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ME-RMUTI Sarthit Toolthaisong
5.Kinematics of Particles
ME-RMUTI Sarthit Toolthaisong
Chapter Outline
- 61 Rectilinear Motion6-2 Angular Motion
6-3 Plane Curvilinear Motion6-4 Plane Relative Motion
ME-RMUTI Sarthit Toolthaisong
6-1 Rectilinear Motion
A study of linear motion of particle
- distance
- velocity
- acceleration
-x +x
x x
ME-RMUTI Sarthit Toolthaisong
6-1 Rectilinear Motion
Average velocity between P- P’
ave
xv
t
Instantaneous velocity
dxv x
dt
ME-RMUTI Sarthit Toolthaisong
6-1 Rectilinear Motion
v = velocity >> m/s
x = distance of motion >> m
a = acceleration >> m/s2
t = time of motion >> sec
Average acceleration between P- P’
ave
va
t
Instantaneous acceleration
2
2
dva v
dt
d xa x
dt
ME-RMUTI Sarthit Toolthaisong
6-1 Rectilinear Motion
Direction of velocity and acceleration
ME-RMUTI Sarthit Toolthaisong
6-1 Rectilinear Motion
Relationship between distance, velocity and acceleration
dx dv
v a
vdv adx
ME-RMUTI Sarthit Toolthaisong
6-1 Rectilinear Motion
The differential equation in each case
1) Constant velocity
0 0 0
x t v
x
dx vdt v dt
0x x vt
ME-RMUTI Sarthit Toolthaisong
6-1 Rectilinear Motion
The differential equation in each case
2) Constant acceleration
0 0 0
v t t
v
dv adt a dt
0v v at
0 0
( )x t
x
dx v at dt
20 0
1
2x x v t at
0 0 0
x t v
x
dx vdt v dt from
ME-RMUTI Sarthit Toolthaisong
6-1 Rectilinear Motion
The differential equation in each case
For x0=0
0
20
0
2 20
1
2
2
2
v v at
x v t at
v vx t
v v x
ME-RMUTI Sarthit Toolthaisong
6-1 Rectilinear Motion
The differential equation in each case
3) For vertical motion (a=g=-9.81 m/s2)
0
20 0
00
2 20 0
1
2
2
2 ( )
v v gt
y y v t gt
v vy y t
v v g y y
ME-RMUTI Sarthit Toolthaisong
6-2 Angular Motion
d
dt
2
2
d d
dt dt
Angular velocity
Angular acceleration
rad/s
rad/s2
Angular displacement rad
ME-RMUTI Sarthit Toolthaisong
6-2 Angular Motion
dd dt
dt
Constant angular velocity
0 t
0 0
0
.t
d dt const
t
ME-RMUTI Sarthit Toolthaisong
6-2 Angular Motion
dd dt
dt
Constant angular acceleration
0 t
0 0
0
.t
d dt const
t
20 0
1
2t t
0
0 0
0
( )t
d t dt t
ME-RMUTI Sarthit Toolthaisong
6-2 Angular Motion
dd dt
dt
Constant angular acceleration
0 t
0 0
0
.t
d dt const
t
20 0
1
2t t
0
0 0
0
( )t
d t dt t
ME-RMUTI Sarthit Toolthaisong
6-2 Angular Motion
d ddt
Constant angular acceleration
2 20 02 ( )
0 0
2 20 0
1( ) ( )2
d d
d d
From relationship
ME-RMUTI Sarthit Toolthaisong
6-3 Plane Curvilinear Motion
Rectangular coordinate x-y
2 2 2x yv v v
2 2x yv v v
2 2 2x ya a a
2 2x ya a a
y
x
vtan
v
x y
= = x y
= = x y
r i j
v r i + j
a = v r i + j
ME-RMUTI Sarthit Toolthaisong
6-3 Plane Curvilinear Motion
Projectile motion
Acceleration
ME-RMUTI Sarthit Toolthaisong
6-3 Plane Curvilinear Motion
Tangential and Normal component (n-t)
velocity
dds d ds d
vdt dt
0nv
tv
ME-RMUTI Sarthit Toolthaisong
6-3 Plane Curvilinear Motion
Tangential and Normal component (n-t)
Acceleration in normal and tangential
22n
n
dv vd va v
dt dt
d
Normal (an)
22
n
va
( )tt
dv da
dt dt
ta
Tangential (at)
ME-RMUTI Sarthit Toolthaisong
6-3 Plane Curvilinear Motion
Tangential and Normal component (n-t)
Acceleration in normal and tangential
For constant
2na r
ta r
ME-RMUTI Sarthit Toolthaisong
6-3 Plane Curvilinear Motion
Polar coordinate (r-)
acceleration
r rera = ( )- 2rθ
( 2 )r r θa θe
2 2ra a a Magnitude
ME-RMUTI Sarthit Toolthaisong
6-3 Plane Curvilinear Motion
Polar coordinate (r-)
velocity
rr r v v e e e e r rv θ θ
2 2rv v v Magnitude
ME-RMUTI Sarthit Toolthaisong
6-3 Plane Curvilinear Motion
Polar coordinate (r-)
acceleration
r rera = ( )- 2rθ
( 2 )r r θa θe
2 2ra a a Magnitude
ME-RMUTI Sarthit Toolthaisong
6-3 Plane Curvilinear Motion
Polar coordinate (r-)
For r = constant
2ra = -r
a r
r v
v r
e e
θv vθ θ
ME-RMUTI Sarthit Toolthaisong
6-5 Plane Relative motion
- Translating reference axes
- Rotating reference axes
ME-RMUTI Sarthit Toolthaisong
6-5 Polar Coordinates (r- )Translating reference axes
Br
B
Ar
/A BrA
X
Y
x
y
/A B A Br r r
/A B A Br r r
/A B A Bv v v
/A B A Bv v v
/A B A Ba a a /B A A Bv v v
/B A B Av v v or
/ /B A A Bv v/B A B Aa a a
ME-RMUTI Sarthit Toolthaisong
6-5 Polar Coordinates (r- )Rotating reference axes
r xi yj
A B Br r r r xi yj
A B relv v v r
/A B relv v r
( ) 2A B rel rela a r r v a
ME-RMUTI Sarthit Toolthaisong
6-6 Relative Motion (Translating Axis)
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