The Physics Philes, lesson 84: My Damp Collection

Last week we complicated the idealized models of pendulums by looking at pendulums as an extended body. Today we’ll further complicate things by adding some other, real-world forces.

So far, the oscillating systems we’ve analyzed have been frictionless. We’ve been dealing with only conservative forces and the total amount of mechanical energy has remained constant. This means that, once set into motion, the oscillating systems we’ve been dealing with will remain oscillating forever with no decrease in amplitude.

That, of course, isn’t how things happen IRL. Unless something or someone replaces the lost mechanical energy, the oscillation will die out. When this happens, we call it damping. The motion that goes along with this is called damped oscillation. Even though this is more complicated than the more idealized models we’ve been using, we can still describe this motion using a simple harmonic oscillator, but in this case it will have a frictional damping force that is directly proportional of the velocity of the oscillating body. (This type of situation occurs in a vehicle’s shock absorbers.)

In this situation, we have to add an additional force, a frictional force. It’s the velocity v times a constant that describes the strength of the damping force:

Newton’s second law for system, then, looks like this:

If the damping force is small enough, the motion can be described as this slightly scary-looking equation:

The angular frequency in this same situation can be expressed as

There are two ways damped oscillation in this case differs from undamped oscillation. First, the amplitude in this damped case isn’t constant. It decreases by the exponential factor of e^-(b/2m)t. Second, the angular frequency isn’t equal to ω = √k/m; it’s a little bit smaller. It actually becomes zero when

When this equation is satisfied, we call the condition critical damping. This means that the pendulum or other oscillating system doesn’t oscillate when it’s moved from it’s equilibrium position. It just returns to it’s equilibrium position.

So what happens when b is more or less than 2√km? If it’s greater, we call that overdamping. In that case, there is also no oscillation when the system undergoes displacement, but it happens more slowly than in in critical damping. When b is less than 2√km it’s called underdamping. In this case the system does oscillate, but the amplitude decreases steadily.

Those are the basics of damped oscillation. Now that we have that background, next week we can move on to an exploration of the energy in damped oscillations. Should be fun!

Featured image credit: Wikimedia Commons