Oscillations and Waves > Forced and damped oscillations > Problem 3.5.30

Statement

Why do the directions of displacement and the driving force coincide when the oscillator is forced to oscillate at a frequency smaller than its natural frequency, and are opposite when its frequency is higher than its natural frequency?

Solution

Assuming that the forced oscillation dominates the free oscillation, when the particular solution is substituted into the equation of motion

we see that the relative magnitude of the two terms on the left are . If , then has the same sign as ; that is, the displacement and the forcing has the same direction. On the other hand, if , then has the same sign as , so and have opposite signs; that is, the displacement and the forcing has the opposite directions.

Answer

See the solution.