Kinetics of Particles: Force and Acceleration - Theory & Concepts

Kinetics of Particles: Force and Acceleration

Learning Objectives

Understand and apply Newton's Second Law of Motion to particles.
Formulate equations of motion in rectangular coordinates.
Analyze central force motion and orbital mechanics using radial and transverse coordinates.
Formulate equations of motion using normal and tangential coordinates.

Kinetics is the study of the relation between the forces acting on a body and the resulting motion. It forms the basis of rigid body dynamics. The fundamental principle of kinetics is Isaac Newton's Second Law of Motion.

Newton's Second Law of Motion

Newton's Second Law states that if an unbalanced force acts on a particle, the particle will accelerate in the direction of the force with a magnitude directly proportional to the force.

The Equation of Motion

Newton's Second Law expressed as a vector equation.

\sum \mathbf{F} = m\mathbf{a}

Variables

Symbol	Description	Unit
$\sum \mathbf{F}$	Vector sum of all external forces (resultant force)	N or lb
$m$	Mass of the particle	kg or slug
$\mathbf{a}$	Acceleration vector	$m/s^2 or ft/s^2$

Units of Motion

SI: Force in Newtons (N), Mass in kilograms (kg), Acceleration in m/s². ( $1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2$ )
US Customary: Force in pounds (lb), Mass in slugs, Acceleration in ft/s². ( $1 \text{ slug} = 1 \text{ lb} / (32.2 \text{ ft/s}^2)$ )

Free-Body and Kinematic Diagrams

The equation of motion, $\sum \mathbf{F} = m\mathbf{a}$ , must be applied using a Free-Body Diagram (FBD) to correctly identify all forces ( $\sum \mathbf{F}$ ) and a Kinematic Diagram (KD) to correctly identify the components of acceleration ( $m\mathbf{a}$ ).

Rectangular Coordinates

When a particle moves in a Cartesian coordinate system, the vector equation of motion can be resolved into three independent scalar equations:

Scalar Equations of Motion

Newton's Second Law resolved into Cartesian coordinates.

\begin{aligned} \sum F_x &= ma_x \\ \sum F_y &= ma_y \\ \sum F_z &= ma_z \end{aligned}

Variables

Symbol	Description	Unit
$\sum F_x, \sum F_y, \sum F_z$	Sum of forces in x, y, and z directions	N or lb
$m$	Mass of the particle	kg or slug
$a_x, a_y, a_z$	Acceleration in x, y, and z directions	$m/s^2 or ft/s^2$

Central Force Motion and Orbital Mechanics

A crucial application of particle kinetics in radial and transverse coordinates is central force motion, where the only force acting on a particle is directed towards or away from a fixed point (the center of force). The most common central force is gravity.

Newton's Law of Universal Gravitation

The gravitational force between two masses.

F = G \frac{M m}{r^2}

Variables

Symbol	Description	Unit
$F$	Gravitational force	N
$G$	Universal gravitational constant	$m^3/(kg \cdot s^2)$
$M$	Mass of the larger body (e.g., Earth)	kg
$m$	Mass of the smaller body (e.g., satellite)	kg
$r$	Distance between the centers of the two masses	m

Because the force is entirely radial (central), there is no force in the transverse ( $\theta$ ) direction. This means $\sum F_\theta = m a_\theta = 0$ .

Zero Transverse Force

The consequence of a purely central force.

a_\theta = r \ddot{\theta} + 2\dot{r}\dot{\theta} = \frac{1}{r} \frac{d}{dt}(r^2 \dot{\theta}) = 0

Variables

Symbol	Description	Unit
$a_\theta$	Transverse acceleration	$m/s^2$
$r$	Radial distance	m
$\ddot{\theta}$	Angular acceleration	$rad/s^2$
$\dot{r}$	Radial velocity	m/s
$\dot{\theta}$	Angular velocity	rad/s

The equation above leads to the conservation of angular momentum, which forms the basis for Kepler's Second Law (equal areas in equal times).

Conservation of Angular Momentum

Angular momentum per unit mass remains constant in central force motion.

r^2 \dot{\theta} = h = \text{constant}

Variables

Symbol	Description	Unit
$r$	Radial distance	m
$\dot{\theta}$	Angular velocity	rad/s
$h$	Angular momentum per unit mass	$m^2/s$

Interactive Simulation

Interact with the simulation below to explore orbital mechanics principles.

Normal and Tangential Coordinates

When a particle moves along a known curved path, it is often best to use normal ( $n$ ) and tangential ( $t$ ) coordinates.

Equations of Motion (n-t)

Newton's Second Law resolved into normal and tangential coordinates.

\begin{aligned} \sum F_t &= m a_t = m \frac{dv}{dt} \\ \sum F_n &= m a_n = m \frac{v^2}{\rho} \end{aligned}

Variables

Symbol	Description	Unit
$\sum F_t$	Sum of forces in the tangential direction. Causes a change in speed.	N or lb
$\sum F_n$	Sum of forces in the normal direction (centripetal force). Causes a change in direction.	N or lb
$m$	Mass of the particle	kg or slug
$a_t$	Tangential acceleration	$m/s^2 or ft/s^2$
$a_n$	Normal acceleration	$m/s^2 or ft/s^2$
$v$	Velocity	m/s or ft/s
$\rho$	Radius of curvature	m or ft

Key Takeaways

Newton's Second Law ( $\sum \mathbf{F} = m\mathbf{a}$ ) is a vector equation that must be applied using Free-Body and Kinematic Diagrams.
Mass vs. Weight: Mass ( $m$ ) is a property of matter, while weight ( $W=mg$ ) is the force of gravity on that mass.
Rectangular Coordinates ( $\sum F_x = ma_x$ , etc.) are used when the path is a straight line or easily defined in Cartesian terms.
Normal and Tangential Coordinates ( $\sum F_t = m\dot{v}$ , $\sum F_n = mv^2/\rho$ ) are used when the path of motion is known.
The normal force ( $\sum F_n$ ) is responsible for changing direction, while the tangential force ( $\sum F_t$ ) is responsible for changing speed.

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