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Two bodies A and B of same mass $m$ are attached with a massless spring and are hanging from a ceiling with a massless rope. They are in same vertical plane but not in same horizontal plane. enter image description here

Now the string that connected A with the ceiling is cut and the system is experiencing free fall.

1. Is there any tension in the spring?

My attempt:

Now the whole system should descend with the acceleration $g$ and the body B (and also A) experiences a gravitational pull $mg$. Let the tension in the spring be T.

Therefore, from the free body diagram of B, $mg - T = mg$,ie. $T=0$.

  1. But A also moves downwards, so puts a force on B, how to take account of that? Will there be an relative acceleration between A and B? I am confused about the free body diagrams of A and B.

  2. Will the tension change if the mass of A and B are different?

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There is tension in the spring. It it extended and hence there is tension! It is the centre of mass that falls with acceleration $g$ rather then each individual mass. So the equation $$mg-T=mg$$ is invalid. As the two masses fall they will oscillate (getting closer and further away) and the tension will cycle.

Let us call the distance fallen by mass $A$, $x_A$ and that fallen by mass B $x_B$ the equation of motion for each mass is given by: $$m \ddot x_A=mg+T$$ $$m \ddot x_B=mg-T$$ $T$ is a function of $x_A$ and $x_B$, ($T=k(x_B-x_A-L)$ where $k$ is the spring constant, and $L$ is the natural length) and we cannot assume that $\ddot x_A=g$ or $\ddot x_B=g$. These sorts of equations are called coupled differential equations and can be solved a number of ways.

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The answer is that it depends on how your initial spring loaded mass is moving. But, the fascinating (but not too fascinating once you phrase it like this) part is that until the compression wave from the top interacts with bottom out on the slinky the dynamics of the bottom half won't change.

If we assume it was at rest, essentially the top mass will move fast enough that the center of mass will accelerate at $9.8 m/s^2$. When it reaches the true equilibrium length of the string without gravity it will start accelerating the bottom mass. At this point if you looked from the COM frame the spring would appear to be oscillating like it normally does. This is because the oscillations occurring here is called an eigenfrequency. The other eigenfrequency (since this problem has 2 independent variables) is the motion of the COM. With the motion of the COM and the motion of both masses about the COM you have all the information needed to reconstruct the dynamics of your masses.

A great demonstration of this is the slinky in this video, which is like a spring with an equilibrium length of zero:

https://www.youtube.com/watch?annotation_id=annotation_314765&feature=iv&src_vid=eCMmmEEyOO0&v=uiyMuHuCFo4

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Is there any tension in the spring?

Setting aside the oscillations mentioned by Joseph and Skyler: yes, because of the tidal force. In normal situations this is so slight that it isn't measurable, but it is there. See the plot of gravitational potential on Wikipeda:

enter image description here CC BY-SA 3.0 image by AllenMcC, see Wikipedia Commons

You could derive this by placing optical clocks throughout an equatorial slice of space through and around the Earth, then plotting the clock rates. The slope of the plot or first derivative of potential relates to the force of gravity. Where it's steepest, the force of gravity is greatest. The curvature of the plot relates to the tidal force. Whilst slight, this the second derivative of potential is related to the Riemann curvature tensor and is said to be the defining feature of a gravitational field, because without it your plot can't off the flat and level in the middle. If your plot was all flat and horizontal, you wouldn't have a gravitational field. So whilst slight, the tidal force is there. You wouldn't notice it for your masses and spring falling down in a room, but you would if they were falling into a stellar black hole. Spaghettification would occur:

"In astrophysics, spaghettification (sometimes referred to as the noodle effect) is the vertical stretching and horizontal compression of objects into long thin shapes (rather like spaghetti) in a very strong non-homogeneous gravitational field; it is caused by extreme tidal forces. In the most extreme cases, near black holes, the stretching is so powerful that no object can withstand it, no matter how strong its components."

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