Fixed pulley with vertical loads

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This article describes a special case of a simple pulley system: a fixed pulley with vertical loads connected by a string passing over the pulley, as depicted in the figure.

Case of a massless pulley and string

In case the pulley and string are massless, the upward tension force exerted by the string on $m_{1}$ equals the upward tension force exerted by the string on $m_{2}$. If we call this value of tension $T$, then the force diagrams of the two masses are as shown below.

Fill this in later

We have the following cases:

Case	Direction of acceleration	Magnitude of acceleration	Magnitude of tension force $T$
$m_{1}>m_{2}$	upward on $m_{2}$, downward on $m_{1}$	$(m_{1}-m_{2})g/(m_{1}+m_{2})$	${\frac {2m_{1}m_{2}g}{m_{1}+m_{2}}}$ -- this is $g$ times the harmonic mean of $m_{1}$ and $m_{2}$
$m_{1}<m_{2}$	downward on $m_{2}$, upward on $m_{1}$	$(m_{2}-m_{1})g/(m_{1}+m_{2})$	${\frac {2m_{1}m_{2}g}{m_{1}+m_{2}}}$ -- this is $g$ times the harmonic mean of $m_{1}$ and $m_{2}$
$m_{1}=m_{2}$	no direction	zero	${\frac {2m_{1}m_{2}g}{m_{1}+m_{2}}}$ -- this is $g$ times the harmonic mean of $m_{1}$ and $m_{2}$

Note that the magnitude of the tension force has the same expression in all cases.

KEY FORCE CONCEPT: There is tension in the string in all cases, even when the system is in equilibrium and hence neither mass is accelerating. Tension is what keeps the string taut and in place. Tension goes to zero only when one of the masses goes to zero, which means that the string is no longer taut at that end.