Kinetics of Rigid Bodies

In planar kinetics, a rigid body’s movement in a two-dimensional plane is fully described by balancing external forces and moments against the body’s mass and rotational inertia.
$\sum \vec{F}=m{\vec{a}}_G\sum M_G=I_G\alpha$

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$G$ — Centre of Mass

Applying a force or moment over a displacement does work that transfers into kinetic energy. Applying a force or moment over a period of time changes the body’s linear momentum and angular momentum.

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Translational Kinetics

A body in translation has zero angular velocity and zero angular acceleration. Every particle in the body moves along a parallel path.
$\omega =\alpha =0$

Rectilinear vs Curvilinear Translation

Rectilinear Translation — $G$ moves in a straight line and $a_G$ has one component. See rectangular coordinates.

$\sum {\vec{F}}_x=m{\vec{a}}_{G_x}\sum {\vec{F}}_y=m{\vec{a}}_{G_y}\sum M_G=0$

Curvilinear Translation — $G$ moves along a curved path, but the body does not rotate. See n-t coordinates.

$\sum {\vec{F}}_n=m{\vec{a}}_{G_n}=m\frac{{\mathbf{v}}_G^2}{\rho }\sum {\vec{F}}_t=m{\vec{a}}_{G_t}=m{\dot{\mathbf{v}}}_G\sum M_G=0$

Even though the body doesn’t rotate, taking moments around $G$ allows us to solve for multiple reaction forces individually, as summing forces only gives us their total.

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Rotational Kinetics

A body rotates about a stationary point/axis known as $O$ (see fixed axis rotation). $O$ may or may not coincide with the body’s centre of mass $G$.

The net external torque about a fixed axis causes angular acceleration ($\alpha$), opposed by the body’s rotational inertia ($I$).
$\sum M_O=I_O\alpha$

Eccentricity

Eccentricity ($e$ or ${\vec{r}}_G$) is known as the linear offset distance between the geometric centre/pivot point $O$ and the centre of mass $G$ ($e=|{\vec{r}}_G|$).

If axis of rotation $O\ne G$, force components at pivot must also satisfy:

$\sum F_n=m{a}_{G_n}=me{\omega }^2\sum F_t=m{a}_{G_t}=me\alpha$

Applying torque over a rotational displacement $(\Delta \theta )$ does work that transfers into rotational kinetic energy, while applying torque over time $(\Delta t)$ changes the body’s angular momentum.

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General Plane Motion for Rigid Bodies

General Plane Motion describes a body undergoing both a translation and rotation. We combine our equations for translation and rotation about $G$.
$\sum {\vec{F}}_x=m{\vec{a}}_{G_x}\sum {\vec{F}}_y=m{\vec{a}}_{G_y}\sum M_G=I_G\alpha$

Rolling Without Slipping

When a body rolls without slipping, the kinematic constraint $a_G=r\alpha$ links the translational and rotational equations, reducing the unknowns needed to solve the system.

Choosing to Sum Moments About $P$

Moments can be summed about any point $P$, but unless $P$ is a fixed axis of rotation or the instantaneous centre of zero velocity/acceleration, you must include the moment of the effective force $m{\vec{a}}_G$ acting at $G$:
$\sum M_P=I_G\alpha \pm m{\vec{a}}_{G}d$

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$d$ is the perpendicular distance from $P$ to line of action of $m{\vec{a}}_G$.

The sign ($\pm$) of the $m{\vec{a}}_{G}d$ term depends on the sense of the moment created by $m{\vec{a}}_G$ about point $P$. If the effective force tends to rotate the body in the positive coordinate direction, it is positive; otherwise, it is negative.

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