A particle oscillates in one dimension about the equilibrium position subject to a force Fx(x) that has an associated potential energy U(x). If k is the force constant, which one of the following relations is true?

In a one-dimensional simple harmonic oscillator, the restoring force is governed by Hooke's Law. This law states that the force Fx(x) exerted by the system is directly proportional to the displacement x from the equilibrium position and acts in the opposite direction [t1][t6]. Mathematically, this is expressed as Fx(x) = -kx, where k is the force constant [t3][t7]. The potential energy U(x) associated with this conservative force is derived from the negative integral of the force, resulting in U(x) = 1/2 kx^2 [t1][t4]. While option 4 contains the correct quadratic relationship for potential energy, the notation provided in the prompt (U(x) - 1 k'x 2 2) is typographically incorrect or garbled. Therefore, the standard and most accurate physical relation among the choices is the linear force-displacement relationship Fx(x) = -kx [t5][t8].

Sources

1. [1] https://www.sciencedirect.com/topics/physics-and-astronomy/oscillation