Equilibrium equations:

The particle A of mass dm will have two components of inertia force due to motion of translation. These components are - ax dm and - a y dm, which are shown in Figure .Similarly, due to rotation about an axis through C there will be tangential inertial force - r α dm and normal inertia force - r ω² dm which are also marked on Figure .

Now, in accordance with D′Alemberts principle, the body is in equilibrium under the action of external forces and inertia forces applied for all the particles in the body. We can write down the following equilibrium equations for this condition of the body :

∑ F_x = 0

X - a_x ∫ dm - α ∫ r sin θ dm + ω² ∫ r cos θ dm = 0

∑ Fy = 0

 Y - α _x∫ dm - α ∫ r cos θ dm+ ω² ∫ r sin θ dm = 0

and ∑ M  = 0

M c - a y ∫ r cos θ dm + a_x ∫ r sin θ dm - α ∫ r² dm = 0

where X, Y and Mc represent the applied external forces.

But as C is the mass centre, we have

∫ r sin θ dm = 0;     ∫  r cos θ dm = 0

Therefore, all of these equations become very simple as

X = m a_x , Y = m a _y  and M _c  = I _c  α .

From the above equations, we can make very significant observations.

In the case of rigid body performing plane motion under the action of external forces X, Y and M_c, the mass centre of the body moves accurately as if the whole mass were concentrated there and acted upon by forces X and Y. The moment Mc of external forces w. r. t. mass centre has no effect on the motion of the point. At the same time, the body rotates around the axis through the moving centre of gravity as if this were a fixed axis.