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Engineering Mechanics deals with the state of a
body at rest or motion of a body caused by the
action of forces.
Statics deals with the action of forces on bodiesat rest.
Dynamics deals with the motion of bodies under theaction of forces.
Kinematics study of motion without reference to
the forces which cause the motion. Deals with
position, velocity, and acceleration in terms of time
Kinetics relates the action of forces on bodies
to their resulting motions
Ken Youssefi 1MAE
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A body , not necessarily small, where its motion can be
characterized without considering its size and orientation.Rotation of the body about its own axis is neglected.
Particle
Rigid bodyThe body is called a rigid body If its rotation about its
own axis cannot be neglected.
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Problem Solving Technique
Identify all given data
Identify the desired results
Draw the necessary diagrams
Identify the dynamics principles applied to the
problem
Clearly state all assumptions
Use your technical judgment (common sense) to
determine if the answer is reasonable.
Report the answer with the same accuracy as the
given data
Careless solutions that cannot be read easily are of
little or no valueKen Youssefi 3MAE
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Position of a point in space
coordinate systems
Rectangular (Cartesian)
coordinates,x, y, andz
Cylindricalcoordinates, r, , andz
Spherical
coordinates,R, ,and
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Planar (2D) Motion
Rectangular (Cartesian) coordinates,x andy
Polar coordinates, rand
xx
y
y
P
O
r
Ken Youssefi 5MAE
x = r cos
y = r sin
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t
ttt
dt
d
t
}{}{lim
0
vvva
Position
Velocity
Acceleration
Kinematics
2
2
dt
d
dt
d rva
Ken Youssefi 6MAE
r = r{t}
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Graphical representation
Velocity at time t is the slope of theline tangent to displacement curve.
Acceleration at time t is the slope of
the line tangent to velocity curve.
Given the displacement curve,
velocity and acceleration curves
could be obtained graphically.
Ken Youssefi 7MAE
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Rectilinear Motion
Particle moving along a straight line from P to P
+s-s
O PP
s s
At time t +t the particle has moved to P
v = lims
t
t 0
}{tv
dt
ds
}{tadt
dv v
ds
dv
dt
ds
ds
dv
dt
dva
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Bdttvts }{}{
t
t
v
v oo
dttadv }{ t
to
o
dttavtv }{}{
t
t
s
s oo
dttvds }{ t
to
o
dttvsts }{}{
Rectilinear Motion
}{tvdt
ds
Relationship between displacementandvelocity
Adttatv }{}{
}{tadt
dv
Relationship between velocity and acceleration
Ken Youssefi 9MAE
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10
Special Cases: Constant Velocity or Acceleration
Ifs =s{t} and v is constant: t
oo dtvsts 0}{tvsts oo}{
ooo ssavv 222
s
s
v
voo odvvdsa
If v = v{s} and a is constant:
t
o dttvsts 0}{}{
221}{ tatvsts ooo
Ifs =s{t}and a is constant:
t
oo dtavtv 0}{
tavtvoo
}{ If v = v{t} and a is constant:
Ken Youssefi MAE
t
to
o
dttavtv }{}{ t
to
o
dttvsts }{}{
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Example
Ken Youssefi MAE
The acceleration of a train (at least during the interval from t= 2
to t= 4) is a = 2t, and it is known that at t= 2 its velocity v = 50.
What is the trains velocity at t= 4? (Assume all units are already
consistently in SI units).
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Example
The acceleration of a trainduring the interval from t= 2to t= 4 is a = 2t, and it isknown that at t= 2 itsvelocity vis 50.
What is the trains velocity att= 4? (Assume all units arealready consistently in SIunits).
Image(s) from Engineering Mechanics: Dynamics, 3rd ed.,
by A. Bedford & W. T. Fowler, Prentice Hall, 2002.
Ken Youssefi MAE
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Example
After deploying its drag parachute, the airplaneshown has an acceleration a = -.004v2 (m/s).
Determine the time required for thevelocity to decrease from 80 m/s to 10 m/s.
What distance does the plane cover duringthat time?
Ken Youssefi MAE
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Example
After deploying its drag parachute, the airplaneshown has an acceleration a = -.004v2 (m/s).
Determine the time required for thevelocity to decrease from 80 m/s to 10m/s.
What distance does the plane coverduring that time?
Image(s) from Engineering Mechanics: Dynamics, 3rd ed.,
by A. Bedford & W. T. Fowler, Prentice Hall, 2002.
Ken Youssefi MAE
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Ken Youssefi MAE 15
Curvilinear Motion 3D
Particle moving along a curved path.
Rectangular coordinates is particularly useful for describing thecurvilinear motion
kjir zyx
kjivzyx
vvv
kjia zyx aaa
kjirvdtdz
dtdy
dtdx
dtd
dt
dxvx
dt
dyvy
dt
dzvz
kjiv
adt
dv
dt
dv
dt
dv
dt
dzyx
dt
dva x
x dt
dva
y
y dt
dva z
z
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Ken Youssefi MAE 17
t
t xxox odttavtv }{)(}{
xox vv )(
t
t xo odttvxtx }{}{
tvxtx xoo )(}{
t
t yyoy odttavtv }{)(}{
gtvtv yoy )(}{
t
t yo odttvyty }{}{
2
21)(}{ gttvyty yoo
Curvilinear Motion Projectile Motion
Vertical motionHorizontal motion
Velocity
Displacement
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Example
The skier leaves the 20ledge with a speed of 10
m/s (along the upper
inclined surface).
How far along the lower
incline does she land?
(What is the distance d?)
Ken Youssefi MAE
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Example
Ken Youssefi MAE 19
(vo )x = vo cos20
(vo )y = -vo sin20
tvxtx xoo )(}{
2
21)(}{ gttvyty yoo
(vo ) = 10 m/s
xo = 0
yo = 0
x = vo cos20 t = 10 cos20 t= 9.4 t
y = - vo sin20 t - gt2 = 10 cos20 t= - 3.42 t - 4.9 t 2
y = - 3 -x
t= 1.6 s
x = 15 m
y = -18 m
The skier leaves the 20 ledge witha speed of 10 m/s (along the upperinclined surface).
How far along the lower inclinedoes she land? (i.e. What is thedistance d?)
x
x
d= 21.3 m
(vo )x
(vo
)y vo
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Ken Youssefi MAE 20
Curvilinear Motion Normal and Tangential coordinates
These coordinates provide a natural description for curvilinear
motion and are frequently the most convenient coordinates to use.
Unit vector ut is tangent to the curve at P,
and points in the direction of increasings.
Unit vector un is always perpendicular to
ut, and points toward the concave side.
Coordinates measures positionof point P along its path with
respect to origin O.
Velocity vis always tangent tounit vector ut.
tuvdt
ds
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Ken Youssefi MAE 22
Radius of Curvature and Normal Component of Acceleration
dds )(
vdt
d
dt
ds
nt
dt
dv
dt
dvuua
nt
v
dt
dvuua
2
dt
dvat
2v
an
Tangential component Normal component
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Ken Youssefi MAE 23
Circular Path
tuvdtds nt
dtdv
dtdv uua
Velocity and acceleration
Rs
Rdt
dR
dt
dsv
dt
d
dt
d
Rdt
dR
dt
dvat
Magnitude of tangentialacceleration
2
Rdt
dvan
Magnitude of normal acceleration
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Example
A motorcycle starts from
rest on a circular trackof radius 400 m, and
accelerates with a
tangentialacceleration
of at= 2 + 0.2tm/s.
How much distance
does the motorcycle
cover in the first 10
seconds?
What is the magnitudeof totalacceleration |a|
at the 10-second mark?
Ken Youssefi MAE
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Ken Youssefi MAE 26
Curvilinear Motion Polar Coordinates
dt
drv
rrur
dt
dva
u
u
dt
d
dt
d r rdt
d
dt
du
u
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Ken Youssefi MAE 27
dt
dr
dt
dr
dt
d rr
uu
rv
uuv
dt
dr
dt
drr
rrur
dt
drvr
dt
drv
Curvilinear Motion Polar Coordinates
Velocity
u
u
dt
d
dt
dr
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Ken Youssefi MAE 28
dt
d
dt
dr
dt
dr
dt
d
dt
dr
dt
d
dt
dr
dt
rd
dt
d rr
uuu
uu
va
2
2
2
2
uua
dt
d
dt
dr
dt
dr
dt
dr
dt
rdr 22
22
2
2
uuvdtdr
dtdr r
Curvilinear Motion Polar Coordinates
Acceleration
r
dt
d
dt
du
u
u
u
dt
d
dt
d r
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Example
A point P moves along the spiral path r= (0.1)
meters, where is in radians. The angular position
= 2trad, where tis in seconds, and r= 0 at t= 0.Determine the magnitudes of the velocity and
acceleration of P at t= 1 s.
Ken Youssefi MAE
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Coupled-Motion
Some problems are disguised as being
more complicated, even though there is
really only one degree of freedom.
These problems can be simplified by using
geometry to relate the motions of each
moving point, then taking derivates to
relate velocity and acceleration.
To express the dependency properly,
position coordinate s needs to be defined
with respect to a fixed datum, with
deliberately assigned positive direction.
Questions:
How are sA, sB, and h related?
How are vA and vB related?
How are aA and aB related?
Ken Youssefi MAE
l
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Relative Motion
If pointA is a moving particle, it
can be assigned a local
reference frame that translateswith respect to a fixed
reference frame with origin O.
From the perspective of point
A, the relative motion of other
points such as B is often moreuseful than motion with
respect to a fixed reference
frame.
Vector analysis provides an
effective way to express therelative motion between
particles, based on the motion
of each with respect to a fixed
reference frame.
ABAB /rrr ABAB /vvv
ABAB /aaa Ken Youssefi MAE
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Example
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Ken Youssefi MAE 34
ExampleUse polar coordinate system
aP =
=
d2
rOP
dt2
+ (u)OPd
2
OPdt
2
d OPdt
rOP +d rOP
dt2
OP length is constantd
2
rOP
dt2 = 0and
d rOP
dt= 0
rOPd OP
dt( )
2(ur)OP
rOP d OP
dt( )2(ur)OPaP =
rQPd Qp
dt( )
2(ur)QPaQ/P =
= rOPOP2 i
rQPQP2
(cos45) i rQPQP2 (sin45)j
aQ = aP + aQ/P = rOPOP2 rQPQP
2(cos45) i rQPQP
2 (sin45)j
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Ken Youssefi MAE 35
aQ = - (2)(4)2 ( 2 )(8)2(cos45) i -( 2 )(8)2(sin45)j
Example
aQ = 96 i 64j
aQ = (96)2
+ (64)2
= 115.38 ft/s2
= tan-1 6496
= 33.7 O
33.7O
aQ
aQ = aP + aQ/P = rOPOP2 rQPQP
2(cos45) i rQPQP
2 (sin45)j
P bl F l ti d R i
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Ken Youssefi MAE 36
Problem Formulation and Review
The ability to select the most appropriate method is a key
to the successful formulation and solution of engineering
problems.
It is important to recognize the following breakdowns:
1. Type of motion Rectilinear motion
(one coordinate needed to describe the motion)
Plane curvilinear motion
(two coordinate needed to describe the motion)
space curvilinear motion
(three coordinate needed to describe the motion)
P bl F l ti d R i
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Ken Youssefi MAE 37
Problem Formulation and Review
2. Reference axes
Fixed reference axesAbsolute motion - axes attached to the surface of the
earth are sufficiently fixed for most engineering problems
Moving reference axes
Relative motion - axes attached to a moving body that is notattached to a fixed reference
Fixed axes
Problem Formulation and Review
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Ken Youssefi MAE 38
Problem Formulation and Review3. Coordinates
A very important step in formulation of a problem
The options: Rectangular (Cartesian) coordinates
x, y 2D motion
x, y, z 3D motion
Normal and tangential coordinatesn, t 2D curvilinear motion
Polar coordinates
r, 2D motion
Cylindrical coordinatesr, , z 3D motion
Spherical coordinates
R, , 3D motion
Problem Formulation and Review
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Ken Youssefi MAE 39
Problem Formulation and Review
The choice is dictated by how the motion is generated or
measured
Straight line (rectilinear) motion
Type of Motion Type of Coordinates
Curvilinear motion
Curvilinear motion, when
measurements are made
along the curve
Normal and tangential (n, t)
Sliding and rotating motion
at the same time
Polar (r, )
Space curvilinear motion Rectangular (x, y, z)
Cylindrical (r, , z)
Spherical (R, , )
Polar (r, )
Rectangular (x, y)
Rectangular, x(or s)
Examples
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Ken Youssefi MAE 40
Examples
Rectangular, x(or s)
Polar (r, )
Rectangular (x, y)
Normal and tangential (n, t)Spherical (R, , )
Examples
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Ken Youssefi MAE 41
ExamplesPolar (r, )
Polar (r, )
Rectangular (x, y)
Cylindrical (r, , z)
Examples
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Ken Youssefi MAE 42
Examples
Cylindrical (r, , z)
Polar (r, )
Rectangular (x, y)
Rectangular, x(or s)
Polar (r, )
Examples
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Examples
Spherical (R, , )
Rectangular, x(or s)
Spherical (R, , )