4. Modeling the vibrations of two-dimensional solids. (a) Consider a spring that obeys Hooke's law and has one end xed at the origin, but whose other end, to which we attach a particle of mass m, is free to move in two dimensions. Write down the force F(r) exerted by the spring on the particle in terms of the spring's length 1", its equilibrium length 10, and its force constant k. Prove that this force is conservative. Consider a particle of mass m resting on a frictionless horizontal table and suspended between two identical springs of the type considered in part (a). Each spring has an undeformed length to and force constant k, and the two springs are attached along the :caxis. At equilibrium, the particle rests at the origin, with each spring having a length a. Show that when the particle moves to a position (:3, y) with :1: and 3; small, the potential energy of this system has the form of the potential energy of an anisotropic harmonic oscillator with force constants km and Icy, and nd km and ky. Show that if a