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Let there be a situation where a force 'F' is acting on one end of an in extensible string which is connected to a box named 'A' resting on a friction less horizontal surface through the other end.

enter image description here

The Force F is transmitted through the string and acts on box 'A'. As a reaction to this box 'A' pulls the string toward itself with force 'F'.

enter image description here

In the FBDs we can see that the string is balanced by the forces but the box is not, so the box starts accelerating towards the right. As the box starts to accelerate, for a very small moment there is slack in the string which makes the tension force 0 for a brief moment.Then the force F being applied on the string is used to make the string taut again and tension forces again begin to accelerate box A.

This keeps repeating itself.

Is this how it works or there something wrong?

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  • $\begingroup$ The solution you propose will only happen if the box is pulled by an extensible string so it would act similar to a spring. $\endgroup$ Commented Aug 22, 2020 at 9:03
  • $\begingroup$ @Adiran Howard can there be waves in inextensible strings? $\endgroup$
    – Protein
    Commented Aug 22, 2020 at 11:48

3 Answers 3

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As the box starts to accelerate, for a very small moment there is slack in the string which makes the tension force 0 for a brief moment.

Wrong.

Tension is zero before the force is applied. Once the force starts acting and the string becomes taut, the block and string gain the same acceleration and the string doesn't slack again.

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  • $\begingroup$ but in the free body diagram of the string it shows that it has no net force on it. Doesn't this mean that there is a time when the string isn't accelerating but the box is. Or is there a mistake in the free body diagram that I have drawn? $\endgroup$
    – whae
    Commented Aug 23, 2020 at 2:18
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There is a mistake in modeling the system. Indeed, the string is not a rigid body and you can't use the rigid body equations of motion and to be expecting that they will work. You consider, for example, in place of the string, a rigid thin (metallic) rod. In this case, you can consider the box and the rod as a unique rigid body on which the only acting force on x direction is the one you have drawn in your fist picture. So the whole body moves with a constant acceleration on x direction. While, how I said, if you want to consider the string, you have to use proper equations that take into account the physical properties of the string and which will lead to a realistic result.

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The string is accelerating to the right with the box. This means the string is subject to a net force equal to its mass multiplied by the acceleration. This means the forces at the 2 ends of the string are not equal, contrary to what is shown in your FBD. The force on the left end is equal to the force on the box and the force on the other end is slightly larger.

Now, the mass of a string in a setup like this is usually very small compared to the mass of the box, and therefore the difference in force is so small that it is not worth considering. So we usually just assume the string is massless and therefore no net force is needed to accelerate it. So the assumption that the forces at each end of the string are equal is appropriate. Just remember that the string is actually accelerating with the box even though our model shows no net force on the string.

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