The force on the mass for the slingshot in Prob. 3.35 and the force on the mass in the Duf...

The force on the mass for the slingshot in Prob. 3.35 and the force on the mass in the Duffing equation obtained from the slingshot equation of motion (and defined by Eqs. (2) and (3)) can be plotted as a function of x. The nature of that force depends on whether or not the springs are initially stretched (L > L0) or initially compressed (L < L0). The figure shows the elastic restoring force on the mass m as a function of the displacement x for four different cases:

— Force for slingshot with L = 2 and L0 = 1.

— Force for slingshot with L = 1 and L0 = 2.

— Force for the Duffing equation with L = 2 and L0 = 1.

— Force for the Duffing equation with L = 1 and L0 = 2.

For small x, the force given by the Duffing equation is a good approximation to the force in the slingshot. Explain which of the curves corresponds to a “hardening” spring (a spring that gets stiffer as you pull it) and which corresponds to a “softening” spring (a spring that gets less stiff as you pull it), and explain physically why we see this behavior.

Note: Concept problems are about explanations, not computations.

Figure P3.38

Figure P3.38