# What drives the free end of a slackened string?

Let's consider a gentle situation. A small puck is attached to a string. The string is sufficiently long in length. The setup is placed on a frictionless surface. One end of the string is nailed to the surface. Initially it was at a slackened state. Now we give the puck some velocity. The string end attached to it starts moving as well. The question arises, if at the slackened state there is no tension in the string, what causes the free end to move?

My thinking- Well, I think that the moment we gave puck an impulse to provide it with the velocity, some impulsive tension could have developed in the string in the length nearby to the puck. This impulse tension could have given the string's free end the velocity.

Then my subsequent question becomes, if there's no tension in the string in the slackened state, what causes the farther parts of the string (which were at rest initially) to start moving in the direction of puck's motion after a considerable time has passed? (and yes, I don't mean the parts nearby to the pinned end, but yes, the part far away from the free end.)

Note that the string is taken sufficiently long for this experiment.

Edit

After thinking a bit more about the problem, I am now certain that the impulsive tension could not develop in the string like I thought before. The string in the situation is taken ideal and slackened. Also, there is no friction or any other external force on the string. Nothing is pulling the string back so it won't have the tendency to "constrain" the puck's motion like a tight string would do.

Two of the three answers I've yet recieved try to imply (directly or indirectly) that even in the slackened state, the string would have some tension. Which would be growing steadily and uniformly. This, according to me, is incoherent with the general notion of slackened state of string meaning absence of tension. Also, this would not go along with the impulsive nature of the tension which occurs when there is a sudden development of tensile force.

I, hence, can't help but disagree with these views on the situation. If anyone has any different view or a clearer justification of the previous answers, you're always welcomed here.

• It would also be a great help if someone could explain what are the cause of those "constraints" as per @Bob D's answer.

When the puck is given an impulse, it will start moving. This provides a tension on the piece of string directly attached to the puck - let's say that's a piece of string with a length of $$dx$$. As the surface is frictionless, the puck will keep moving, and so that tension will then be extended to the next $$dx$$ of the string, and so on. Eventually, after enough time has passed, the entire string will be under tension. The process of going from slack to tight will cause the string to move slightly, as it wasn't completely straight to begin with.

Imagine coiling up a piece of thin chain, and then pulling on one end. That's how I picture this explanation.

Hope this helps!

• If, according to you, whilst the sting turns tight from the slackened state, tension develops gradually in the subsequent parts of the string; why does the puck not slow down? I mean, both end of the "part" of the string that has been made taught are subject to equal and opposite forces (assume a massless string, then no mass, no inertia, no force). So by Newton's third law, should not be the same tension acting on the puck which should slow it down. Jan 13 at 17:31

The question arises, if at the slackened state there is no tension in the string, what causes the free end to move?

The answer is in the statement preceding this one, "Now we give the puck some velocity" Giving the puck velocity means there was a force applied to the puck, even if briefly. When the force is removed, the puck continues at constant velocity until and unless another force acts on it, per Newton's first law. This assumes the mass of the string is negligible (massless string).

My thinking- Well, I think that the moment we gave puck an impulse to provide it with the velocity, some impulsive tension could have developed in the string in the length nearby to the puck. This impulse tension could have given the string's free end the velocity.

When the impulse was given, the string was "slack", so there would be no tension. Only when the velocity of the puck eliminates that slack will there be tension in the string.

Then my subsequent question becomes, if there's no tension in the string in the slackened state, what causes the farther parts of the string (which were at rest initially) to start moving in the direction of puck's motion after a considerable time has passed?

If the string is considered to be massless, those farther parts of the string move simply because they are constrained to move due to be connected to the puck.

If the string is not considered to be massless, you would then simply add the mass of the slack portion of the string that moves with the puck to the mass of the puck. That would, of course, effect the initial velocity of the puck due to the applied force.

Hope this helps.

• "If the string is considered to be massless, those farther parts of the string move simply because they are constrained to move due to be connected to the puck."- What sort of physical interaction causes this constraint? There must me some interaction right? Otherwise, how could the motion be constrained? Jan 13 at 17:25

There is a second factor, the nature of the string. Consider a "U" shaped section of the string. We move the top left end of the "U" - perhaps push it in some direction or another.

The surface is frictionless, but the string has elasticity and its present shape. Will the top left part of the "U" (and oerhaps left side) move but the rest stay absolutely motionless, or will the entire "U" tend to move slightly? Clearly the latter.

So you have to consider the elasticity of the string and its stiffness, however small. Not just the surface and friction forces.

• The phenomena of elasticity and the restoring forces come into the play only when there is already some force of deforming nature is developed in the cross-section of the string. So, if you consider elasticity, are not you indirectly implying the fact that tension is being developed in the string. Right? If tension develops, how is the string still considered slackened? Jan 15 at 6:19
• True, the two are really the same thing. But I didn't feel that previous answers gave a good enough sense to someone not accustomed to thinking about tension within strings. I remember from my earliest teenage years in physics classes quite a lot of people found it hard to think around or even unintuitive how to think about it. I've tried to fix that gap a little by this Jan 15 at 10:23
• But I still don't understand why does this tension develops, if slackening of string is taken same as vanishing of tensile force in it? More frankly to say, I find this understandable, but rather indigestible Jan 15 at 10:30