Serway & Jewett — Energy of a System: Problem 50.

A particle of mass \( m = 1.18 \text{ kg} \) is attached between two identical springs on a frictionless, horizontal tabletop. Both springs have spring constant \( k \) and are initially unstressed, and the particle is at \( x = 0 \). (a) The particle is pulled a distance \( x \) along a direction perpendicular to the initial configuration of the springs as shown in Figure P7.50. Show that the force exerted by the springs on the particle is \(\vec{F} = -2kx \left( 1 - \frac{L}{\sqrt{x^2 + L^2}} \right) \hat{i}\). (b) Show that the potential energy of the system is \( U(x) = kx^2 + 2kL(L - \sqrt{x^2 + L^2}) \). (c) Make a plot of \( U(x) \) versus \( x \) and identify all equilibrium points. Assume \( L = 1.20 \text{ m} \) and \( k = 40.0 \text{ N/m} \). (d) If the particle is pulled \( 0.500 \text{ m} \) to the right and then released, what is its speed when it reaches \( x = 0 \)?