Kinematics of a Particle - ## 전산설계자동화 실험실 ## 방문해...
Transcript of 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 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)
, , z
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Plane Rectilinear Motion
◆ What is plane motion? 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.
' r r r
's s s
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Rectilinear Motion(Velocity)Average velocity is defined as a displacement from P to P’ during the time
interval .t
avg t
rv
0limt
dt dt
r 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( ) TS
sp tv
avg
stv
Average velocity
t
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Rectilinear Motion (Acceleration)Average acceleration is defined as velocity changes from P to P’ during
the time interval .t
Instantaneous acceleration is defined as time derivatives of velocity
avg t
va
2
20limt
d dt dt dt
v 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
2 20 02 ( )cv v a s s
0 0 0
0
c
c
v t t
c cv
c
dva constdt
dv a dt
dv a dt a dt
v v a t
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
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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)