tension and acceleration

[reburbia]

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A hanging mass is attached to a mass resting above on a frictionless table. As the hanging mass falls, the mass on the table begins to slide (due to tension in the string).

The question is: What is the maximum tension that can be achieved?
The correct answer is "g," which makes sense to me intuitively but not mathematically. Intuitively, I see that nothing can fall faster than the speed of gravity regardless of its mass.


However mathematically: The hanging mass produces the tension in the rope, which translates to the force pulling down on the mass resting above on the table. If F=ma, then the heavier the hanging mass is, the larger its force. If the hanging mass force is infinitely large, then by F=ma (m here now being the mass resting on the table), then "a" seems like it should be infinitely large.

 

[syoung]

it's not speed of gravity!
T is = mg, and so it cannot exceed the acceleration of gravity (10 m/s/s), especially since it's frictionless and the mass hanging is just in free fall

If F=ma, and you have a larger M, it will STILL accelerate towards the earth at g (10 m/s/s)
Remember, if you drop a cannon ball and a ping pong ball, with out air resistance, both reach the ground at the same time!

 

[milski]

If the resting mass is M and the hanging m, you can derive that the tension, T=gmM/(m+2M).

For M->infinity, T-> gm/2
For m->infinity, T->gM

In other words, for m significantly larger than M, the tension will be close to mg. Which makes sense - if the mass of M is insignificant compared to m, the tension will be about the same as it would be with m only - mg.

You could also use that F=mg-T and since F always has the same sign as mg, T cannot be larger than mg, thus mg is its maximum. That does not give you the conditions under which it will be close to mg.