Kinetics of Particles: Force and Acceleration

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

∑F=ma\sum \mathbf{F} = m\mathbf{a}
Where:
  • ∑F\sum \mathbf{F} is the vector sum of all external forces acting on the particle (the resultant force).
  • mm is the mass of the particle.
  • a\mathbf{a} is the acceleration vector of the particle.
Units:
  • SI: Force in Newtons (N), Mass in kilograms (kg), Acceleration in m/s². (1 N=1 kg⋅m/s21 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2)
  • US Customary: Force in pounds (lb), Mass in slugs, Acceleration in ft/s². (1 slug=1 lb/(32.2 ft/s2)1 \text{ slug} = 1 \text{ lb} / (32.2 \text{ ft/s}^2))

Important

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

∑Fx=max\sum F_x = ma_x∑Fy=may\sum F_y = ma_y∑Fz=maz\sum F_z = ma_z

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).

Orbital Mechanics

The most common central force is gravity. According to Newton's Law of Universal Gravitation, the force FF between two masses MM (e.g., Earth) and mm (e.g., a satellite) separated by a distance rr is:
F=GMmr2F = G \frac{M m}{r^2}
where GG is the universal gravitational constant.
Because the force is entirely radial (central), there is no force in the transverse (θ\theta) direction:
∑Fθ=maθ=0  ⟹  aθ=rθ¨+2r˙θ˙=1rddt(r2θ˙)=0\sum F_\theta = m a_\theta = 0 \implies a_\theta = r \ddot{\theta} + 2\dot{r}\dot{\theta} = \frac{1}{r} \frac{d}{dt}(r^2 \dot{\theta}) = 0
This leads to the conservation of angular momentum:
r2θ˙=h=constantr^2 \dot{\theta} = h = \text{constant}
where hh is the angular momentum per unit mass. This forms the basis for Kepler's Second Law (equal areas in equal times).
Interact with the simulation below to explore orbital mechanics principles.

Orbital Mechanics: Conic Sections

Orbit Type: Elliptical

Polar Equation of Orbit:

r=a(1−e2)1+ecos⁡θr = \frac{a(1 - e^2)}{1 + e \cos \theta}

Normal and Tangential Coordinates

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

Equations of Motion (n-t)

∑Ft=mat=mdvdt\sum F_t = m a_t = m \frac{dv}{dt}∑Fn=man=mv2ρ\sum F_n = m a_n = m \frac{v^2}{\rho}
Where:
  • ∑Ft\sum F_t is the sum of forces in the tangential direction. It causes a change in the particle's speed.
  • ∑Fn\sum F_n is the sum of forces in the normal direction. It causes a change in the particle's direction of motion. This force is often referred to as the centripetal force.
  • ρ\rho is the radius of curvature.
Key Takeaways
  • Newton's Second Law (∑F=ma\sum \mathbf{F} = m\mathbf{a}) is a vector equation that must be applied using Free-Body and Kinematic Diagrams.
  • Mass vs. Weight: Mass (mm) is a property of matter, while weight (W=mgW=mg) is the force of gravity on that mass.
  • Rectangular Coordinates (∑Fx=max\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 (∑Ft=mv˙\sum F_t = m\dot{v}, ∑Fn=mv2/ρ\sum F_n = mv^2/\rho) are used when the path of motion is known.
  • The normal force (∑Fn\sum F_n) is responsible for changing direction, while the tangential force (∑Ft\sum F_t) is responsible for changing speed.
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