Question

The classical turning points of a harmonic oscillator occur at the displacements at which all of the energy is potential energy. When is this point reached? A) At the maximum displacement B) At the minimum displacement C) At the equilibrium position D) At half the maximum displacement

Answer 1

Choice A: At the maximum displacement.

Step-by-step explanation:

In a harmonic oscillator, the potential energy
`U` of the system is proportional to the square of displacement
`x` from the equilibrium position:

`\displaystyle U = (1)/(2)\, k\, x^(2)`,

Where the value of
`k` does not depend on the position of the oscillator. The value of
`k` is fixed for the particular oscillator. For example, for an object attached to a horizontal spring on a horizontal frictionless surface,
`k\!` would be equal to the spring constant.

In a harmonic oscillator, the total mechanical energy of the system (sum of potential and kinetic energy) is fixed. If all the mechanical energy of this oscillator is in the form of potential energy, the potential energy of this system would be maximized. Because potential energy is proportional to the square of displacement
`x^(2)`, displacement
`x` also needs to be maximized.

In other words, when all the mechanical energy of this harmonic oscillator is in the form of potential energy, the displacement of the system from equilibrium also needs to be maximized.