physics young's modulus

Question

A steel cable whose effective cross section is 1.20 cm2
supports a 200-kg elevator. When the elevator is at rest at the second
floor the length of the supporting cable is 40.000 meters. What would be the length of the cable if it were not under stress?

Answer 1

This can be solved using the definition of Young's modulus, as the ratio of applied stress to compressive or tensile strain.  This can be written as:

 

E = stress/strain = (F/A)/(ΔL/L)

 

where

 

F = applied force

A = cross-sectional area of the material (the area you would see if you cut perpendicularly across the material and looked at either cut piece head-on)

ΔL = change in length under the stress

L = original (rest) length

 

The stress is basically applied pressure (as it is in defining other modulus values as well, such as bulk modulus or shear modulus), represented as applied force over the cross-sectional area (F/A).  The strain represents the fractional change in length (change in length over original length) caused by the stress.

 

You can look up Young's modulus for steel.  Then you can use the above equation to solve for a value for the strain (since you are given F and A):

 

(ΔL/L) = [value]

 

Effectively, this is one equation with two unknowns, since neither ΔL nor L is known individually.  Hence, we cannot solve for the rest (unstretched) length (L) using this equation alone.

 

However, you also have a second relationship between the two values:  you know that that overall stretched length of the cable is 40 meters:

 

(L + ΔL) = 40 

 

This becomes a second equation.  Between the two, you can solve for L in whichever method of solving a system of two equations you prefer.

 

I hope this can still help!