Why does, at one time, tension balances the weight but at the other time don't balance it?

Question

Everyone is accustomed to many set-ups of Atwood machine; sometimes the question asks about acceleration of the blocks or so; sometimes they ask about tension etcetera.

But one thing I couldn't understand is how at sometimes, tension balances the weight of a mass while at other times, it can't.

I was viewing some worked-out problems of my previous-year class & saw this one:

The mass of the part of the string below a certain point $A$ is $m$. A block of mass $M$ is attached to the lower end of the string. Find the tension in the string at the lower end & at $A$ when

$\bullet$ $M$ is at rest.

$\bullet$ $M$ descends with acceleration $a$.

The answers are quite simple; for first part the tensions at $A$ & at the lower end are $(M+m)g$ & $Mg$ respectively & for the second case tension at $A$ is $(M + m)(g -a)$ & at the lower point is $M(g-a)$.

The author wrote for the first case as the block is in equilibrium, using the 1st law & for the later case, as the block descends with acceleration, using Newton's 2nd law. Okay, he is right.

But why is the tension different for the two cases? Anyone should say that at the later case an additional downward force is applied on the block so that it descends with acceleration $a$, but why can't then the rope balance that extra force? When the extra force is applied on the block, definitely it stretched the rope which would create tension; can't it balance that extra force on the block & prevent it from accelerating?

So, why does, at one case, tension balanced $M$ but in the other case didn't balance it?

What I am missing. Could anyone explain?

Answer 1

But why is the tension different for the two cases? Anyone should say that at the later case an additional downward force is applied on the block so that it descends with acceleration a, but why can't then the rope balance that extra force?

I believe you are thinking "backwards". The acceleration actually tells us what is going on. The downward force on the block (weight and/or pull from a different rope) is a different interaction than the rope tension pulling up. There is no reason to expect them to be the same because they are from different sources. The acceleration is what actually happens when the two forces are acting on the same object. The acceleration tells us that the forces cannot be equal at that moment.

When the extra force is applied on the block, definitely it stretched the rope which would create tension; can't it balance that extra force on the block & prevent it from accelerating??

Yes, it can if something is done to change the upward tension force to become equal to the downward force. Consider this based on your statement: A rope holds a 1 kg object in equilibrium. The tension in the rope equals the weight of the object. A 500 g object is added to the 1 kg object. The rope stretches, the tension increases, then the system returns to equilibrium. In order for the rope to stretch, the system briefly accelerated downward because the downward force was greater than the tension force at the moment the 500 g was added. After accelerating downward, the tension force changed and began pulling up with a greater force and stopped the downward motion.

On the other hand, imagine the rope is a piece of stretchy plastic or gum. It holds the 1 kg object in equilibrium, but when the 500 g is added, the objects accelerate downward continually because the stretching doesn't increase the tension, in fact, it might decrease the tension, depending on the gum. From the instantaneous acceleration we can calculate what the tension in the gum is, but by observation is can't be equal to the weight.

The tension is NOT a reaction or Newton 3rd Law force of the gravity. It is a separate force.

Answer 2

The rope might be able to add an additional force, but it depends on what's on the other end of the rope. Maybe it can't. And if it could, the forces would then be balanced again, and the block would not accelerate.

Focus on the block and its motion. You know that there are two forces on it: tension and gravity. In the second case it's accelerating downward. Therefore, there must be a net unbalanced force on the block. One of those forces, gravity, is fixed in magnitude. The other is tension. One must conclude that the magnitude of the tension force is less than the magnitude of the gravitational force. There's no other possible explanation.

It is irrelevant whether or not the rope can supply more or less force. The laws of mechanics dictate what the force of tension must be in this case.