kinematics of a particle - ## 전산설계자동화 실험실 ## 방문해...
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Kinematics of a Particle
◆ What is kinematics?Kinematics: study of motion without reference to the forces which cause motion.
◆ What is motion of a particle?Movement of a particle à need position changes
Thus, when we call motion, it is related to position change, velocity, acceleration.
◆ How to describe the motion of a particle?- Need reference frame and coordinate systems- Coordinate systems
Cartesian coordinates (x, y, z)Spherical coordinates (R, )Cylindrical coordinates (r, )Normal-Tangential coordinates (n-t)
,q f, zq
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Plane Rectilinear Motion
◆ What is plane motion? à a particle moves in a single plane.
◆ Type of motionLectilinear motion Curvilinear motion
Position, velocity, accelerations are vector quantities. However, sense is important for rectilinear motion, because direction is fixed.
Position vector r is used to specify the location of the particle P at any given instant.
Note: r is always along the s axis,direction is never changed
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Rectilinear Motion (displacement)
Displacement of the particle is defined as the change in its position.
'D = -r r r
's s sD = -
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Rectilinear Motion(Velocity)Average velocity is defined as a displacement from P to P’ during the time
interval .tD
avg tD
ºDrv
0limt
dt dtD ®
Dº =
Dr rv
Instantaneous velocity is defined as time derivatives of position
Average speed is defined as the total distance traveled by a particle during the time interval .
TS
avg( ) TSsp tv =
D avg
stv -D
=D
Average velocity
tD
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Rectilinear Motion (Acceleration)Average acceleration is defined as velocity changes from P to P’ during
the time interval .tD
Instantaneous acceleration is defined as time derivatives of velocity
avg tD
=Dva
2
20limt
d dt dt dtD ®
D= = =
Dv v ra
2
2
dv d sadt dt
= =
For rectilinear motiondsv dsdt
dva dvdt
= = vdv ads=
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Constant acceleration
0 0 0
2 20 0
1 ( ) ( )2
cv s s
c cv s s
c
vdv a ds
vdv a ds a ds
v v a s s
=
= =
- = -
ò ò ò
2 20 02 ( )cv v a s s= + -
0 0 0
0
c
cv t t
c cv
c
dva constdt
dv a dt
dv a dt a dt
v v a t
= =
=
= =
- =
ò ò ò
0 cv v a t= +2
0 012 cs s v t a t= + +
0
0
0
00
20 0
( )
( )
12
c
cs t
cs
c
ds v v a tdtds v a t dt
ds v a t dt
s s v t a t
= = +
= +
= +
- = +
ò ò
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Erratic Motion(1)
Slope of s-t graph
= velocity
Given the s-t graph, construct the v-t graph
dsvdt
=
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Erratic Motion(2)Given the v-t graph, construct the a-t graph
Slope of v-t graph
= acceleration
dvadt
=
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Erratic Motion(3)Given the a-t graph,
construct the v-t graphGiven the v-t graph,
construct the s-t graph
displacement= area under v-t graph
Change in velocity
= area under a-t graph
dv adt= ds vdt=
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Erratic Motion(3)Given the a-t graph,
construct the v-t graphGiven the v-t graph,
construct the s-t graph
displacement= area under v-t graph
Change in velocity
= area under a-t graph
dv adt= ds vdt=
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Erratic Motion(4)Given the a-s graph, construct the v-s graph
Given the v-s graph, construct the a-s graph
( )dva vds
=
vdv ads=
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Example (rectilinear motion – constant acceleration)
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Solution (rectilinear motion – constant acceleration)
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Example (rectilinear motion – variable acceleration)
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solution (rectilinear motion – variable acceleration)
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Plane Curvilinear Motion
◆ What is plane motion? à a particle moves in a single plane.
Coordinate system for a plane motion à x-y, (z)Position, velocity, accelerations are vector quantities.We have to consider “direction” and “magnitude” àcurvilinear
motionWe have to consider “sense” and “magnitude” àrectilinear motion
Position vector r measuredfrom some fixed origin
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Plane Curvilinear Motion
◆ Displacement: Vector change of position. Independent of the choice of origin
◆ Distance: Path from one point to the other point.◆ Velocity:
avg tD
=Drv
0limt
dt dtD ®
D= =
Dr rv
0 0
r dslim limdtt t
svt tD ® D ®
D D= = =
D D
Speed=mag. of velocityVelocity is tangent to the path
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Plane Curvilinear Motion
◆ acceleration
avg tD
=Dva
2
20limt
d dt dt dtD ®
D= = =
Dv v ra
Hodograph
Acceleration is tangent to Hodograph
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Curvilinear Motion: Rectangular coordinates
◆ Position= (t) = + +
( ) + ( ) + ( )x y z
x t y t z t=r r i j k
i j k
2 2 2r x y z= + +
Magnitude of position vector
Direction of position vector
r r= ru
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Curvilinear Motion: Rectangular coordinates
◆ Velocity
2 2 2x y zv v v v= + +
Magnitude of velocity vector
Direction of velocity vector
v v= vu
{ } { } { }= (t) = ( ) + ( ) + ( )d d d dx t y t z tdt dt dt dt
dx d dy d dz dx y zdt dt dt dt dt dt
=
= + + + + +
rv v i j k
i j ki j k
x y zdx dy dz v v vdt dt dt
= + + = + +i j k i j k
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Curvilinear Motion: Rectangular coordinates
◆ Acceleration
2 2 2x y za a a a= + +
Magnitude of acceleration vector
Direction of acceleration vector
a a= au
yx zx y z
dvdv dv a a adt dt dtx y z
= + + = + +
= + +
i j k i j k
i j k&& && &&
{ } { } { }= (t) = ( ) + ( ) + ( )x y zd d d dv t v t v tdt dt dt dt
=va a i j k
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Curvilinear Motion: Rectangular coordinates
When is it convenient using Cartesian coordinate systems?X, y components of acceleration are independently generated or determined.
0,x ya a g= = -0x
xdv adt
= = 0( )x xv const v= =
yy
dva g
dt= = -
0( )y yv v gt= -
0( )y ydy v v gtdt
= = - 20 0
1( )2yy y v t gt= + -
0( )xdx vdt
= 0 0( )xx x v t= +
x component: Horizontal motion
y component: Vertical motion
Motion of Projectile
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Example (curvilinear motion – Cartesian coordinates)
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Solution (curvilinear motion – Cartesian coordinates)