Imagine that a person is seated in a chair that is suspended by a rope that goes over a pulley. The person holds the other end of the rope in his or her hands. Assume that the combined mass of the person and chair is M.
What is the magnitude of the downward force the person must exert on the rope to raise the chair at a constant speed? Express your answer in terms of M and g. (Hint: the answer is not Mg!)
What is the magnitude of the required force if the person is accelerating upward with mag(a ⃑)=0.10g? 
The tension in the rope acts upwards with respect to the chair and upwards with respect to the force of the man's hands. Both tension forces are equal and in the same direction. Ma=-Mg+tension1+tension2. Tension 1 and 2 are equal so the equation reads Ma=-Mg+2tension so Ma+Mg=2F(t) so M(a+g)/2=F. The chair moves at a constant speed so acceleration is zero implying 1/2(Mg)=F(t).
To determine the magnitude of the downward force the person must exert on the rope to raise the chair at a constant speed, we need to consider the forces acting on the system.
At a constant speed, we know that the net force on the system is zero. This means that the force the person exerts on the rope must balance the force of gravity acting on the person and the chair.
Let's break it down step by step:
1. First, let's consider the forces acting on the person and the chair. There are two forces at play:
a) The force of gravity acting downwards, which is given by the equation F_gravity = M * g, where M is the combined mass of the person and chair, and g is the acceleration due to gravity.
b) The force that the person exerts on the rope, which we're trying to determine.
2. Next, let's consider the forces acting on the rope. Since the system is at a constant speed, the tension in the rope must balance the force of gravity. Thus, the magnitude of the tension in the rope is equal to the magnitude of the force of gravity, but in the opposite direction.
3. Now, let's put it all together. The downward force the person must exert on the rope is equal in magnitude but opposite in direction to the tension in the rope. So, the magnitude of the downward force the person must exert is:
-Magnitude of the force of gravity = -M * g
Therefore, the magnitude of the downward force the person must exert on the rope to raise the chair at a constant speed is -M * g.
Now, let's move on to the second part of the question.
If the person is accelerating upward with magnitude (a ||) = 0.10g, we'll need to consider the additional force required to overcome this acceleration.
1. Calculate the net force required to accelerate the system. This can be done using Newton's second law, which states that the net force is equal to the mass of the system multiplied by its acceleration:
Net force = M * a ||
2. Since the rope is the only force acting on the system, the net force must be provided by the tension in the rope.
3. So, the magnitude of the required force the person must exert is equal to the net force, which is given by M * a ||.
Therefore, the magnitude of the required force if the person is accelerating upward with magnitude (a ||) = 0.10g is M * 0.10g.