Consider the pulley arrangement as shown in the figure. If all pulleys are massless and frictionless

Question

Consider the pulley arrangement as shown in the figure. If all pulleys are massless and frictionless, what is the value of force $F$ required to keep the mass $m$ in static equilibrium?

Professor PC Thomas Classes · Answer

Let the tension in the string be $T=F$. Consider the movable pulley P1: the string passes over it, pulling it down with a force of $2T$, while the end of the string attached to its axle pulls it up with a force of $T$. For P1 to be in equilibrium, $T = 2T$, which implies $T = 0$. Now consider P2 carrying mass $m$: the upward force is $2T$. For its equilibrium, $2T = mg$, meaning $mg = 0$. Thus, for any real load ($m > 0$), the system can never remain static regardless of the applied force $F$.