The Physics Philes, lesson 85: Losing My Religion Mechanical Energy

Last week we started learning about what happens when oscillations lose energy and slow to a stop. We call this damping. This week we’ll focuses specifically on the mechanical energy in these damped oscillations.

It’s probably obvious, but in oscillating systems we’re dealing with mechanical energy. In the real world, this mechanical energy will gradually melt away. If someone or something doesn’t replenish the mechanical energy in the oscillating system, the oscillation will eventually stop. In short, the damping force is nonconservative.

How fast does the mechanical energy dissipate? We can derive an equation for the rate of change of energy from the total mechanical energy equation:

We can find the rate of change of anything by taking the derivative. In this case, to find the rate of change over time, we’ll take the derivative of the mechanical energy equation with respect to time. It looks like this:

The dv/dt and dx/dt indicate derivatives v with respect to t and x with respect to t. Respectively. Notice, too, that dv/dt and dx/dt are equal to some other things we’ve looked at recently: acceleration and velocity. We can use this knowledge to make our equation look less scary and fraction-y:

That’s already looking much more manageable. But we’re not done. Remember Newton’s second law of motion for damped oscillating systems? It’s -kx – bv = ma. We can make some more substitutions to get the final equation:

This equation is the rate of change of mechanical energy in damped oscillating systems!

That’s great and all, but what does it mean? The right side of the equation will be negative if the body is oscillating, regardless of whether it’s velocity is positive or negative. This indicates that the system is losing energy. The product -bv^2 is the damping power; the rate at which force does negative work.

That’s it for now. There are only two more posts about oscillation in the works. Don’t miss it!

Featured image credit: Shannon Kokoska