Particle Kinetics: Equation of Motion for a Particle and a System of Particles

Newton's Laws of Motion

The motion of a particle is governed by Newton's 3 Laws of Motion:

  1. A particle originally at rest, or moving in a straight line at a constant velocity, will remain in this state if the resultant force acting on the particle is zero
  1. If the resultant force on the particle is not zero, the particle experiences an acceleration in the same direction as the resultant force. This acceleration has a magnitude proportional to the resultant force.
  1. Mutual forces of action and reaction between two particles are equal, opposite, and collinear

The 1st and 3rd laws stated were used in developing the concept of statics. Newton's 2nd law forms the basis for the study of dynamics.

Mathematically, Newton's 2nd law of motion can be written: F = ma

Where F is the resultant unbalanced force acting upon the particle, and a is the acceleration of the particle. The positive scalar m is called the mass of the particle

Mass and Weight

It is very important to understand the difference between mass and weight of a body

  1. Mass is an absolute property of a body. It is independent of the gravitational field in which it is measured. The mass provides a measure of the resistance of a body to a change in velocity, as defined by Newton's 2nd law of motion (m = F/a
  1. The weight of a body is not absolute, since it depends on the gravitational field in which it is measured. Weight is defined as: W = mg where g is the acceleration due to gravity

Kinetics: Equation of Motion

The motion of a particle is governed by Newton's second law, relating the unbalanced forces on a particle to its acceleration. If more than one force acts on the particle, the equation of motion can be written as:
ΣF = F(resultant) = ma
Where F(resultant) is the resultant force, which is a vector summation of all the forces.

This can be illustrated by considering a particle acted on by two forces:

  1. Draw the particle's free-body-diagram, shwoing all forces acting on the particle.

image

  1. Draw the kinetic diagram showing the inertial force ma acting in the same direction as the resultant force F(resultant)

image

Kinetics: Inertial Frame of Reference

The equation model shown above is only valid if the acceleration is measured in a Newtonian or inertial frame of reference. What does this mean?

A Newtonian (inertial) frame of reference does not rotate and is either fixed or translates in a given direction with constant velocity (i.e. zero acceleration)

For problems concerned with motions at or near the Earth's surface, we typically assume out "inertial frame" to be fixed to the Earth. We neglect any acceleration effects from the Earth's rotation.

For problems involving satellites or rockets for example, the inertial frame of reference is often fixed to the stars

Equation of Motion For a System of Particles

The equation of motion can be extended to include systems of particles. This includes the motion of solids, liquids, or gas systems

As in statics, there are internal forces and external forces acting on the system. What is the difference between them?

Using the definitions of m = Σm(subscript i) as the total mass of all particles and a(subscript G) as the acceleration of the center of mass G of the particles, then:
ma(subscript G) = Σm(subscript i)a(subscript i)

For a system of particles: ΣF = ma(subscript G) where ΣF is the sum of the external forces acting on the entire system

Unbalanced forces cause the acceleration of objects. THIS CONDITION IS FUNDAMENTAL TO ALL DYNAMICS PROBLEMS

Procedure For The Application of The Equation of Motion:

  1. Select a convenient inertial coordinate system. Rectangular/Cartesian, Normal/Tangential, or Cylindrical (polar) coordinates may be used.
  1. Draw a Free-Body-Diagram showing all external forces applied to the particle. Resolve force into their appropriate components, consistent with the reference frame sign convention. Friction (µ(subscript k)) opposes motion. A spring for example can be modeled F = ks where k is stiffness (N/m).
  1. Draw the kinetic diagram, showing the particle's inertial force, ma. Resolve this vector into its appropriate components, consistent with the reference frame sign convention.
  1. Apply the Equations of Motion in their scalar component form and solve these equations for the unknowns.
  1. It may be necessary to apply the proper kinematic relations to generate additional equations (relative velocity analysis, velocity, and position via integration of acceleration etc.)