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.
∑ Fx = m ax , ∑ Fy = m a y
∑ M = I α
Keeping in mind that ax, ay 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 ax 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 an and ar, 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
∑ Fx = mr ω2
∑ Fy = 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
∑ Fx = 0, ∑ Fy = 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
∑ Fx = m a, ∑ Fy = 0, ∑ M = Iα
If the body rolls without slipping, an additional equation can be used
∑ M c = I c α