0 Dynamics includes:
- Kinematics: study of the eometry of motion" inematics is
used to
relate dislacement, %elocity, acceleration, and time ithout
reference to
the cause of motion"
- Kinetics: study of the relations e3istin !eteen the forces
actin on a
!ody, the mass of the !ody, and the motion of the !ody"
inetics is used
to redict the motion caused !y i%en forces or to determine the
forces
re4uired to roduce a i%en motion"
0 Rectilinear motion: osition, %elocity, and
acceleration of a article as it
mo%es alon a straiht line"
0 Curvilinear motion: osition, %elocity, and
acceleration of a article as it
Rectilinear Motion: Position, Velocity & Acceleration
0 Particle mo%in alon a straiht line is said
to !e in rectilinear motion"
0 Position coordinate of a article is defined
!y ositi%e or neati%e distance of article
from a fi3ed oriin on the line"
0 he motion of a article is 5non if the
osition coordinate for article is 5non for
e%ery %alue of time t " Motion of the article
may !e e3ressed in the form of a function,
e"", #26 t t x −=
0 *onsider article ith motion i%en !y
#26 t t x −=
dx v −==
t dt
−===
0 at t 9 /, x 9 /, v 9 /, a 9 12
ms2
0 at t 9 2 s, x 9 16 m, v 9 vmax 9
12 ms, a 9 /
Determination of te Motion of a Particle
0 Recall, motion of a article is 5non if osition is 5non for
all time t "
0 yically, conditions of motion are secified !y the tye of
acceleration
e3erienced !y the article" Determination of %elocity and osition
re4uires
to successi%e interations"
0 hree classes of motion may !e defined for:
time t ,
corresondin time, and
corresondin %elocity"
from indo 2/ m a!o%e round"
S)@$I).:
0 Interate tice to find v<t = and y<t ="
0 Sol%e for t at hich %elocity e4uals
ero <time for ma3imum ele%ation=
and e%aluate corresondin altitude"
0 Sol%e for t at hich altitude e4uals
ero <time for round imact= and
e%aluate corresondin %elocity"
0 +or articles mo%in alon the same line, time
should !e recorded from the same startin
instant and dislacements should !e measured
from the same oriin in the same direction"
=−= A B A B x x x relati%e
osition of B
ith resect to A A B A B
x x x +=
!am"le Pro#lem $$%+ 0 Pulley % has uniform rectilinear
motion" *alculate
chane of osition at time t "
( )
t v x x
0 ?loc5 B motion is deendent on motions of collar
A and ulley %" Brite motion relationshi and
sol%e for chane of !loc5 B osition at time t "
otal lenth of ca!le remains constant,
( ) ( ) ( )
( )[ ] ( )[ ] ( )[ ]
( ) in"16/ −=− B B x x
ra"ical !olution of Rectilinear-Motion Pro#lems
0 (i%en the x't cur%e, the v't cur%e is
e4ual to
the x't cur%e sloe"
0 (i%en the v't cur%e, the a't cur%e is e4ual
to
the v't cur%e sloe"
ra"ical !olution of Rectilinear-Motion Pro#lems
0 (i%en the a't cur%e, the chane in %elocity !eteen
t ( and t ) is
e4ual to the area under the a't cur%e !eteen
t ( and t )"
0 (i%en the v't cur%e, the chane in osition !eteen
t ( and t ) is
=
=
0 *onsider %elocity of article at time t and
%elocity
at t . t ,
0 In eneral, acceleration %ector is not tanent to
article ath and %elocity %ector"
Motion Relati)e to a /rame in Translation 0 Desinate one frame as
the fixed frame of reference"
All other frames not riidly attached to the fi3ed
reference frame are moving frames of reference"
0 Position %ectors for articles A and B ith
resect to
the fi3ed frame of reference -xy are "and B A
r r
0 Vector Coinin A and B defines the osition
of
B ith resect to the mo%in
frame Ax’y’’ and A Br
A B A B r r r
+=
0 Differentiatin tice,
= A Bv %elocity of B relati%e
to A" A B A B vvv +=
= A Ba acceleration of B relati%e
to A" A B A B aaa +=
0 A!solute motion of B can !e o!tained !y com!inin
motion of A ith relati%e motion of B ith resect
to
mo%in reference frame attached to A"
t evv =0 Bith the %elocity %ector e3ressed as
the article acceleration may !e ritten as
dt
ds
ds
d
d
chane of seed and normal comonent reflects
chane of direction"
0 anential comonent may !e ositi%e or
neati%e" .ormal comonent alays oints
toard center of ath cur%ature"
0 Relations for tanential and normal acceleration
also aly for article mo%in alon sace cur%e"
0 Plane containin tanential and normal unit
%ectors is called the osculating plane"
nt " eee ×=
"inormal e
normal principale
Radial and Trans)erse Com"onents 0 Bhen article osition is i%en in
cylindrical
coordinates, it is con%enient to e3ress the
%elocity and acceleration %ectors usin the unit
%ectors "and,, k ee R θ
0 Position %ector,
r d v R ++== θ θ
0 Acceleration %ector,
vd a R +++−==
θ θ θ θ 22