Equations of motion for plane motion

For a rigid body of mass $m$ in plane motion with mass centre G, acceleration $\mathbf{a}_G$ and angular acceleration $\alpha$, acted on by external forces and couples in the plane, the equations of motion (Newton-Euler equations) are:

$$\Sigma F_x = m\,a_{Gx} \qquad \Sigma F_y = m\,a_{Gy} \qquad \Sigma M_G = I_G\,\alpha$$

The first two give the motion of the mass centre (it moves as if all the mass and forces were at G); the third gives the rotation about G. $I_G$ is the mass moment of inertia about the axis through G perpendicular to the plane. Alternatively, the inertia force $-m\mathbf{a}_G$ (through G) and the inertia couple $-I_G\alpha$ can be added and the body treated as being in static equilibrium (D'Alembert's principle).

(a) Translation

$\alpha = 0$, so $\Sigma M_G = 0$ and the resultant force passes through G: $\Sigma F = m\mathbf{a}_G$. Example: a crate sliding on a surface (check tipping with $\Sigma M_G = 0$).

(b) Rotation about a fixed axis O

G moves in a circle, with $a_{Gt} = \alpha\bar r$ and $a_{Gn} = \omega^2\bar r$. Taking moments about O:

$$\Sigma M_O = I_O\,\alpha \qquad I_O = I_G + m\bar r^2$$

The reactions at O follow from $\Sigma F_t = m\bar r\alpha$ and $\Sigma F_n = m\bar r\omega^2$.

(c) General plane motion

Use the three equations with $a_G$ and $\alpha$ unknown, plus kinematic relations (for example, rolling without slipping, $a_G = \alpha r$). Moments may be taken about any point P if the moment of $m\mathbf{a}_G$ is included:

$$\Sigma M_P = I_G\alpha + (\text{moment of } m\mathbf{a}_G \text{ about P})$$

For a body rolling without slipping on a fixed surface, taking P as the contact point C gives $\Sigma M_C = I_C\alpha$ only when G moves parallel to the surface (such as a homogeneous round body on a plane).

Procedure

1. Draw the free-body diagram with all forces and the unknown reactions.
2. Show $m\mathbf{a}_G$ and $I_G\alpha$ in a kinetic diagram.
3. Write the three equations and add the kinematic constraint equations.
4. Solve for the unknowns.