Confusion about Acceleration of an Object

[user14245]

(I am a newbie in physics. I did not use the template when posting as this isn't really a homework question but about something that makes me confused in elementary physics).
Suppose I have two objects (with masses m<M respectively) connected vertically by a wire such that the one with mass m is above. Now I drop this thing and obviously it will speed up with the acceleration g.
However, if I look at the forces acting on the smaller object with mass m, there are two forces: the Earth's gravitational force mg and the force exerted by the bigger object Mg. So the total net force on the smaller object would be $F=(M+m)g$. By the Newton's second law, it would accelerate with $a=g(M+m)/m$
This is paradoxical and it confuses me. Can you explain why?

 

[chipotleaway]

I believe the net force on the smaller mass would be mg + the tension, not Mg.

Mg - T = Ma (net force on M)
mg + t = ma (net force on m)

Combining the two gives Mg + mg = Ma + ma and so a = g

 

[haruspex]

I believe the net force on the smaller mass would be mg + the tension, not Mg.

Mg - T = Ma (net force on M)
mg + t = ma (net force on m)

Combining the two gives Mg + mg = Ma + ma and so a = g

... and so T=0. In short, there's no tension.
If you were holding the system by the upper mass only, just before letting go, there would have been tension (and some small extension of the wire). As soon as you let go, that will result in the upper mass accelerating a little faster than g and the lower mass a little slower. But pretty soon the wire will have returned to its unstretched length, so at that point there will be no tension and the two will accelerate equally. But since the upper mass got a bit of a kick start it will be traveling faster and will eventually catch up with the lower mass.

 

[ehild]

(I am a newbie in physics. I did not use the template when posting as this isn't really a homework question but about something that makes me confused in elementary physics).
Suppose I have two objects (with masses m<M respectively) connected vertically by a wire such that the one with mass m is above. Now I drop this thing and obviously it will speed up with the acceleration g.
However, if I look at the forces acting on the smaller object with mass m, there are two forces: the Earth's gravitational force mg and the force exerted by the bigger object Mg.

The objects are connected with the wire, so they feel force from the wire and no force from each other. Mg is the force the Earth exerts on M, and that object exerts a force of the same magnitude on the Earth, and not on the other object.
The wire exerts force (tension, T) when it is taut, and then the two objects move together, with the same acceleration. The net force on m is mg+T=ma, the net force on M is MG-T=Ma.
If you add that two equations together, you get (m+M)g=(m+M)a, a=g. Replacing that back into the original equations, you get that the tension is zero. In free fall, both objects move with the same acceleration g as if they were not in contact.

ehild

 

[user14245]

Thank you. Now I got it. So, to tie an object with something of mass M is not equivalent to exerting a force of Mg on it.