Final answer:
The potential energy in a horizontal mass-on-a-spring system is largest at the turning points, where the spring is either fully compressed or fully extended (maximum displacement from equilibrium), and where the mass has zero velocity.
Step-by-step explanation:
In a horizontal mass-on-a-spring system, the potential energy (U) is largest when the spring is at its maximum displacement from the equilibrium position, which occurs at the points of maximum compression or extension (turning points). This is because the potential energy of a spring is given by U = 1/2kx², where k is the spring constant and x is the displacement from equilibrium.
At the turning points, labeled as x = ±A (where A is the amplitude of the oscillation), the speed of the oscillator is zero and all its energy is stored as potential energy. Therefore, the potential energy is maximum at the positions x = ±A, and it can be calculated as U = kA²/2. No kinetic energy is present at these points, as the velocity of the mass is zero.