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Particle Kinetics: Equation of Motion for a Particle and a System of…
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
:
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
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
.
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
= m
a
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
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
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:
Σ
F
=
F(resultant)
= m
a
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:
Draw the particle's
free-body-diagram
, shwoing all forces acting on the particle.
Draw the
kinetic diagram
showing the
inertial force
m
a
acting in the same direction as the resultant force
F(resultant)
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:
m
a(subscript G)
= Σm(subscript i)
a(subscript i)
For a system of particles: Σ
F
= m
a(subscript G)
where Σ
F
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:
Select a convenient
inertial coordinate system
. Rectangular/Cartesian, Normal/Tangential, or Cylindrical (polar) coordinates may be used.
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).
Draw the
kinetic diagram
, showing the particle's inertial force, m
a
. Resolve this vector into its appropriate components, consistent with the reference frame sign convention.
Apply the
Equations of Motion
in their scalar component form and solve these equations for the unknowns.
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.)
