D’Alembert’s Principle Derivation

D’Alembert’s Principle:

Consider a rigid body acted upon by a system of forces. This system may be reduced to a single resultant force acting on the body whose magnitude is given by the product of the mass of the body. The linear acceleration of the centre of mass of the body. According to Newton’s second law of motion,

F = m.a ….equ(i)

F = Resultant force acting on the body

m = Mass of the body

a = Linear acceleration of the centre of mass of the body

The equation (i) may also be written as:

F -m.a = 0 ….equ(ii)

A little consideration will show that if the quantity -m.a be treated as a force, equal, opposite and with the same line of action as the resultant force F. It includes this force with the system of forces of which F is the resultant, then the complete system of forces will be in equilibrium. This principle is known as D’Alembert’s Principle.

The equal and opposite force -m.a is known as reverted effective force or the inertia force. The equation (i) and (ii) may be written as:

F + F1 = 0

Thus, D’Alembert’s Principle states that the resultant force acting on a body together with the reversed effective force are in equilibrium. This principle is used to reduce a dynamic problem into an equivalent static problem.