Chapter 13: Kinetics of a Particle: Force and Acceleration
13.1 Newton’s Laws of Motion
 The three basic laws governing the motion of a particle can be stated as follows:
o First Law: A particle originally at rest, or moving in a straight line with a constant
velocity, will remain in this state provided the particle is not subjected to an unbalanced
force.
o Second Law: A particle acted upon by an unbalanced force F experiences an acceleration
a that has the same direction as the force and a magnitude that is directly proportional to
the force.
o Third Law: The mutual forces of action and reaction between two particles are equal,
opposite, and collinear.
 In this course (Dynamics), attention is focused on the second law. The second law can be
expressed mathematically as
F = ma
Note: Newton’s second law does not apply when the particle’s speed approaches the speed of
light (3×108 m/s) or when particles are the size of an atom and move close to one another.
 There is a mutual attraction between any two particles. According to Newton’s law of
gravitational attraction, the force of attraction between two particles is
F  G
m1 m2
r2
(1)
where m1, m2 = mass of each of the two particles
r = distance between the centers of the two particles, and
G = universal constant of gravitation (G = 66.73 ×10-12 m3/kg.s2)
 The gravitational force between the earth and a particle located at or near the surface of the
earth is the weight of the particle. The weight of a body is expressed in newtons (N) in SI
system of units and Ib in FPS system. It is obtained from
W = mg
where m = mass of body (kg in SI; slug in FPS), and
g = acceleration due to gravity (= 9.81 m/s2 = 32.2 ft/s2)
13.2 The Equation of Motion
 When more than one force acts on a particle, the resultant force (FR) is determined by a
vector summation of all the forces. The equation of motion may be written as
FR = ΣF = ma
(2)
 Application of the equation of motion involves drawing the particle’s free body diagram
and kinetic diagram (se Fig. 13-2).
 Whenever the equation of motion is applied, it is required that measurements of the
acceleration be made from a Newtonian or inertia frame of reference. Such a coordinate
system does not rotate and is either fixed or translates in a given direction with constant
velocity (zero acceleration).
13.3 Equation of Motion for a System of Particles
 Consider a system of n particles isolated within an enclosed region in space, as shown in Fig.
13-4 of the textbook.
o Each particle experiences internal forces that other particles exert on it. The summation of
these internal forces will equal zero.
o The resultant external forces on the system (from, for example, gravitational, electrical,
magnetic, or contact forces between the system and the surroundings) results in
acceleration of the system.
o If the center of mass of the system is represented by G, the equation of motion of the
system is
F
 ma G
(3)
i.e. the sum of the external forces acting on a system of particles is equal to the total mass
of the particles times the acceleration of its center of mass G.
13.4 Equations of Motion: Rectangular Coordinates
 The equation of motion (ΣF = ma) can be expressed in the rectangular coordinate system as
follows:
ΣFxi + ΣFyj + ΣFzk = m(axi + ayj + azk)
(4)
 From equation (4), we may write the following three scalar equations:
ΣFx = max
ΣFy = may
ΣFz = maz
(5)
Procedure for Analysis
1.
Draw the free-body diagram of the particle. Here, the particle is isolated from its
surroundings and all external forces acting on the particle are shown. Some of the forces
come from the following sources:
a. Gravitational force: This is the weight of the particle (W = mg).
b. Friction: If the particle contacts a rough surface, calculate frictional force as Ff = µN,
where N is the normal force acting between the particle and the surface and µ is the
coefficient of friction. Ff acts opposite to the direction of motion of the particle.
c. Spring: If the particle is connected to an elastic spring, calculate spring force as Fs = ks.
Here, k is the spring constant and s is the distance from the undisturbed position of the
spring.
2.
Draw the kinetic diagram. Here, the acceleration vector is shown.
3.
Apply the equation of motion either in the scalar component form (equation 4) or vector
form (equation 5).
4.
If the velocity or position of the particle is to be found, apply the equations for rectilinear
motion (chapter 12) in each component direction.
5.
If the problem involves the dependent motion of several particles, use the method outlined
in section 12.9 to relate their accelerations.
13.5 Equations of Motion: Normal and Tangential Coordinates
 When a particle moves over a curved path which is known, the equation of motion for the
particle may be written in the tangential, normal, and binormal directions (see Fig. 13-11).
Thus
ΣFtut + ΣFnun + ΣFbub = mat + man
(6)
Note: The binormal axis b is perpendicular to the t and n axes (see more details on page 51).
There is no motion of the particle in the binormal direction, since the particle is constrained
to move along the path.
 From equation (6), we may write the following three scalar equations:
ΣFt = mat
ΣFn = man
ΣFb = 0
 Recall that
(7)
o at (=dv/dt) represents the time rate of change in the magnitude of velocity.
o an (=v2/ρ) represents the time rate of change in the velocity’s direction. It acts toward the
path’s center of curvature.
 When the particle is constrained to travel in a circular path with a constant speed, the force Fn
exerted by the constraint is often referred to as the centripetal force.
13.6 Equations of Motion: Cylindrical Coordinates
 When all the forces acting on a particle are resolved into cylindrical components, the
equation of motion may be expressed as
ΣFrur +
ΣFθuθ + ΣFzuz
=
marur + maθuθ
+ mazuz
(8)
 From equation (8), we may write the following three scalar equations:
ΣFr = mar
ΣFθ = maθ
ΣFz = maz
(9)
 Recall that
o The r coordinate extends outward from the fixed origin O to the particle (see Fig. 13-17).
o The θ coordinate is perpendicular to the r coordinate.
o
a r  r  r 2
o
a  r  2r
o
a z  z
 When the particle is constrained to move along a path r = f(θ) (as shown in Fig. 13-17) the
following should be observed about the forces acting on the particle:
o The normal force N which the path exerts on the particle is always perpendicular to the
tangent of the path.
o The frictional force F always acts along the tangent in the opposite direction of motion.
o The angle ψ between the extended radial line and the tangent to the curve can be
determined from
tan  
r
dr d
(10)