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Lets say we have a frictionless and massless pulley with a massless string over it. There are two masses attached to it $m_1$ and $m_2$ where $m_2>m_1$.

So $m_2$ would an experience an acceleration down while $m_1$ experiences an acceleration up and massless string would also move accordingly.

However, if the net force on a massless string is always zero, it shouldn't experience any acceleration and stay at rest. But the string does move. How does it move?

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    $\begingroup$ You may be partly missing the point. Whenever the problem statement uses a massless string, massless and frictionless pulley, etc., that component is not part of the mathematical model that is used to solve the problem, and the physics of that component can be ignored. This simplifies the problem and makes the problem solution somewhat easier to obtain. $\endgroup$
    – David White
    Commented Sep 19, 2020 at 13:27
  • $\begingroup$ Think of the string as a medium via which a force on $m_1$ is transmitted to $m_2$, rather than an entity that experiences any such force itself. $\endgroup$
    – chepner
    Commented Sep 19, 2020 at 17:51

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Newton's second law is

$$ F=ma $$

For a massless object with no force on it, this equation becomes

$$ 0=0*a $$

which is clearly true for any value of $a$, and so any value of acceleration is consistent with Newton's second law. Indeed, if the force was non-zero the acceleration is undefined. A massless object can have any acceleration, but it can never have a net force applied to it.

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    $\begingroup$ A net force of 0 can cause an acceleration so long as the object is massless. $\endgroup$
    – Chris ♦
    Commented Sep 19, 2020 at 10:17
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    $\begingroup$ @Chris your last comment confused me - how does an “object” with zero net force applied to it, which is massless accelerate? Can you give me an example please? $\endgroup$
    – joseph h
    Commented Sep 19, 2020 at 10:47
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    $\begingroup$ @Drjh Let's say you have an imaginary basketball that is imaginarily glued to the palm of your right hand. To the extent that it exists, it will always move with your hand, yet it does not change the amount of force that your muscles exert at all. Thus, you can move it without exerting a nonzero net force on it. No physical object exists that is actually massless and can be regarded as being moved around; photons are massless, but the possibility of controlling the instantaneous location of one is limited. It just so happens that an imaginary object is consistent with the laws of physics here $\endgroup$
    – Post169
    Commented Sep 19, 2020 at 19:11
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    $\begingroup$ This answer is somewhat misleading, IMO. I mean, saying that it's true for any value of $a$ is not really that much of a feature. You could say that the usefulness of it breaks down for zero mass, since you can't flip it around, and express it as $m \vec{a}/m = \vec{F}/m =\{\vec{a} \text{ for } m \neq 0 \text{, else undefined}\}$. You could legitimately decide to treat it as not applicable in this case. The reason an imaginary massless string can move is because it's a part of an idealized model that assumes that it can, without justification, and it works, both with the math and in practice. $\endgroup$
    – Filip Milovanović
    Commented Sep 20, 2020 at 1:08
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    $\begingroup$ @FilipMilovanović If you prefer you can take the limiting case as $m\to 0$. For any fixed acceleration, it is clear that $F\to 0$. So, again, regardless of acceleration, $F=0$ for a massless object. Regardless of acceleration, it is clear from Newton's second law that there can be no force. And it's easily justifiable as a limiting case of massive objects as $m\to 0$. $\endgroup$
    – Chris ♦
    Commented Sep 20, 2020 at 3:29
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Your problem arises from the negligence of the fact that when we say massless, frictionless, etc. then we are only talking about the limiting case of these quantities tending to zero and not the absolute zero.

This in the given scenario means that you can neglect the mass of the string. But that doesn't imply that the string doesn't have a mass.

You might ask why mass cannot be zero (absolute one) here? Because force is the cause of acceleration and without any net force no body can accelerate, whether it be massless or not. As soon as the net force stops acting on a body the acceleration of the body ceases.

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The fact that the question states an assumption of massless pulley and string is to “idealise the situation”. Lots of physics problems ask you to ignore factors that are not particularly relevant to the point of the question. Here, you are asked to consider the forces and motion of the masses only. From there you are to determine the motion of the masses. If the question stated otherwise then this would create another level of complexity which is not consistent with level of physics you are studying and the physics being currently taught to you.

But to answer your question, the string is attached to a mass and the mass experiences a force (and therefore acceleration) meaning the string will accelerate as well.

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