D’Alembert’s Principle of Equating Forces

I understand your confusion (at least I think I do). It looks like it's just Newton's law rewritten, right?

F=ma (Newton's Second Law)
F-ma=0 (D'Alembert's Principle)

So what is gained by adding (-ma) to each side to get the new formulation of Newton's Second Law?

Well, in the new formulation you can view ma as an inertial force (see below)

F-ma=0
F_applied - F_inertial = 0

Why is that usefull?

Well it reconciles the static and dynamic reference frames of the same problem. Allow me to explain (Here's where my explanation came from)

Imagine you are driving a car. You slam on the gas and the car moves much faster, as you do you feel yourself sink into your seat. Basically the car accelerated and the chair, attached to the car, accelerated too, but since the chair is made of softer material, as it tried to accelerate you the seat cushion gave a little so you sunk into the seat. Eventually you are brought up to speed (once you let off the gas) and you are no longer sinking into your seat.

That's Newton's Seconds Law way of looking at it.

Imagine now that the car had black windows so you couldn't see out of it, and the gas was controlled by someone else so that you were just sitting there. When the car accelerated, you wouldn't see anything, to you, you're in a black box, the only thing you'll notice is you sink into your seat. It feels like someone is pushing you into the seat. The force pushing you into the seat is the inertial force we described above. This situation is described as static (no motion) whereas the earlier one was kinematic. Both are equally correct and equivalent, it's just that they are looking at the same problem from different reference frames (stationary and accelerating with the car).

So there's an example, there are many more if you look around. The point here again that in reformulating Newton's Second Law, D'Alembert demonstrated the equivalence of the Static and Dynamic reference frames of a problem.

At least, that's how I understand it. Any corrections to any mistakes I made above would be most welcome.

Roger

Here's an example (scroll down to find it) that might help. The language in this link is weird, but?

http://kr.cs.ait.ac.th/~radok/physics/c1.htm