Why Use D'Alembert's Principle in Circular Motion Analysis?

[almirza]

Homework Statement

I am just not getting the benefit or the application of this principle. Why are we using it?
For example, in my textbook 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 consider 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.

Homework Equations

The Attempt at a Solution

 

[almirza]

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[Moetasim]

D'Alembert's principle is a fundamental principle in mechanics that is used to simplify the analysis of systems in motion. It states that the net sum of the external forces and the inertial forces acting on a system is equal to the mass of the system multiplied by its acceleration. This principle is based on the concept of virtual work, where the virtual work done by the inertial forces is equal to zero.

In the example given, the particle is moving in a circular path with constant angular velocity. This means that the only acceleration acting on the particle is the centripetal acceleration, directed towards the center of the circle. In this case, the only external force acting on the particle is the tension in the string. By applying D'Alembert's principle, we can treat the inertial force as an external force acting in the opposite direction of the acceleration, and solve for the tension in the string using the equation T-ma=0. This approach simplifies the analysis and allows us to solve for the tension without considering tangential and radial accelerations separately.

The use of an imaginary force may seem counterintuitive, but it is a mathematical tool that allows us to apply D'Alembert's principle and simplify the analysis of systems in motion. In this case, the imaginary force represents the inertial force, which is a reaction to the acceleration of the particle. This approach is valid as long as the acceleration is constant, and the only external force acting on the system is the tension in the string.

In conclusion, D'Alembert's principle is a useful tool for simplifying the analysis of systems in motion. It allows us to treat the inertial forces as external forces and solve for unknown quantities, such as the tension in the string in the given example, without having to consider tangential and radial accelerations separately. This approach saves time and effort in solving complex problems in mechanics.