The problem is a rope through a single pulley attached to the ceiling. A person sits in a chair (system mass

**) which is attached to one end of the rope, and pulls down on the other rope to lift himself. I need to explain mathematically how much force he must apply to remain in position and also come up with a formula for how much force to apply to achieve upward acceleration**

*M***.**

*a*From experience, I know that you will have to apply half of your weight to achieve zero acceleration. This is because by Newton's third, for every dF of force delivered downwards by your muscles on the rope, an equal force pushes you up, decreasing the effective "weight" of the chair person system. So when you apply a force equal to half the weight of the system, you effectively cancel your weight. Beyond that (I'm a little less certain here) every bit of force you apply will only count once towards the tension in the rope.

I must be missing forces, because I can't put this in terms that make mathematical sense. What equation can I set up that will justify F(applied) = F(weight)/2 when tension = weight? How do I show that the effective weight of the system is equal to the actual weight minus the 2x(applied force) but only until the two are equal?