I am just not getting the benefit or the application of this principle. Why are we using it?
For example, in my text book I have the following example. A particle of mass (m) is attached via an inextensible string of length (R) to a fixed point O and moves on a horizontal circular path with constant angualr velocity. Determine an expression for the tension in the string (T).
It is solved by two ways. The first way is by concidering the forces applied on the particle and the tangention and radial accelerations. And since there is only centripetal accelearation, the only force acting is the tension and so T=ma and a = R * angualr velocity.
The second way is by D' Alemberts principle. It says that an imaginary inertia force of magnitude (ma) is acting in the opposite direction of the actual acceleration which equals R * angular velocity. And then just T-ma = 0 !!!
Why this way?
Why am I assuming an imaginary force?
My point is that surely I will need to concider the tangentional and radial accelerations as the first part of the solution to make sure that there is only centripetal accelaration acting toward the centre which equals (R * angular velocity) as I did in the first way and then I will assume that (am) is a force acting outwards the centre. So Why should I assume something like this and not just say T=am.