Breaking Strength of a Cable

Newton's second law is F = ma, where

• F = force in Newtons (kg·m/s²)
• m = mass in kg
• a = acceleration in m/s²

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

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:

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.

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:

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!