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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?

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  • $\begingroup$ What do you mean by "not attached to a ceiling?" Are you saying that the whole setup is just free-falling straight down? If it's not attached to a ceiling, then what object is exerting the force F on the pulley? $\endgroup$ – Ben Crowell Oct 22 '14 at 15:09
  • $\begingroup$ Clearly, since there is a net upward force, the pulley itself will accelerate upwards. Huh? Why doesn't the whole apparatus just drop? $\endgroup$ – Ben Crowell Oct 22 '14 at 15:19
  • $\begingroup$ Someone's pulling the pulley upward by a hook of some sort... $\endgroup$ – ODP Oct 22 '14 at 15:27
  • $\begingroup$ Quite clear by "a force pulls the pulley upward". $\endgroup$ – ODP Oct 22 '14 at 15:27
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    $\begingroup$ Good homework question in my opinion. Effort has been put in, and the OP is asking about a conceptual interpretation of their derived equation. $\endgroup$ – BMS Oct 22 '14 at 17:47
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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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