Imagine a massless and frictionless pulley with two weights hanging either side of the pulley by a massless string.
Like this except not attached to a ceiling
Rather than being fixed to a ceiling, the pulley is being pulled upward by an external force F, with the weights and string still attached.
Due to Newton's 2nd Law,
$\Sigma F_y=F-2T=ma$,
where $T$ is the tension in the string on either side of the pulley and $a$ is the vertical acceleration of the pulley.
Clearly, since there is a net upward force, the pulley itself will accelerate upwards.
But because the $m=0$,
$F-2T=0$.
Does this not then suggest that the pulley has a constant velocity?
Answer
In the equation $F_{net}=ma$, normally we would assume that $F_{net}=0$ implies $a=0$ on the right-hand side. However, for a massless object, we can satisfy the equation by having $F_{net}=0$, $m=0$, and $a\ne0$. In reality, of course, the pulley is not massless, so $m$ is small, $a$ is some nonzero number, and $F_{net}$ is small.
The above reasoning is the justification for the usual assumption that low-mass objects transmit forces unchanged, e.g., that the tension in a rope is the same value throughout the length of the rope.
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