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 ω^{2}
∑ 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