We have seen how the spring constant ( k ) varies in proportion to the magnitude of force ( Fs ) acting within a springy system. Consider the two systems below, where three physically identical spring systems are used to create two systems, one on the left with the other to the right of the diagram:
The spring in each system has changed in length ( Lo ) by equal increments ( Δ L ). If we consider the two masses to the right as being a single system, a value of ( 2 Fw ) corresponds to a spring constant of ( 2 k = k’ ) as opposed to the spring constant ( k ) of the system to the left. Notwithstanding, the normal tensile stress ( σt ) of each spring is the same. For this reason, it will be useful to develop a relationship between each system based upon ( Lo ) and ( Δ L ) alone. This task is accomplished with the equation for strain:
Strain = ( change in length / original length )
ϵ = ( Δ L / Lo )
If the interatomic spacing between neighboring atoms in each spring changes by ( Δ L ), the overall length will change by the same increment. Furthermore, ( Δ L ) of this spacing gives rise to the counterforce that opposes ( Fw ) in each system.
It is important to remember that regardless of how springy a system may be, each system encountered has an elastic limit. The elastic limit is directly proportional to the degree to which a given system has been deformed:
ϵs = ɣ ≅ tan ɣ = ( Δ L / Lo )
Gamma ( ɣ ) is measured in radians ( θ ), and thus, it is a dimensionless quantity. The equivalency above is based on the assumption that the angle of distortion is generally very, very small. Many hard solids such as bone, metal, stone, and concrete display elastic shear strain of less than 1°. Regardless of the system at hand, stress and strain go hand-in-hand.
RENAISSANCE PHYSICS 