Breaking Strength of a Cable

Newton's second law is F = ma, where

We want the acceleration to be one earth gravity which is 9.8 m/s2.   To solve this problem:

  • convert the mass (100 tonnes) into kg by multiplying by 1000 (1 tonne is 1000 kg), so the mass is 100000 kg.  Substituting this into the formula will give us the answer in N
F = (100000 kg)(9.8 m/s2) = 980000 N
  • the strength of cable would usually be given in units of kiloNewtons (kN).  Divide the above answer by 1000
F = 980 kN

 

Rockclimbing rope has high strength, and high stretch
Rapelling on Mt. Buffalo, Australia

How great a force is this?  It is actually quite large.  Rock climbing rope is obviously pretty strong, and is usually 12.5 mm (1/2 inch) diameter.  How does its strength compare to some other materials that could be used for a cable?  The table below lists some characteristic properties of various cables:

Material

Breaking Strength (kN)
12.5 mm (1/2 inch) diameter

Mass (g/m)
Stainless steel 95 3000
Kevlar 65 60
Nylon rock climbing rope 40 100

While ten 12.5 mm steel cables would be just about able to stand the force, it is pretty obvious that this would leave no safety margin.  Generally in non-critical applications, a safety factor of 5 to 10 times is required.  In a critical application (the occupants die if the cable breaks), 20 or more times would be needed as a safety margin.  It is obvious that steel is out of the question.  To get this required strength would be an enormous mass.  Kevlar is much more likely a choice.  Nylon would not be used because, although it is reaonably strong, it has enormous stretch (which is why it is actually used for rock climbing, since it causes a much lower shock should you fall).

We can calculate the size of the cable required to stand a 980 kN force, with a 20 times safety factor as follows.  The required safety margin breaking strength = (20)(980 kN) = 19600 kN.   The breaking strength of a cable depends directly on its cross sectional area.   Assuming that a cable is circular, then its cross sectional area is r2, so a larger cable would have a cross sectional area of  R2 (where R is the second cable's radius).  This means that the breaking strength of two cables is the ratio of their radii (or diameters) squared.   The following formula could then be used to calculate the required cable diameter.

gravity15.gif (181 bytes) where:
  • f and d are the smaller cable
  • F and D are the larger cable

breaking strength and diameter respectively

The mass of the cable will also increase by the ratio of the diameters squared.  Because the strength goes up as the square of the cable diameter, the cable doesn't have to actually increase in size by a great quantity.  This information is summarized in this table:

Material

Required diameter to stand
19600 kN force (mm)

Mass of a 500 m cable
(kg)

Stainless steel 180 310000
Kevlar 220 9000
Nylon rock climbing rope 280 24500

It is very obvious that a steel cable is totally impractical.  The mass of a strong enough cable would be more than 3 times the mass of the spaceship!