Answer to Question #89122 in Mechanics | Relativity for Shivam Nishad

Question

A periodic force applied on a weakly damped
oscillator produces sustained oscillations. What happens to the energy supplied ?

Answer 1

The energy of damped oscillations is

$$E(t)=\frac{1}{2}m\Big(\frac{\text{d}x}{\text{d}t}\Big)^2+\frac{1}{2}kx^2,$$

it consists of kinetic energy and potential energy. As we see, this energy (energy of the oscillator) decreases with time because of damping. If we apply an external periodic force on a weakly damped oscillator, our expression will be transformed to

$$E(t)=\frac{1}{2}m\Big(\frac{\text{d}x}{\text{d}t}\Big)^2+\frac{1}{2}kx^2+Q(t),$$

where $Q(t)$ - heat lost to overcome damping.

Imagine a damped oscillator - a car shock absorber with a coil spring: when a car hits a bump, it oscillates a bit and then stops oscillating. The oscillator had both kinetic and potential energy and some heat produced we neglect.

Imagine another situation: someone periodically pushes a car hood down to oscillate the car. The absorbers with the springs have not only kinetic but potential energy as well as there is sufficient heat produced, because the energy of a person's hands must produce spring's compression (potential energy), motion (kinetic energy) and oil or gas flows in the shock absorber (heat, friction).