What happens when I grab a rope after jumping down? If a body has a mass of 54 kg and falls straight down for 11.33m before grabbing a rope, what are the forces acting here? What part does the coefficient of friction between the rope and hand play?
 A: After falling $11.33$ m a body of mass 54 kg would have a kinetic energy of 6000 joules.
The specific heat capacity of water is 4200 joules per kg per degree Celcius. So I guess the specific heat capacity of human hands is similar to this. A quick web search gave values between 3500 and 4000 for skin. The whole of a human hand has a mass about half a kg, so with two hands that's 1 kg. So if the 6000 joules is dissipated as heat in the hands then it will only raise their temperature a little ($1.5^\circ$ C).
However in practice anyone who ever grabbed a rope while falling knows that they get skin burns unless they have a good pair of gloves. The reason is that the falling person needs to slow their rate of descent, and they have the following dilemma. Slow too quickly and there is not enough time to conduct the heat away. Slow too slowly and you have more energy to get rid of because you continue to fall (and there may be some ground below which you need to avoid hitting).
Let's say the time constant for heat conduction in the hand is of order seconds. In this case the falling person needs to spend some seconds reducing their rate of descent in order to avoid skin burns. But this in turn means they will fall a further distance of order 50 metres! So now the total energy they need to get rid of is increased 5-fold and we have a vicious circle.
My main conclusion is that without gloves it can't be done, but with good enough gloves it should be ok from a heat dissipation point of view. But as a further check let's think about how much force the human arm and shoulder can withstand. A quick web search says 100 N is ok but larger than that risks a dislocation. With two hands maybe 200 N then. On the other hand, most of us have done pull-ups which shows that we can sustain forces of order 500 to 700 N, but 1000 N would be difficult. Since most of us would struggle to lift twice our own body weight, it follows that we would find it hard to get an overall deceleration equal to $g$. The lesson here is that if you first fall freely for a given distance, then you would need at least that same distance to come to a stop using forces provided by your hands and arms.
One lesson from the above is that if you ever see someone in a movie fall freely for 10 metres and then grab something to break their fall and overall sustain no injury then very likely it is not physically possible. If they managed to use their legs in some way, or wrap a rope around their torso, then maybe.
But I still have not answered the question which asked about coefficient of friction. The friction force between surfaces is, to reasonable approximation, $\mu f$ where $f$ is the force squeezing the surfaces together and $\mu$ is the coefficient of friction. To slow one's descent we need $\mu f$ to be of order 500 N as we just saw. Values of order $0.5$ are reasonable for $\mu$ involving leather and some other materials. So we want $f$ of order 1000 N. So now we ask, can a human grip be that strong? It is equivalent to picking up a 100 kg object using your grip: the answer is no. A more reasonable number would be 200 N, and then the deceleration is more modest; you end up falling a further 27 metres rather than 11.
If the question is asking if a human grip can stop the descent in a short distance, then the answer is no.
A: After falling a certain height, the body gains kinetic energy due to positive work being done by the gravitational force. If the person now grabs the rope, there will be a kinetic friction force between his hand and the rope. Friction force always acts in the direction opposite to the motion.
The kinetic friction force is defined as
$$f_k = \mu_k n$$
where $\mu_k$ is coefficient of kinetic friction between the hand and the rope, and $n$ is the normal force the rope exerts on the hand. Stronger you hold the rope, by action-reaction principle the more normal force you get.
Since the friction force acts in the direction opposite to the motion, from the definition of work
$$W = \vec{F} \cdot d\vec{x}$$
the work done by friction force will be negative and it will tend to decrease kinetic energy of the body. At the same time the gravitational force does positive work on the body since it points in the same direction as the motion (displacement).
If you want to completely stop the body from falling any further, the friction force must be greater in magnitude than the gravitational force. More friction you have, less displacement is needed to dissipate the kinetic energy of the body. Please note that when I say dissipate I actually mean to convert mechanical energy of the system to some other forms such as heat and material (skin) deformation. In reality, it is very unlikely that you would be able to stop from falling by grabbing a rope - human hands (skin) just cannot dissipate much energy.
