Since we are not sure what’s going to happen, we let the frequency possibly be different, but we assume the frequency is not a function of time. (You can check out the details at Damped Oscillators – The Math (Technical) ) But the idea is straightforward: we assume that it will still oscillate, so we still have a cosine term, but now we let the amplitude become a function of t. This small change – the extra v term – makes the math a lot messier. So net force on the mass and vane system is Recall our discussion of drag from 1st semester physics, drag is proportional to the velocity. We will assume the amplitude of the oscillations are small and we don’t induce any turbulence in the liquid (the velocity is not too high). This could be the mass itself, or, as shown in the figure at the right, something attached to the mass that is moving in the resistive medium, in this case a vane (something with a lot of surface area) in a liquid. A simple model of a damped oscillator is a hanging mass on a spring attached to something that is moving through a resistive medium. Overdamping, and critical dampling (where there are few or no oscillations) are important for engineering in the macroscopic world, but are not terribly relevant for biology. We will focus on the case of small damping here, also called underdamping. Oscillators are also damped on the molecular level, though the damping is usually small. This is called a Damped Oscillator and as you might guess, it is due to some kind of dissipative force such as friction, air resistance or fluid drag. If you pull a mass on a spring and let go, eventually the mass will stop moving, with the amplitudes of the oscillations getting continually smaller. As you have probably noticed in everyday life, oscillators don’t oscillate forever.
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