How much force would someone falling with a rope experience?

Question

I was trying to figure out how much force a rock climber would experience when they fall, at first I thought the rock climber would have some potential energy ($E_p$) given they are a certain distance up ($h$), have mass ($m$) and are under the force from gravity ($g$). $E_p = mgh$. I then thought that potential energy would be converted to kinetic energy, then to work done. Assuming that the transfer of energy is 100% efficient then I can just skip the kinetic energy part and convert right from potential energy to work done ($E_w$). So $E_w = E_p$, work done is the force ($F$) applied over a certain distance. I’m guessing in this situation the distance would be how much the rope stretches when the faller falls, lets call that ($d$).

$$Fd = mgh$$

or

$$F = \frac{mgh}{d} \tag{1}$$

(1) Feels right as it meets my sanity checks :-

1. The more stretching ($d$), the less force ($F$)
2. The more mass ($m$) the more force ($F$)
3. The higher the fall ($h$) the more force ($F$)

There is just one problem. When I go online to find dummy values to check for ($d$) I find out that the stretch of the rope is always given as a percentage... Which implies the longer the rope is the more it stretches. If we are falling from a larger distance ($h$) we would need more rope, we can conclude $h \propto d$. If we add in the constant of proportionality we end up with $$d = kh \tag{2}$$ subbing (2) into (1) we get

$$F = \frac{mgh}{kh}$$ Which simplifies down to $$F = \frac{mg}{k}\tag{3}$$

If I then do a sanity check on that…

1. More stretching probably means higher ($k$), so lower ($F$) which is fine

2. More mass ($m$) is more force ($F$), fine

3. Uhm, yeah… Turns out falling from higher ($h$) doesn’t make a difference here...

So this implies that someone falling from say $1\cdot10^{-3}$ m is experiencing the same force as someone falling $100m$. Where have I gone wrong in my thinking? Thanks

Answer 1

To a first approximation the rope acts as a spring. The force is linearly proportional to the amount of stretch. It starts at $0$ when the rope just goes taut. We usually write $F=kx$ where $k$ is the spring constant and $x$ is the current stretch. The energy absorbed is then $\frac 12kx^2$, which you can set equal to the potential energy loss from the fall. Then $E=mgh=\frac 12kx^2=\frac {F^2}{2k}$. The maximum force rises as the square root of the height of the fall and is smaller when the $k$ factor is reduced because the energy is dissipated over a longer distance.

The stretch is given as a percentage because the rope stretches a given percentage when subjected to a given force. At a given force a longer rope will stretch more.

Answer 2

While a rope is not in tension, it applies no force to the best of my knowledge. So yes, no matter how high you fall, you'll experience the exact same force $W = mg$.

But yes this probably belongs on physics stack exchange.

Answer 3

I think your error is in assuming that the rope length is the same as the fall length. A rope will stretch a given percentage of its own length, not a percentage of the length you fell. If you fall through a 1mm drop with only a 1mm long rope, you will feel a sudden jerk at the bottom since the rope has virtually no room to stretch - even though you're not moving fast with such a short fall, the jerk is sharp, since you will likely come to a stop within a fraction of a millimeter. But there's no reason to think that since you fell 1mm, the rope must also be only 1mm long - you can fall 1mm at the end of a 100m rope. A longer rope will have more stretch regardless of how far you actually fall. By equating the fall distance and rope length, you've found that the increased fall speed is exactly cancelled out by the increased stretch distance of the longer rope - you have more kinetic energy, but more time/distance to dissipate it, resulting in the same forces.

If you fix the rope length, you will find that falling from a larger height does increase the forces experienced. Falling 50m at the end of a 100m rope will result in larger forces than falling 1m at the end of a 100m rope, which is probably the intuitive result you're looking for.