Kinetics of Rigid Bodies:

For the bodies undergoing plane motion, a common scheme for solutions is to apply the equations of dynamic equilibrium as below.

∑ F_x  = m a_x ,         ∑ F_y  = m a _y

∑ M = I α

Keeping in mind that a_x, a_y and I are calculated with reference to centroidal axes. Inertia forces and inertia couple should be applied to the body before writing down these equilibrium equations. The relation between a_x and α is normally obtained by using kinematic relationships.

When motion of rotation takes place about an axis which is not coinciding with centroidal axis, it is convenient to select x and y axis through the mass centre which are positive in the direction of a_n and a_r, respectively. Also ∑ M _A  - I _A  α = 0 is the equation to be utilized which normally gets rid of the unknown reaction at the axle A. Then these equations get transformed into

∑ F_x = mr ω²

∑ F_y = mr α

∑ M = I α; ∑ M _A = I _A α

In case of centroidal rotation A coincides with mass centre, r = 0 and the above equations reduce to

∑ F_x  = 0,         ∑ F_y  = 0,           ∑ M = Iα

For homogeneous rolling bodies in which mass centre has a rectilinear motion parallel to the surface on which it rolls it is suitable to select reference axes at the mass centre with the x axis parallel to the surface and positive in the initial direction of motion. The equations become

∑ F_x = m a, ∑ F_y = 0, ∑ M = I_α

If the body rolls without slipping, an additional equation can be used

∑ M _c = I _c α