The term "spring with a force constant" refers to the proportional constant k. It gauges the stiffness of the spring.
A spring produces a force in the direction of its equilibrium position when it is stretched or compressed to a length that differs by an amount x from its equilibrium length. Constant force springs, which are frequently constructed of steel, are created by coiling a steel ribbon over itself or by winding a flat sheet of steel around a drum (with a virtually constant radius). One end of the spring is fastened to a weight, and the spring is installed in its intended use. If the masses and springs are the same, frictionless vertical and horizontal spring-mass systems oscillate indistinguishably around an equilibrium point.
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