I'm a little confused with the application of laws of motion on a man climbing a rope. Suppose a man of mass Mg is climbing a rope with an acceleration a. Rope is massless. Now if look through the frame of the piece of rope held by the man, there is a force Mg downward by man, ma downward applied by man and T upward. This balances out as the piece is at rest. This equation is correct, though my way of looking at it maybe incorrect. Now if we look through the man's frame, we have Mg downward, T upward, ma upward the reaction by rope, and as we are in an accelerated frame, ma downward. What am I doing wrong and can you explain how is the man able to climb upward i.e. how the forces are acting to give this motion.
Your question looks a bit confusing, but still I would try to answer it (as per my understanding of the question). Clearly in any frame of reference the only force that are acting on the object are:
- Gravitational force on the man,$F_g$
Tension from the rope, $T$(See the note)
- Static Friction (from the rope, assuming that he is moving upward by pulling ),$f_s$
Now these forces act together to cause a net acceleration $a$. The net force is then $ma$ ( Yes! the man does not apply this force its just the effect of all the other forces.).
Therefore the equation can be written as $$F_g + f_s = ma $$.
Here you just have to account for the pseudo force $-ma$ and the net force equation is as follows: $$F_g + f_s - ma =m0=0$$.
Hope this clarifys your confusion.
I have not considered the question with the following picture in my mind
If that isn't the case but like the one in which the man is attached to the rope and is being pulled up then you must consider tension and erase friction from the term.