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An Overview of Mechanics
Mechanics: The study of how bodies
react to forces acting on them.
Statics: The study of
bodies in equilibrium.
Dynamics:
1. Kinematics – concerned
with the geometric aspects of
motion
2. Kinetics – concerned with
the forces causing the motion
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
RECTILINEAR KINEMATICS: CONTINIOUS MOTION
(Section 12.2)
A particle travels along a straight-line
path defined by the coordinate axis s.
The position of the particle at any
instant, relative to the origin, O, is
defined by the position vector r, or
the scalar s. Scalar s can be positive
or negative. Typical units for r and s
are meters (m) or feet (ft).
The displacement of the particle is
defined as its change in position.
Vector form:  r = r’ - r
Scalar form:  s = s’ - s
The total distance traveled by the particle, sT, is a positive
scalar that represents the total length of the path over which
the particle travels.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
VELOCITY
Velocity is a measure of the rate of change in the position of a
particle. It is a vector quantity (it has both magnitude and
direction). The magnitude of the velocity is called speed, with
units of m/s or ft/s.
The average velocity of a particle during
a time interval t is
vavg = r / t
The instantaneous velocity is the time-derivative
of position.
v = dr / dt
Speed is the magnitude of velocity:
v = ds / dt
Average speed is the total distance traveled divided by
elapsed time: (vsp)avg = sT / t
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
ACCELERATION
Acceleration is the rate of change in the velocity of a particle.
It is a vector quantity. Typical units are m/s2.
The instantaneous acceleration is the
time derivative of velocity.
Vector form: a = dv / dt
Scalar form: a = dv / dt = d2s / dt2
Acceleration can be positive (speed
increasing) or negative (speed
decreasing).
As the text indicates, the derivative equations for velocity
and acceleration can be manipulated to get
a ds = v dv
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
SUMMARY OF KINEMATIC RELATIONS:
RECTILINEAR MOTION
• Differentiate position to get velocity and acceleration.
v = ds/dt ;
a = dv/dt
or
a = v dv/ds
• Integrate acceleration for velocity and position.
Position:
Velocity:
v
t
v
s
 dv =  a dt or  v dv =  a ds
s
t
 ds =  v dt
vo
o
vo
so
so
o
• Note that so and vo represent the initial position and
velocity of the particle at t = 0.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
CONSTANT ACCELERATION
The three kinematic equations can be integrated for the
special case when acceleration is constant (a = ac) to obtain
very useful equations. A common example of constant
acceleration is gravity; i.e., a body freely falling toward earth.
In this case, ac = g = 9.81 m/s2 downward. These equations
are:
v
t
 dv =  a
dt
yields
v = vo + act
 ds =  v dt
yields
s = s o + v ot + (1/2) a c t 2
yields
v 2 = (vo )2 + 2ac(s - so)
vo
o
s
t
so
v
c
o
s
 v dv =  ac ds
vo
so
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
GENERAL CURVILINEAR MOTION
(Section 12.4)
A particle moving along a curved path undergoes curvilinear motion. Since
the motion is often three-dimensional, vectors are used to describe the
motion.
A particle moves along a curve defined
by the path function, s.
The position of the particle at any instant is designated by the vector
r = r(t). Both the magnitude and direction of r may vary with time.
If the particle moves a distance s along the
curve during time interval t, the displacement is
determined by vector subtraction:  r = r’ - r
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
VELOCITY
Velocity represents the rate of change in the position of
a particle.
The average velocity of the
particle during the time increment
t is
vavg = r/t .
The instantaneous velocity is the
time-derivative of position
v = dr/dt .
The velocity vector, v, is always
tangent to the path of motion.
The magnitude of v is called the speed. Since the arc
length s approaches the magnitude of r as t→0, the
speed can be obtained by differentiating the path function
(v = ds/dt). Note that this is not a vector!
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
ACCELERATION
Acceleration represents the rate of change in
the velocity of a particle.
If a particle’s velocity changes from v to v’
over a time increment t, the average
acceleration during that increment is:
aavg = v/t = (v - v’)/t
The instantaneous acceleration is the timederivative of velocity:
a = dv/dt = d2r/dt2
A plot of the locus of points defined by the
arrowhead of the velocity vector is called a
hodograph. The acceleration vector is tangent
to the hodograph, but not, in general, tangent
to the path function.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
CURVILINEAR MOTION: RECTANGULAR COMPONENTS
(Section 12.5)
It is often convenient to describe the motion of a particle in
terms of its x, y, z or rectangular components, relative to a
fixed frame of reference.
The position of the particle can
be defined at any instant by the
position vector
r=xi+yj+zk .
The x, y, z components may all
be functions of time, i.e.,
x = x(t), y = y(t), and z = z(t) .
The magnitude of the position vector is: r = (x2 + y2 + z2)0.5
The direction of r is defined by the unit vector: ur = (1/r)r
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
RECTANGULAR COMPONENTS: VELOCITY
The velocity vector is the time derivative of the position vector:
v = dr/dt = d(xi)/dt + d(yj)/dt + d(zk)/dt
Since the unit vectors i, j, k are constant in magnitude and
direction, this equation reduces to v = vx i + vy j + vz k
•
•
•
where vx = x = dx/dt, vy = y = dy/dt, vz = z = dz/dt
The magnitude of the
velocity vector is
v = [(vx)2 + (vy)2 + (vz)2]0.5
The direction of v is tangent
to the path of motion.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
RECTANGULAR COMPONENTS: ACCELERATION
The acceleration vector is the time derivative of the
velocity vector (second derivative of the position
vector):
a = dv/dt = d2r/dt2 = ax i + ay j + az k
where
••
•
•
ax = vx = x = dvx /dt, ay = vy = y = dvy /dt,
••
••
az = v• z = z = dvz /dt
The magnitude of the acceleration vector is
a = [(ax)2 + (ay)2 + (az)2 ]0.5
The direction of a is usually
not tangent to the path of the
particle.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
NORMAL AND TANGENTIAL COMPONENTS
(Section 12.7)
When a particle moves along a curved path, it is sometimes
convenient to describe its motion using coordinates other than
Cartesian. When the path of motion is known, normal (n) and
tangential (t) coordinates are often used.
In the n-t coordinate
system, the origin is located
on the particle (the origin
moves with the particle).
The t-axis is tangent to the path (curve) at the instant
considered, positive in the direction of the particle’s motion.
The n-axis is perpendicular to the t-axis with the positive
direction toward the center of curvature of the curve.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
NORMAL AND TANGENTIAL COMPONENTS
(continued)
The positive n and t directions are
defined by the unit vectors un and ut,
respectively.
The center of curvature, O’, always
lies on the concave side of the curve.
The radius of curvature, r, is defined
as the perpendicular distance from
the curve to the center of curvature
at that point.
The position of the particle at any
instant is defined by the distance, s, along the curve from
a fixed reference point.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
VELOCITY IN THE n-t COORDINATE SYSTEM
The velocity vector is
always tangent to the path
of motion (t-direction).
The magnitude is determined by taking the time derivative of
the path function, s(t).
.
v = v ut
where
v = s = ds/dt
Here v defines the magnitude of the velocity (speed) and
ut defines the direction of the velocity vector.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
ACCELERATION IN THE n-t COORDINATE SYSTEM
Acceleration is the time rate of change of velocity:
.
.
a = dv/dt = d(vut)/dt = vut + vut
.
Here v represents the change
in the magnitude of velocity
.
and ut represents the rate of
change in the direction of ut.
After mathematical manipulation,
the acceleration vector can be
expressed as:
.
a = v ut + (v2/r) un = at ut + an un.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
ACCELERATION IN THE n-t COORDINATE
SYSTEM (continued)
So, there are two components to the
acceleration vector:
a = at ut + an un
• The tangential component is tangent to the curve and in the
direction of increasing or decreasing velocity.
.
at = v
or
at ds = v dv
• The normal or centripetal component is always directed
toward the center of curvature of the curve.
an = v2/r
• The magnitude of the acceleration vector is
a = [(at)2 + (an)2]0.5
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
SPECIAL CASES OF MOTION
There are some special cases of motion to consider.
1) The particle moves along a straight line.
.
2
r

=>
an = v /r = 0
=>
a = at = v
The tangential component represents the time rate of
change in the magnitude of the velocity.
2) The particle moves along a curve at constant speed.
.
at = v = 0
=>
a = an = v2/r
The normal component represents the time rate of
change in the direction of the velocity.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
SPECIAL CASES OF MOTION (continued)
3)
The tangential component of acceleration is constant, at =
(at)c.
In this case,
s = so + vo t + (1/2) (at)c t2
v = vo + (at)c t
v2 = (vo)2 + 2 (at)c (s – so)
As before, so and vo are the initial position and velocity of
the particle at t = 0. How are these equations related to
projectile motion equations? Why?
4) The particle moves along a path expressed as y = f(x).
The radius of curvature, r, at any point on the path can be
calculated from
[ 1 + (dy/dx)2]3/2
r = ________________
d2y/dx2
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
CYLINDRICAL COMPONENTS
(Section 12.8)
We can express the location of P in polar coordinates as r = r
ur. Note that the radial direction, r, extends outward from the
fixed origin, O, and the transverse coordinate, q, is measured
counter-clockwise (CCW) from the horizontal.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
VELOCITY in POLAR COORDINATES)
The instantaneous velocity is defined as:
v = dr/dt = d(rur)/dt
dur
.
v = rur + r
dt
Using the chain rule:
dur/dt = (dur/dq)(dq/dt)
.
We can prove that
. dur/d. q = uθ so dur/dt = quθ
Therefore: v = rur + rquθ
Thus,
the velocity vector has two components:
.
.
r, called the radial component, and rq called
the transverse component. The speed of the
particle at any given instant is the sum of the
squares of both components or
v=
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
.
.
(r q)2 + ( r )2
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
ACCELERATION (POLAR COORDINATES)
The instantaneous acceleration is defined as:
.
.
a = dv/dt = (d/dt)(rur + rquθ)
After manipulation, the acceleration can
be expressed as
..
.2
..
..
a = (r – rq )ur + (rq + 2rq)uθ
.2
..
The term (r – rq ) is the radial
acceleration or ar .
..
..
The term (rq + 2rq) is the transverse
acceleration or aq .
..
.2
The magnitude of acceleration is a = (r – rq
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
)2
..
..
+ (rq + 2rq)2
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
CYLINDRICAL COORDINATES
If the particle P moves along a
space curve, its position can be
written as
rP = rur + zuz
Taking time derivatives and using
the chain rule:
Velocity:
vP
.
= ru
..
.
r
.
+ rquθ + zuz
.
..
..
..
Acceleration: aP = (r – rq2)ur + (rq + 2rq)uθ + zuz
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
NEWTON’S LAWS OF MOTION (Section 13.1)
The motion of a particle is governed by Newton’s three laws
of motion.
First Law: A particle originally at rest, or moving in a straight
line at constant velocity, will remain in this state if the
resultant force acting on the particle is zero.
Second Law: 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.
Third Law: Mutual forces of action and reaction between two
particles are equal, opposite, and collinear.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
NEWTON’S LAWS OF MOTION
(continued)
The first and third laws were used in developing the
concepts of statics. Newton’s second law forms the
basis of the study of dynamics.
Mathematically, Newton’s second law of motion can be
written
F = ma
where F is the resultant unbalanced force acting on the
particle, and a is the acceleration of the particle. The
positive scalar m is called the mass of the particle.
Newton’s second law cannot be used when the particle’s
speed approaches the speed of light, or if the size of the
particle is extremely small (~ size of an atom).
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
NEWTON’S LAW OF GRAVITATIONAL ATTRACTION
Any two particles or bodies have a mutually attractive
gravitational force acting between them. Newton postulated
the law governing this gravitational force as
F = G(m1m2/r2)
where
F = force of attraction between the two bodies,
G = universal constant of gravitation ,
m1, m2 = mass of each body, and
r = distance between centers of the two bodies.
When near the surface of the earth, the only gravitational
force having any sizable magnitude is that between the earth
and the body. This force is called the weight of the body.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
MASS AND WEIGHT
It is important to understand the difference between
the 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 second 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.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
UNITS: SI SYSTEM VS. FPS SYSTEM
SI system: In the SI system of
units, mass is a base unit and weight
is a derived unit.
Typically, mass is specified in
kilograms (kg), and weight is
calculated from W = mg.
If the gravitational acceleration (g) is
specified in units of m/s2, then the
weight is expressed in newtons (N).
On the earth’s surface, g can be taken as g = 9.81 m/s2.
W (N) = m (kg) g (m/s2) => N = kg·m/s2
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
EQUATION OF MOTION (Section 13.2)
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
F = FR = ma
where FR is the resultant force, which is a vector summation
of all the forces.
To illustrate the equation, consider a
particle acted on by two forces.
First, draw the particle’s freebody diagram, showing all
forces acting on the particle.
Next, draw the kinetic diagram,
showing the inertial force ma
acting in the same direction as
the resultant force FR.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
INERTIAL FRAME OF REFERENCE
This equation of motion is only valid if the
acceleration is measured in a Newtonian or inertial
frame of reference. What does this mean?
For problems concerned with motions at or near
the earth’s surface, we typically assume our
“inertial frame” to be fixed to the earth. We
neglect any acceleration effects from the earth’s
rotation.
For problems involving satellites or rockets, the
inertial frame of reference is often fixed to the
stars.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
EQUATION OF MOTION FOR A SYSTEM OF PARTICLES
(Section 13.3)
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 = mi
as the total mass of all particles
and aG as the acceleration of the
center of mass G of the particles,
then m aG = mi ai .
The text shows the details, but for a system of particles:
F = m aG where F is the sum of the external forces
acting on the entire system.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
KEY POINTS
1) Newton’s second law is a “law of nature”-experimentally proven, not the result of an analytical
proof.
2) Mass (property of an object) is a measure of the
resistance to a change in velocity of the object.
3) Weight (a force) depends on the local gravitational field.
Calculating the weight of an object is an application of
F = m a, i.e., W = m g.
4) Unbalanced forces cause the acceleration of objects.
This condition is fundamental to all dynamics problems!
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
PROCEDURE FOR THE APPLICATION OF THE
EQUATION OF MOTION
1) Select a convenient inertial coordinate system. Rectangular,
normal/tangential, or cylindrical coordinates may be used.
2) Draw a free-body diagram showing all external forces
applied to the particle. Resolve forces into their
appropriate components.
3) Draw the kinetic diagram, showing the particle’s inertial
force, ma. Resolve this vector into its appropriate
components.
4) Apply the equations of motion in their scalar component
form and solve these equations for the unknowns.
5) It may be necessary to apply the proper kinematic
relations to generate additional equations.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
RECTANGULAR COORDINATES
(Section 13.4)
The equation of motion, F = m a, is best used when the
problem requires finding forces (especially forces perpendicular
to the path), accelerations, velocities, or mass. Remember,
unbalanced forces cause acceleration!
Three scalar equations can be written from this vector
equation. The equation of motion, being a vector equation,
may be expressed in terms of its three components in the
Cartesian (rectangular) coordinate system as
F = ma
or Fx i + Fy j + Fz k = m(ax i + ay j + az k)
or, as scalar equations, Fx = max, Fy = may, and Fz = maz.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
PROCEDURE FOR ANALYSIS
• Free-Body Diagram (always critical!!)
Establish your coordinate system and draw the particle’s
free-body diagram showing only external forces. These
external forces usually include the weight, normal forces,
friction forces, and applied forces. Show the ‘ma’ vector
(sometimes called the inertial force) on a separate diagram.
Make sure any friction forces act opposite to the direction
of motion! If the particle is connected to an elastic linear
spring, a spring force equal to ‘k s’ should be included on
the FBD.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
PROCEDURE FOR ANALYSIS
(continued)
• Equations of Motion
If the forces can be resolved directly from the free-body
diagram (often the case in 2-D problems), use the scalar
form of the equation of motion. In more complex cases
(usually 3-D), a Cartesian vector is written for every
force and a vector analysis is often best.
A Cartesian vector formulation of the second law is
F = ma or
Fx i + Fy j + Fz k = m(ax i + ay j + az k)
Three scalar equations can be written from this vector
equation. You may only need two equations if the motion is
in 2-D.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
PROCEDURE FOR ANALYSIS
(continued)
• Kinematics
The second law only provides solutions for forces and
accelerations. If velocity or position have to be found,
kinematics equations are used once the acceleration is
found from the equation of motion.
Any of the kinematics tools learned in Chapter 12 may
be needed to solve a problem.
Make sure you use consistent positive coordinate
directions as used in the equation of motion part of
the problem!
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
NORMAL & TANGENTIAL COORDINATES
(Section 13.5)
When a particle moves along a
curved path, it may be more
convenient to write the equation
of motion in terms of normal
and tangential coordinates.
The normal direction (n) always points toward the path’s center
of curvature. In a circle, the center of curvature is the center of
the circle.
The tangential direction (t) is tangent to the path, usually set
as positive in the direction of motion of the particle.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
EQUATIONS OF MOTION
Since the equation of motion is a
vector equation, F = ma,
it may be written in terms of the n
& t coordinates as
Ftut + Fnun+ Fbub = mat+man
Here Ft & Fn are the sums of the force components acting in
the t & n directions, respectively.
This vector equation will be satisfied provided the individual
components on each side of the equation are equal, resulting
in the two scalar equations: Ft = mat and
Fn = man.
Since there is no motion in the binormal (b) direction, we can
also write Fb = 0.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
NORMAL AND TANGENTIAL ACCERLERATIONS
The tangential acceleration, at = dv/dt, represents the time
rate of change in the magnitude of the velocity. Depending
on the direction of Ft, the particle’s speed will either be
increasing or decreasing.
The normal acceleration, an = v2/r, represents the time rate
of change in the direction of the velocity vector. Remember,
an always acts toward the path’s center of curvature. Thus,
Fn will always be directed toward the center of the path.
Recall, if the path of motion is
defined as y = f(x), the radius of
curvature at any point can be
obtained from
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
SOLVING PROBLEMS WITH n-t COORDINATES
• Use n-t coordinates when a particle is moving along a
known, curved path.
• Establish the n-t coordinate system on the particle.
• Draw free-body and kinetic diagrams of the particle. The
normal acceleration (an) always acts “inward” (the positive
n-direction). The tangential acceleration (at) may act in
either the positive or negative t direction.
• Apply the equations of motion in scalar form and solve.
• It may be necessary to employ the kinematic relations:
at = dv/dt = v dv/ds
an = v2/r
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
CYLINDRICAL COORDINATES
(Section 13.6)
This approach to solving problems has
some external similarity to the normal
& tangential method just studied.
However, the path may be more
complex or the problem may have other
attributes that make it desirable to use
cylindrical coordinates.
Equilibrium equations or “Equations of Motion” in cylindrical
coordinates (using r, q , and z coordinates) may be
expressed in scalar form as:
.2
..
 Fr = mar = m (r – r q )
..
. .
 Fq = maq = m (r q – 2 r q)
..
 Fz = maz = m z
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.
CYLINDRICAL COORDINATES
(continued)
If the particle is constrained to move only in the r – q
plane (i.e., the z coordinate is constant), then only the first
two equations are used (as shown below). The coordinate
system in such a case becomes a polar coordinate system.
In this case, the path is only a function of q.
.
 Fr = mar = m(r – rq 2)
.. . .
 Fq = maq = m(rq – 2rq )
..
Note that a fixed coordinate system is used, not a
“body-centered” system as used in the n – t approach.
Mechanics for Engineers: Dynamics, 13th SI Edition
R. C. Hibbeler and Kai Beng Yap
© Pearson Education South Asia Pte Ltd
2013. All rights reserved.