Page 206 - J. C. Turner "History and Science of Knots"

196 History and Science of Knots

be considered to equal the initial potential energy, mgh.
If the rope is not to break, the energy of the falling weight must be fully
absorbed by the rope stretching throughout its length. When the weight stops
falling, all its initial potential energy is contained within the rope as a product
of the maximum tensile (arresting) force, Fa, which could also be called the
shock force or, in popular parlance, the impact force, and the maximum stretch
in the rope, S. Thus we can say mgh = Fa x S.
The fall factor f is the ratio of the vertical height fallen to the length of
the rope from the weight to the anchorage, that is, f = h/l. If the weight
is dropped from the anchorage point, the fall factor is 1. If the rope is ex-
tended to its full length above the anchorage before releasing, the fall factor
is 2. Intermediate values are obtained by starting at some intermediate point,
perhaps with some slack in the rope or passing the rope through a directional
pulley or the like at some point above the anchorage, limiting the distance
fallen for the amount of rope paid out. Fall factors greater than 2 can only be
obtained by such exceptional circumstances as snatching the rope close to the
falling object after it has already fallen some distance.

0
U.

Time

Fig. 3. Duration of arresting forces [8, 181. A mass of 80 kg was dropped with a fall
factor of 1.78. Solid line: rope length 2.8 m, distance fallen 5.0 m; the small curve
to the right shows the first oscillation . Broken line: rope length 11 . 2 m, distance
fallen 20 m
It can be shown [33] that, ignoring minor complications,

Fa = mg + mg 1 + 2(f M/mg) ,

where M is the modulus of the rope relating the stretch to the load. That is,
the shock force applied to the rope by arresting the falling weight is indepen-
dent of the absolute distance fallen, h,, but is a function of the fall factor, f
only. This does not mean that all falls of the same fall factor are equivalent
to the falling object. At the same fall factor, the greater the length of the
rope l (and therefore the greater distance fallen h), the longer will it take to
reach the maximum impact force Fa (Fig. 3) and the longer the time the rope,