A mass is hanging by a thread from another mass, which starts to fall. What happens and why

free-body-diagramnewtonian-mechanicsstring

Please consider the following situation (ignoring air friction, and assuming all strings have no mass):

Ball A with mass m1 is hanging from the ceiling by the blue string.

Ball B with mass m2 is hanging by another string, from ball B. So currently, the system is at rest.

Now, somebody cuts the blue string.

Which of the following scenarios would happen, and exactly why? Also, does the answer depend on the values of m1 and m2 at all?

  1. Both ball A and ball B will start to accelerate to the ground (with acceleration g)?

  2. Ball A will start to accelerate downwards (with acceleration g). Ball B, however, will stay exactly in its place, until ball A collides with it from above.

  3. Something else?

My intuition seems to be around option 1. In order to stay at rest, ball B needs some force to cancel out the weight m2g. But once ball A is accelerating downwards, the string connecting the two balls is no longer fully stretched. So I can't see why ball B won't start to accelerate downwards at this point, at the same rate as ball A (a = g m/s^2).

Please explain the correct intuition and show the relevant equations (in terms of Newton's laws).

Best Answer

Before the blue string is cut there is tension $T=m_2g$ in the black string which supports the weight of the lower ball.

Once the blue string is cut, the upper ball starts to accelerate downwards, the black string becomes slack, and the tension in that string falls to zero. Once the tension in the black string has reached zero, the only forces acting on the two balls are their respective weights and they will both accelerate downwards with acceleration $g$.

In real life the tension in the string cannot fall to zero instantaneously, so there will be some (very short) interval between when the blue string is cut and when the lower ball reaches an acceleration of $g$. However, since we are instructed to ignore air friction and the mass of the black string, we can also probably assume that the black string is an ideal string that does not stretch. With an ideal string then the tension will fall to zero instantaneously and both balls will immediately accelerate downwards with acceleration $g$. So the answer that you are probably expected to give is option $1$.

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