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Introduction Mechanics
Mechanics = science which
describes and predicts the
conditions of rest or motion of
bodies under the action of forces
It is divided into three parts:
1. Mechanics of rigid bodies
2. Mechanics of deformable bodies
3. Mechanics of fluids
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Introduction
Mechanics of rigid bodies is subdivided into:
1. Statics: deals with bodies at rest
2. Dynamics: deals with bodies in
motion
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Introduction
Dynamics is subdivided into:
1. Kinematics
study of geometry of motion.
relating displacement, velocity,
acceleration, and time without reference to the cause of motion
2. Kinetics
study of the relation existing between the
forces acting on a body, the mass of the body, and the motion of the body
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Introduction
A dynamic study could be done on two levels:
1. Particle
an object whose size and shape can
be ignored when studying its motion.
2. Rigid Body
a collection of particles that remain at
fixed distance from each other at all
times and under all conditions of
loading.
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Motion of Particles
Motion of Particles:
1. Rectilinear Motion
2. Curvilinear Motion
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Rectilinear Motion of Particles
Position
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Velocity
t
xv
t
x
t
0lim
Average velocity
Instantaneous
velocity
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Rectilinear Motion of Particles
Acceleration
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Instantaneous
acceleration t
va
t
0lim
t
v
Average acceleration
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Rectilinear Motion of Particles
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• Consider particle with motion given by
326 ttx 2312 ttdt
dxv
tdt
xd
dt
dva 612
2
2
• at t = 0, x = 0, v = 0, a = 12 m/s2
• at t = 2 s, x = 16 m, v = vmax = 12 m/s, a = 0
• at t = 4 s, x = xmax = 32 m, v = 0, a = -12 m/s2
• at t = 6 s, x = 0, v = -36 m/s, a = 24 m/s2
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Three classes of motion may be defined:
1.Acceleration is a function of time, a = f(t)
2.Acceleration is a function of position, a = f(x)
3.Acceleration is a function of velocity, a = f(v)
Determination of Motion of a Particle
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Determination of the Motion of a Particle
Mohammad Suliman Abuhaiba,Ph.D., P.E.
11
1. Acceleration is a function of time, a = f(t)
tttx
x
tttv
v
dttvxtxdttvdx
dttvdxtvdt
dx
dttfvtvdttfdv
dttfdvtfadt
dv
0
0
0
0
0
0
0
0
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Determination of the Motion of a Particle
Mohammad Suliman Abuhaiba,Ph.D., P.E.
12
2. Acceleration is a function of position, a = f(x)
x
x
x
x
xv
v
dxxfvxv
dxxfdvvdxxfdvv
xfdx
dvva
dt
dva
v
dxdt
dt
dxv
0
00
2
0212
21
or or
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Determination of the Motion of a Particle
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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3. Acceleration is a function of velocity, a = f(v)
tv
v
tv
v
tx
x
tv
v
ttv
v
vf
dvvxtx
vf
dvvdx
vf
dvvdxvfa
dx
dvv
tvf
dvdt
vf
dv
dtvf
dvvfa
dt
dv
000
00
0
0
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Sample 11.2
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Determine:
a. velocity & elevation above
ground at time t
b. highest elevation reached by
ball and corresponding time
c. time when ball will hit the
ground & corresponding
velocity
Ball tossed with 10 m/s vertical
velocity from window 20 m
above ground.
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Sample 11.3
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Brake mechanism used to reduce gun recoil consists
of piston attached to barrel moving in fixed cylinder
filled with oil. As barrel recoils with initial velocity
v0, piston moves and oil is forced through orifices in
piston, causing piston and cylinder to decelerate at
rate proportional to their velocity; that is a = -kv
Determine v(t), x(t), and v(x).
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Assignment #11.1
1, 6, 11, 17, 22, 29 Due Wednesday
11/2/2015
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Uniform Rectilinear Motion
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Acceleration is zero and velocity is constant
vtxx
vtxx
dtvdx
vdt
dx
tx
x
0
0
00
constant
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Uniformly Accelerated Rectilinear Motion
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Acceleration of the particle is constant
atvv
atvvdtadvadt
dv tv
v
0
000
constant
221
00
221
000
00
0
attvxx
attvxxdtatvdxatvdt
dx tx
x
020
2
020
221
2
constant
00
xxavv
xxavvdxadvvadx
dvv
x
x
v
v
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Motion of Several Particles Relative Motion
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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ABAB xxx relative position of B wrt A
ABAB xxx
ABAB vvv relative velocity of B wrt A
ABAB vvv
ABAB aaa relative acceleration of B wrt A
ABAB aaa
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Ball thrown vertically from 12 m
level in elevator shaft with initial
velocity of 18 m/s. At same
instant, open-platform elevator
passes 5 m level moving upward
at 2 m/s.
Determine
a. when and where ball hits the
elevator
b. relative velocity of ball wrt
levator at contact
Sample 11.4
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Motion of Several Particles: Dependent Motion
Position of block B depends on
position of block A.
Since rope is of constant length, it
follows that sum of lengths of
segments must be constant.
BA xx 2 constant (one DOF)
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Motion of Several Particles: Dependent Motion
CBA xxx 22 constant (2 DOF)
022or022
022or022
CBACBA
CBACBA
aaadt
dv
dt
dv
dt
dv
vvvdt
dx
dt
dx
dt
dx
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Sample 11.5 Pulley D is attached to a collar
which is pulled down at 3 cm/s.
At t = 0, collar A starts moving
down from K with constant
acceleration and zero initial
velocity. Knowing that
velocity of collar A is 12 cm/s
as it passes L, determine the
change in elevation, velocity,
and acceleration of block B
when block A is at L.
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Assignment #11.2
33, 38, 42, 47, 52, 57
Due Saturday 14/2/2015
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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• Given x-t curve, v-t curve = x-t curve slope
• Given v-t curve, a-t curve = v-t curve slope
Graphical Solution of
Rectilinear-Motion Problems
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Given a-t curve, change in velocity between t1 &
t2 = area under a-t curve between t1 & t2.
Given v-t curve, change in position between t1 &
t2 = area under v-t curve between t1 & t2.
Graphical Solution of
Rectilinear-Motion Problems
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Sample Problem 11.6 A subway car leaves station A; it gains speed at
the rate of 4 ft/s2 for 6 s and then at the rate of 6
ft/s2 until it has reached the speed of 48 ft/s. The
car maintains the same speed until it approaches
(car does not reach B yet) station B; brakes are
then applied, giving the car a constant
deceleration and bringing it to a stop in 6 s. The
total running time from A to B is 40 s. Draw the a−t,
v−t, and x−t curves, and determine the distance
between stations A and B.
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Assignment #11.3
61, 67, 73, 79, 87 Due Monday 16/2/2015
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Curvilinear Motion: Position, Velocity & Acceleration
• Curvilinear motion: Particle moving along a curve
other than a straight line
• Position vector of a particle at time t
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Curvilinear Motion: Position, Velocity & Acceleration
dt
ds
t
sv
dt
rd
t
rv
t
t
0
0
lim
lim
instantaneous velocity (vector)
instantaneous speed (scalar)
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Curvilinear Motion: Position, Velocity & Acceleration
dt
vd
t
va
t
0lim
instantaneous acceleration
(vector)
• In general, acceleration vector is
not tangent to particle path &
velocity vector. Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Rectangular Components of Velocity & Acceleration
kzjyixr
kvjviv
kzjyixkdt
dzj
dt
dyi
dt
dxv
zyx
kajaia
kzjyixkdt
zdj
dt
ydi
dt
xda
zyx
2
2
2
2
2
2
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Rectangular Components of Velocity & Acceleration
Motion of a projectile
00 zagyaxa zyx
initial conditions:
0000 zyx
Integrating twice:
0
02
21
00
00
zgttvytvx
vgtvvvv
yx
zyyxx
Motion in horizontal direction is uniform
Motion in vertical direction is uniformly accelerated
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Motion Relative to a Frame in Translation
xyz = fixed frame of reference
moving frames of reference: frames not rigidly
attached to the fixed reference frame
Position vectors for particles A and B wrt to the
fixed frame of reference Oxyz are
: position of B wrt
moving frame Ax’y’z’ ABr
ABAB rrr
Mohammad Suliman Abuhaiba,Ph.D., P.E.
. and BA rr
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Motion Relative to a Frame in Translation
ABv
velocity of B wrt A ABAB vvv
ABa
acceleration of B wrt A ABAB aaa
Absolute motion of B =
combined motion of A and
relative motion of B wrt
moving reference frame
attached to A.
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Sample Problem 11.7 A projectile is fired from edge of a 150-m cliff with
an initial velocity of 180 m/s at an angle of 30°
with the horizontal. Neglecting air resistance,
find:
a. horizontal distance from the gun to the point
where the projectile strikes the ground,
b. greatest elevation above the ground reached
by the projectile.
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Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Sample Problem 11.9
Automobile A is traveling east
at the constant speed of 36
km/h. As automobile A crosses
the intersection shown,
automobile B starts from rest 35
m north of the intersection and
moves south with a constant
acceleration of 1.2 m/s2.
Determine the position,
velocity, and acceleration of B
relative to A 5 s after A crosses
the intersection.
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Assignment #11.4
89, 95, 101, 107, 113, 120, 126
Due Wednesday 18/2/2015
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Tangential and Normal Components Velocity vector is tangent to path.
= tangential unit
vectors for particle path at P &
P’
tt ee and
ttt eee
d
ede
eee
e
tn
nnt
t
2
2sinlimlim
2sin2
00
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Tangential and Normal Components
dt
ds
ds
d
d
edve
dt
dv
dt
edve
dt
dv
dt
vda tt
22 va
dt
dvae
ve
dt
dva ntnt
vdt
dsdsde
d
edn
t
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Tangential and Normal Components Tangential component of acceleration reflects
change of speed
Normal component reflects change of direction.
Tangential component may be positive or negative.
Normal component always points toward center of
path curvature.
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Tangential and Normal Components
22 va
dt
dvae
ve
dt
dva ntnt
Relations for tangential & normal acceleration
also apply for particle moving along space curve.
Osculating plane: Plane containing
tangential & normal unit vectors
ntb eee
binormale
normalprincipal e
b
n
No Acceleration component along
binormal. Mohammad Suliman Abuhaiba,Ph.D., P.E. 2/20/2015 11:21 AM 42
Sample 11.10
A motorist is traveling on
curved section of highway at 88
m/s. The motorist applies
brakes causing a constant
deceleration rate.
Knowing that after 8 s the speed
has been reduced to 66 m/s,
determine the acceleration of
the automobile immediately
after the brakes are applied.
Mohammad Suliman Abuhaiba,Ph.D., P.E. 2/20/2015 11:21 AM 43
Radial and Transverse Components
rr e
d
ede
d
ed
dt
de
dt
d
d
ed
dt
ed rr
dt
de
dt
d
d
ed
dt
edr
erer
edt
dre
dt
dr
dt
edre
dt
drer
dt
dv
r
rr
rr
The particle velocity vector is
rerr
Mohammad Suliman Abuhaiba,Ph.D., P.E. 2/20/2015 11:21 AM
Radial and Transverse Components
The particle acceleration vector is
errerr
dt
ed
dt
dre
dt
dre
dt
d
dt
dr
dt
ed
dt
dre
dt
rd
edt
dre
dt
dr
dt
da
r
rr
r
22
2
2
2
2
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Radial and Transverse Components – 3D
kzeRr R
kzeReRdt
rdv R
kzeRReRR
dt
vda
R
22
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Sample 11.12 The rotation of the 0.9 m arm OA about
O is defined by the relation 0.15t2
where is expressed in radians and t in
seconds. Collar B slides along the arm
in such a way that its distance from O
is r = 0.9-0.12t2, where r is expressed in
meters and t in seconds. After the arm
OA has rotated through 30o , determine
a. total velocity of the collar
b. total acceleration of the collar
c. relative acceleration of the collar wrt the arm
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