ME-RMUTI Sarthit Toolthaisong 5.Kinematics of Particles

ME-RMUTI Sarthit Toolthaisong

5.Kinematics of Particles


Chapter Outline

- 61 Rectilinear Motion6-2 Angular Motion

6-3 Plane Curvilinear Motion6-4 Plane Relative Motion


6-1 Rectilinear Motion

A study of linear motion of particle

- distance

- velocity

- acceleration

-x +x

x x



Average velocity between P- P’

ave

xv

t

Instantaneous velocity

dxv x

dt



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



Direction of velocity and acceleration



Relationship between distance, velocity and acceleration

dx dv

v a

vdv adx



The differential equation in each case

1) Constant velocity

0 0 0

x t v

x

dx vdt v dt

0x x vt




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




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




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


6-2 Angular Motion

d

dt

2

2

d d

dt dt

Angular velocity

Angular acceleration

rad/s

rad/s2

Angular displacement rad


6-2 Angular Motion

dd dt

dt

Constant angular velocity

0 t

0 0

0

.t

d dt const

t


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


6-2 Angular Motion

dd dt

dt


0 t

0 0

0

.t

d dt const

t

20 0

1

2t t

0

0 0

0

( )t

d t dt t


6-2 Angular Motion

d ddt


2 20 02 ( )

0 0

2 20 0

1( ) ( )2

d d

d d

From relationship


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



Projectile motion

Acceleration



Tangential and Normal component (n-t)

velocity

dds d ds d

vdt dt

0nv

tv




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)




Acceleration in normal and tangential

For constant

2na r

ta r



Polar coordinate (r-)

acceleration

r rera = ( )- 2rθ

( 2 )r r θa θe

2 2ra a a Magnitude




velocity

rr r v v e e e e r rv θ θ

2 2rv v v Magnitude




acceleration

r rera = ( )- 2rθ

( 2 )r r θa θe

2 2ra a a Magnitude




For r = constant

2ra = -r

a r

r v

v r

e e

θv vθ θ


6-5 Plane Relative motion

- Translating reference axes

- Rotating reference axes


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


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


6-6 Relative Motion (Translating Axis)

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ME-RMUTI Sarthit Toolthaisong 5.Kinematics of Particles