# In the case of tension on a massless string

The string can be thought of as made of little elements the size of elements can be made arbitrarily small. The forces acting on each element when the string is pulled are equal and oppositely directed, the net force on each element is therefore zero. So does any of the elements move (i.e. why does the string move)? Do massless objects obey Newton's laws that when there is force then there is acceleration, not otherwise?

• – Qmechanic Aug 22 '17 at 15:10
• Why do you say that the forces on each element are equal and oppositely directed? Is it because you are assuming that the elements themselves are massless? If so, then any acceleration is possible with zero net force, and any state of motion is possible. Note, however, that ultimately the question makes no sense because massless strings and massless elements are unphysical: they serve as limits that are valid approximations under certain conditions. – garyp Aug 22 '17 at 15:46

## 2 Answers

As garyp pointed out in his comment, your mistake is that you applied the limits incorrectly. You assumed that the force on an element is zero, and its mass is zero. In this case the acceleration is indeterminate, it can be anything :
$a=\frac{F}{m}=\frac{0}{0}=?$

The difference $\delta F$ in the forces at the ends of the element is small, like the mass $\delta m$ of the element, but neither are zero. You can neglect them when considering the motion of the far more massive objects attached to the ends of the string, but you cannot neglect either when considering the motion of the element itself. The ratio of a small but non-zero force to a small but non-zero mass is a finite acceleration :
$a= \frac{\delta F}{\delta m}$.

Newton's Laws apply to elements of a "massless" string just as they to do "massive" objects. All strings have some mass. They are only "massless" when compared with the masses they are attached to.

One could answer the question in this way,
Since we assume that the string does not have mass it must neither have any forces. F=MA. Here f=0 and m=0, hence the equation is satisfied for any value of a. Hence it is not necessary that the sting does not accelerate. Newtons second law of motion is applied here too.

:-)