an acceleration appropriate to this force and her mass.

The rope itself will accelerate in the direction of Astronaut A, with an acceleration appropriate to its mass and a force of $12 \text{ lb-f}$

• This doesn't really answer the question. The questions asks what force pulls the astronauts together. Commented May 17, 2016 at 0:34
• The question is: " What happens with to (sic) the astronauts?" . This answer explicitly answers that question... Commented May 18, 2016 at 2:15
• Yes, but it doesn't fit the multiple choice answers. All of them were about the force between the astronauts. Commented May 18, 2016 at 14:56

The existing answers are good, but i try to present an answer that encompasses all the concepts together.

Step 1: Consider the astronaut B and the rope to be a system, and the astronaut A to be the other system. The astronaut A applies a force of $$20 \text{ lb-f}$$ on the other system, and hence, by Newton's Third Law, the other system applies a force of $$20 \text{ lb-f}$$ on the astronaut A system, towards the center of the rope.

Step 2: Applying similar reasoning to astronaut B, we find that the astronaut B experiences a force $$8 \text{ lb-f}$$ towards the center of the rope.

Bonus Step 3: Consider the rope to be a system. It has two forces acting on it, $$20 \text{ lb-f}$$ towards astronaut A and $$8 \text{ lb-f}$$ towards astronaut B. It experiences a net force of $$12 \text{ lb-f}$$ towards astronaut A.

## Physical Interpretation

The astronaut A would see the astronaut B coming towards him, and the rope collecting behind him. On the other hand, astronaut B would see the astronaut A coming towards him, but the rope would be slipping out of his hands, towards A. The tugging force by the rope at the tethering point on the astronaut B (along with a little friction at the rope) would be the force actually pulling the astronaut B.

Also, in reality, no person can apply a continous constant force, but like a boatman, has to pull at intervals. So the rope can momentarily collect even near B, but would be soon pulled back by the greater force by A.

Finally, the (hypothetical) answer to the science test question:

(C) Astronaut A is pulled by 20 lbs. force and astronaut B is pulled by 8 lbs. force, both towards each other

• How do you reconcile "tethered together" from the OP and "slipping out of his hands" from your answer? Commented May 18, 2016 at 2:20
• @DJohnM Thank you for pointing that out! It had slipped my notice. Only the physical interpretation changes a little then. The friction force by the rope (very small now), and the tugging force by it at the tethering point on B will pull the astronaut. The rope will now collect only near A, not near B as i said earlier. I will edit the answer to reflect this. Commented May 18, 2016 at 4:35

Let's suppose first that the astronauts are tethered by a rope to their suits, and the rope is tight, in that initially there is no slack. Put another way, the length of the rope is the distance between them.

They both grab the rope and pull with their respective forces and keep holding those points on the rope.

Now astronaut A pulls with 20lbs of force so he moves with 20 pounds of force toward B and the rope (with some slack on his side) towards A with the same force.

Now B pulls with 8lbs of force so the same applies as above, some slack, but now the rope has a net force force of 12lbs toward A and so moves toward A.

But B is holding the rope so the tension in the rope, a net 12lbs also pulls B toward A.

So B moves toward A with 20lbs of force (8+12), A moves toward B with 20lbs of force.

Does that make sense?

If by 'pulled together' it means the net force on the rope, then you get 12lbs as your answer.