Kinetics of Particles

Kinetics studies the relationship between motion and forces. Primarily we apply Newton’s Second Law to derive equations for this relationship.

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Equations of Motion

$\sum \vec{F}=m\vec{a}$
In rectangular coordinates, this can be split into three scalar equations:
$\sum F_x=m{a}_x\sum F_y=m{a}_y\sum F_z=m{a}_z$

Curvilinear Motion and Force

We can apply $\sum F=ma$ component by component for a particle in curvilinear motion using any coordinate system.

Coordinates	Equations of motion	Acceleration
Rectangular	$\sum F_x=m{a}_x$ $\sum F_y=m{a}_y$	$a_x=\ddot{x}$ $a_y=\ddot{y}$
Normal and tangential	$\sum F_n=m{a}_n$ $\sum F_t=m{a}_t$	$a_n=\rho {\dot{\beta }}^2=\frac{v^2}{\rho }=v\dot{\beta }$ $a_t=\dot{v}v=\rho \dot{\beta }$
Polar	$\sum F_r=m{a}_r$ $\sum F_{\theta }=m{a}_{\theta }$	$a_r=\ddot{r}-r{\dot{\theta }}^2$ $a_{\theta }=r\ddot{\theta }+2\dot{r}\dot{\theta }$

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Primary Analytical Methods

Method	Relates	Used when	Note
Force-acceleration $\sum F=ma$	Forces ↔ acceleration at an instant	Need forces or $a$, motion is constrained	Newton’s Second Law
Work-energy $U_{1-2}=\Delta T$	Forces ↔ speed over a displacement	Force depends on position, time not needed	Work-Energy Principle
Impulse-momentum $G_1+\int \sum Fdt=G_2$	Forces ↔ velocity over time	Force depends on time, or an impact or collision occurs	Linear Momentum, Impact
Angular impulse-momentum $\int \sum M_{O}dt=\Delta H_O$	Moments ↔ rotation about a point	Central force (moment about $O$ is zero)	Angular Momentum

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