Final answer:
The magnitude of the force on a particle moving under the influence of a spring force is maximized when the particle is at a position farthest from equilibrium, which occurs when x is maximum or minimum.
Step-by-step explanation:
The potential energy of a particle moving under the influence of a spring force can be given by the equation U(x) = kx² + C, where k is the spring constant and C is a constant. The force on the particle is related to the potential energy through the equation F(x) = -dU/dx, where dU/dx is the derivative of the potential energy with respect to position. In this case, the derivative of U(x) with respect to x is 2kx, so the force on the particle is F(x) = -2kx.
At any given position x, the magnitude of the force is given by the absolute value of F(x), which is equal to 2kx. Therefore, the magnitude of the force is maximized when the particle is at a position farthest from equilibrium, which occurs when x is maximum or minimum. When x = 2.0 m, 5.0 m, 8.0 m, and 12 m, the magnitude of the force is approximately 4k, 10k, 16k, and 24k respectively.