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Chapter 11
Mohammad Suliman Abuhaiba,Ph.D., P.E.1
Kinematics of Particles
2/8/2019 3:49 PM
First Exam
Wednesday
2/3/2019
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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IntroductionMechanics
Mechanics = science which
describes & predicts conditions of
rest or motion of bodies under
action of forces
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: bodies at rest
2. Dynamics: 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 withoutreference to cause of motion
2. Kinetics
study of relation existing between
forces acting on a body, mass of body,and motion of body
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Introduction
A dynamic study could be done on twolevels:
1. Particle: object whose size & 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 & under all conditions of
loading.
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Motion of Particles
1. Rectilinear Motion
2. Curvilinear Motion
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Rectilinear Motion of Particles
Position
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Rectilinear Motion of Particles
Mohammad Suliman Abuhaiba,Ph.D., P.E.
9
Velocity
t
xv
t
x
t
0lim
Average velocity
Instantaneous
velocity
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Rectilinear Motion of Particles
AccelerationMohammad Suliman Abuhaiba,Ph.D., P.E.
10
Instantaneous
acceleration t
va
t
0lim
t
v
Average acceleration
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Rectilinear Motion of Particles
Mohammad Suliman Abuhaiba,Ph.D., P.E.
11
Consider particle with motion given by
326 ttx
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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 Motion of a Particle
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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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 Motion of a Particle
Mohammad Suliman Abuhaiba,Ph.D., P.E.
14
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 Motion of a Particle
Mohammad Suliman Abuhaiba,Ph.D., P.E.
15
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
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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1. velocity & elevation above
ground at time t
2. highest elevation reached by
ball and corresponding time
3. time when ball will hit the
ground & corresponding
velocity
Ball tossed with 10 m/s vertical
velocity from a window 20 m
above ground. Determine:
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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; a = - k v
Determine v (t ), x (t ), and v (x ).
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Assignment #11.1
1, 6, 11, 17, 22, 29
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Uniform Rectilinear Motion
Mohammad Suliman Abuhaiba,Ph.D., P.E.
19
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.
20
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
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Uniformly Accelerated Rectilinear Motion
Mohammad Suliman Abuhaiba,Ph.D., P.E.
21
Acceleration of the particle is constant
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.
22
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
1. When & where ball hits the
elevator
2. Relative velocity of ball wrt
elevator at contact
Sample 11.4
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Motion of Several Particles: Dependent Motion
Position of B depends on
position of A
Rope length = constant
Sum of lengths of
segments = constant
constxx BA 2
(one DOF)
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Motion of Several Particles: Dependent Motion
constxxx CBA 22
(2 DOF)
022
022
CBA
CBA
aaa
vvv
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Sample 11.5Pulley 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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A
B
Assignment #11.2
33, 38, 42, 47, 52, 57
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 11.6A 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
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Curvilinear Motion: Position, Velocity & Acceleration
• Curvilinear motion: Particle moving along a
curve
• 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
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
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Rectangular Components of Velocity & Acceleration
Motion of a projectile
00 zyx agaa
initial conditions:
0000 zyx
Integrating twice:
0
02
21
00
00
zgttvytvx
vgtvvvv
yx
zyyxx
Mohammad Suliman Abuhaiba,Ph.D., P.E.36
Sample Problem 11.7A 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 point where
projectile strikes the ground
b. greatest elevation above ground reached by
the projectile
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Mohammad Suliman Abuhaiba,Ph.D., P.E.
37
Motion Relative to a Frame in Translation xyz = fixed frame of reference
x’y’z’ = moving frame
position of B wrt
moving frame Ax’y’z’ABr
ABAB rrr
Mohammad Suliman Abuhaiba,Ph.D., P.E.
38
Motion Relative to a Frame in Translation
ABAB vvv
ABAB aaa
Absolute motion of B = motion of A + relative motion
of B wrt moving reference frame attached to A
Mohammad Suliman Abuhaiba,Ph.D., P.E.39
Sample Problem 11.9
Car A is traveling east at
constant speed of 36 km/h.
As car A crosses the
intersection, car B starts
from rest 35 m north of the
intersection & moves south
with a constant
acceleration of 1.2 m/s2.
Determine position,
velocity, and acceleration
of B relative to A 5 s after A
crosses the intersection.
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Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Assignment #11.4
89, 95, 101, 107, 113, 120, 126
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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Tangential and Normal Components
Velocity vector is tangent
to path
ttt eee
d
ede
ee
e
tn
nt
t
2
2sinlimlim
2sin2
00
Mohammad Suliman Abuhaiba,Ph.D., P.E.42
Tangential and Normal Components
dt
ds
ds
d
d
edve
dt
dv
dt
edve
dt
dv
dt
vda
tt
t
2
2
va
dt
dva
ev
edt
dva
nt
nt
vdt
dsdsde
d
edn
t
Mohammad Suliman Abuhaiba,Ph.D., P.E.43
Tangential and Normal Components Tangential component of
acceleration reflects
change of speed
Normal component reflects
change of direction
Tangential component may
be +ve or -ve
Normal component always
points toward center of
path curvatureMohammad Suliman Abuhaiba,Ph.D., P.E.
44
Tangential and Normal Components – 3D Path
22 va
dt
dvae
ve
dt
dva ntnt
Osculating plane: Plane
containing tangential &
normal unit vectors
ntb eee
binormale
normalprincipal e
b
n
No Acceleration component
along binormalMohammad Suliman Abuhaiba,Ph.D., P.E.
45
Sample 11.10A 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.46
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
rerr
Mohammad Suliman Abuhaiba,Ph.D., P.E.
Radial and Transverse Components
erer
edt
dre
dt
dr
dt
edre
dt
drer
dt
dv
r
rr
rr
Mohammad Suliman Abuhaiba,Ph.D., P.E.
errerr
dt
ed
dt
dre
dt
dr
edt
d
dt
dr
dt
ed
dt
dre
dt
rd
edt
dre
dt
dr
dt
da
r
rr
r
22
2
2
2
2
Radial and Transverse Components
Mohammad Suliman Abuhaiba,Ph.D., P.E.
49
Radial and Transverse Components – 3D
kzeRr R
kzeReRdt
rdv R
kzeRReRR
dt
vda
R
22
Mohammad Suliman Abuhaiba,Ph.D., P.E.
50
Sample 11.12The 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 - 0.12 t2, where r is expressed in
meters and t in seconds. After the arm OA has rotated through
30o, determine
Mohammad Suliman Abuhaiba,Ph.D., P.E.
51
a. Total velocity of collar
b. Total acceleration of collar
c. Relative acceleration ofcollar wrt arm
Assignment #11.5
133, 140, 146, 153, 167
Mohammad Suliman Abuhaiba,Ph.D., P.E.
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