Image from the Wikipedia Commons

To begin our math, the equation for the force of friction:

F=uN

Where u is the coefficient of friction, and N is the normal force. What this means for climbing is that the more force you apply (in a direction perpendicular to the surface of contact), the more friction you get!

Let's look again at u. We call it the coefficient of friction, but what does that really mean? Different surfaces have different coefficients of friction. The greater value of u, the more friction you get for a given normal force$$. In other words, sticky surfaces have a large u$\mu$ and slippery surfaces have a small u$$. This explains why we "chalk up." Chalk increases the coefficient of friction, giving us more friction and therefore better grip. This also explains why the bottom of climbing shoes are made out of rubber: rubber has a high coefficient of friction (along with other advantages).

I want to emphasize one last time that equation for friction. It tells us that the friction is dependent upon the NORMAL force, as in the force PERPENDICULAR to the surface. If you apply a large force to hold in a direction that is not perpendicular to the surface, you're not doing yourself any favors. In fact, this force will only pull you off of the hold.

Elasticity and Impulse:

We talk about elasticity for on major reason: ropes. In order to truly understand the qualities and effects of a dynamic rope, we have to touch on the impulse momentum theorem. Momentum describes an object's resistance to stop moving. Mathematically, momentum (p) is defined as mass times distance:

p=mv

We have another quantity to talk about, and that's impulse. Impulse tells us about the effect of a net force acting on an object. Mathematically, impulse if equal to the average force an object applies, multiplied by the time this force is applied for:

J = FAVE TDELTA

These two quantities can be related by the Impulse-Momentum Theorem. This simply states that impulse changes an objects momentum, and more specifically the value of the change in the objects momentum is equal to the impulse:

J = pDELTA

This tells tells us something important: no matter how long we apply the force for, if the objects change in momentum is always the same between cases, the impulse is always the same between cases. Note that your total change in momentum for a given fall will always be the same, no matter the rope. You will have the same momentum the moment before the rope becomes taut, and you will have the same momentum once your are rest (0). Therefore, the change in momentum will always be the same.

So what does a dynamic rope do for us? It increases the amount of time that the force is applied. Since our impulse must remain the same across cases, the average force must decrease. Not only does the average force decrease, but so does the maximum force. In other words, you feel force for a longer amount of time, but it's not as strong of a force (we call this the impact force in climbing).

In climbing, we measure the amount of stretch that occurs (and therefore the increased time duration of the force) as the dynamic elongation. We climbers generally like a fairly high dynamic elongation, as this makes the fall less painful or prone to injury. However, dynamic elongation also increases the distance that we fall. This can, of course, be a bad thing, especially when you are close to ground. The desired elongation of a rope depends a lot on the task at hand.

The moral of the story:

Never climb with a static rope! A static rope has very little elasticity, and it will hurt like all hell when you fall. Pay attention to the dynamics (and statics) of your ropes. Ropes are rated for a certain amount of falls, make sure you look into this number and other properties when buying ropes. Understand the limitations of your gear, and keep it in good maintenance. Your life is important. See the gearx link in the resources for more information on ropes (its worth it). I've also included a link to a life in the vertical article on gear maintenance and lifetimes. Be well read.